An introduction to group representation theory

An Introdaction t o

Grozlp Represent& n Theoy R. KEOWN Department of Mathematics University of Arkansas Fayetteville, Arkansas

@

1975

ACADEMIC PRESS

New York San Francisco London

A Subsidiary of Harcourt Brace Jovanovich, Publishers

COPYRIGHT 0 1975, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

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United Kingdom Edition published b y

ACADEMlC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWI

Library of Congress Cataloging in Publication Data

Keown, R . An introduction to group representation theory. Bibliography: p Include3 index. 1. Finite groups. 2. I. Title. QA 17 1 .K417 512'.2 ISBN 0--12-404250-3

Representations of groups.

74-27783

PRINTED IN THE UNITED STATES OF AMERICA

To my wife, Jean

Contents Preface

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

.

Chapter 1 Groups and Modules 1. Fundamental Group Concepts .

. . . . . . . . . . 2. Rings and Fields . . . . . . . . . . . . . . . . 3. Abelian Groups. Modules. and Vector Spaces . . . . 4. Linear Transformations on Vector Spaces . . . . . . 5. Invariants of Linear Transformations . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . .

.

Chapter 2

2 10

16

33 40 55

The Representation Theory of Finite Groups 1. Basic Concepts and Definitions in the Representation Theory of Finite Groups . . . . . . . . . . . . .

2. The Group Algebra KG of a Finite Group G . . . . 3. The Structure of the Group Algebra KG . . . . . . .

4. The Simple Components of the Group Algebra KG . . 5 . Introduction to Group Characters . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . vii

65

84 96

106 115

134

viii

Contents

Chapter 3. The Computation of Representations and Characters of Finite Groups 1. Basic Concepts of Tensor Products of Group Representations . . . . . . . . . . . . .

. . . . 143

2. Representations and Characters of a Group Induced from Those of a Subgroup . . . . . . . . . . . .

156

3. The Group of Euclidean Motions of Three-Dimensional Euclidean Space and Some of Its Subgroups . . . . . . . . . . . . . . . . . .

167

4. The Irreducible Representations of Certain Point and Space Groups . . . . . . . . . . . . . . . . . . 186 206 Problems . . . . . . . . . . . . . . . . . . . . . Chapter 4. The Representation Theory of Several Special Groups 1. The Representation Theory of the Symmetric Group . 214 2. Modules over Symmetric Algebras . . . . . . . . . 234 3. The Integral Representations of the General Linear Groups . . . . . . . . . . . . . . . . . . . . . 251 4. General Remarks about the Representation Theory of Certain Matrix Groups. . . . . . . . . . . . . 278 Problems . . . . . . . . . . . . . . . . . . . . . 314 References

Index..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 19 321

Preface This book is designed for an introductory course in group representation theory at the senior or first-year graduate level not only for students of mathematics but also for those in other disciplines, such as physics and chemistry, where significant applications of representation theory are made. The nominal prerequisites for students at this level are a one-semester course in linear algebra and a one-semester course in general algebra introducing the concepts of group, ring, and field. More mature students from outside of mathematics can probably proceed on the basis of a one-semester course in linear algebra and a serious study of the material summarized in Chapter 1. The book is intended more as a prerequisite to than as a competitor with most of the books on representation theory which have appeared during the past decade. The bookswritten by pure mathematicians are one or two semesters beyond this one; those written by chemists and physicists are three or four. The discussion is almost entirely restricted to the case of real or complex representations. This restriction is primarily for pedagogical rather than technical reasons. Most of the arguments work with little or no change for any field whose characteristic is not a divisor of the order of the group under consideration. Nevertheless, the teaching experience of the author has been that fields of finite characteristic are received with some suspicion by students outside of mathematics and with little enthusiasm by the nonalgebraically minded ones within the discipline. Our arguments tend to be of a computational nature. It is true, of course, that mathematicians as a rule do not enjoy computations. Given a certain mastery of a field, we find it more pleasant to talk in structural generalities rather than in terms of calculations and complex examples. Nevertheless, it is evident that the average student cannot really understand such an overview without a substantial background of experience. In addition, there are far more people outside of mathematics than inside that are interested in group

X

Preface

representation theory. Many of these prospective users of representation theory need to carry out complicated technical calculations rather than to obtain a global view of representation theory. A recent survey of nonmathematicians holding advanced degrees and using mathematics in their research found that most of these people dismissed abstract algebra courses as of little worth. The single exception to this blanket charge was a group representation course with a broad coverage of the field. We now turn to a discussion of the contents of the book following these remarks about its philosophy. Chapter 1 is a collection of algebraic facts needed in group representation theory and with which students having had standard one-semester courses in linear algebra and general algebra are commonly acquainted. An instructor with a section of well-prepared students can treat this material in a cursory fashion or perhaps pass over it altogether. On the other hand, this chapter can serve as a summary for students with less algebraic preparation using the book. In particular, more advanced students from physics and chemistry with only a course in linear algebra in their background can find here a summary of material they must read up on or accept on faith. With many group representation theory books, such students would be in an awkward situation due to their lack of algebraic sophistication. Chapters 2 and 3 form the nucleus of a one-semester introductory course in group representation theory. Chapter 2 contains the basic definitions, introduces the concept of the group algebra KG of a finite group G over the complex numbers K , and determines the structure of KG using Maschke’s theorem as the principal tool. In an effort to keep the discussion computational and in line with the use of the group algebra in applications, a development based on an analysis of semisimple rings with minimum condition has been eschewed. The basic results on group characters are discussed here. Chapter 3 introduces the concept of tensor product in several different forms. The concept of induced representation is defined by means of the tensor product. The group of Euclidean motions of three-space is discussed along with certain of its subgroups, called space groups, which leave three-dimensional lattices invariant. These space groups are of considerable importance in various calculations of solid state physics. Their representation theory is an interesting and useful application of the method of induced representations. Several specific examples of point and space groups and some of their representations are presented in Chapter 3. Chapter 4 attempts to satisfy some of the needs of students and research workers outside of mathematics for a quick presentation of various facts about the representation theory of special groups. Unfortunately, the number of such groups needing discussion is large, and the representation theory of a single class among them is sometimes the subject of a not only long, but

Preface

xi

difficult treatise. Consequently, we can treat only a few of the groups of interest in a very sketchy fashion. Fortunately, what is frequently desired by these users outside of mathematics is some exposure to groups and their representations, some computational skill with them, and perhaps the generation of sufficient momentum to attack the more special monographs. Generally such individuals are much less interested in proofs as a way of understanding than in nontrivial examples that illustrate the ideas and computational methods. Chapter 4 approaches the representation theory of the symmetric group, the general linear group, and some subgroups of the general linear group from such a point of view. The author has received aid over the years from both the National Science Foundation and the National Aeronautics and Space Administration. He records here his appreciation for this financial support of research and writing. Professor J. S. Frame read an earlier version of the manuscript of this book and made numerous useful suggestions. Former students P. G. Ruud, C. W. Conatser, and J. R. Talburt have been helpful in many ways.

Chapter I

Groaps und Modztles

This chapter presents those definitions and elementary results from the theories of groups, modules, and vector spaces which are especially useful in an introduction to the theory of group representations. Various results are given without proofs and others are given with only sketchy arguments. Many facts about these algebraic systems are found only in the exercises which should be studied seriously by readers with a minimum preparation in abstract algebra. The following summary is presented for the benefit of the more experienced reader who may wish to proceed immediately to the following chapter where the theory of group representations begins. In the first section of this chapter we define the concepts of groups, subgroups, conjugacy classes, and group homomorphisms. The fundamental group homomorphism theorems are established. In the second section, the terminology of rings, subrings, ideals, and ring homomorphisms are introduced. The basic homomorphism theorems of the theory of groups are extended to the theory of rings. The special concepts of integral domain, division ring, and field are presented. In Section 3 we consider the elementary aspects of the theory of abelian groups, modules, and vector spaces. The fundamental theorem on finitely generated abelian groups is established. The definitions of the ascending and descending chain conditions on subsystems are given. The concept of a composition series is presented. These ideas are developed primarily for finite-dimensional K-spaces, that is, vector

2

1. Groups and Modules

spaces over the field K rather than for the more general systems of groups and modules. In Section 4 we present the notation and basic facts about the set Hom,(M, N) of linear transformations of a finite-dimensional Kspace M into a finite-dimensional K-space N. The dual space M* of a Kspace M is introduced. The relationship between a linear transformation T and its matrix with respect to a pair {B, C} of bases of M and N, respectively, is determined. In Section 5 we discuss various invariants of a linear transformation T including its eigenvectors, eigenspaces, and invariant subspaces. Bilinear and hermitian forms on a K-space M are defined along with the set Homf(M, M) of elements of Hom,(M, M) which leave a form f invariant. The Jordan and other canonical forms of the matrices of a linear transformation T a r e developed. 1. FUNDAMENTAL GROUP CONCEPTS

We recall that a group is a pair (G, o)consisting of a set G together with a binary operation o on G. The image o(s, t ) , {s, t } c G, is denoted by the juxtaposition st and called the product of s and t. The following relations are required to hold: (i) x(y-4 = (xy)z, {x, Y , 4 = G. (ii) There exists a unique 1E G such that IX = x l = X,

x E G.

(iii) For every x E G there exists a unique y and written x-', such that

E

G, called the inverse of x

xy = y x = 1.

The notation G is used rather than (G, a}to denote either a group or its underlying set. When G contains only a finite number n of elements, it is a finite group and its order n is denoted by [G : 11. A subset H of a group G is called a complex. A complex H of a group G is a subgroup iff H is a group under the binary operation o restricted to H x H. A useful criterion for a subset of a group to be a subgroup is the following (1.1) LEMMA. A nonempty complex H of the group G is a subgroup of G iff xy-' E H for every subset {x, y } c H . This lemma makes straightforward a proof of the fact that the intersection of any family 8 of subgroups of a group G is a subgroup of G. However, the union of a family 5 of subgroups of G need not be a subgroup of G . Given a nonempty complex K of a group G, the symbol ( K ) denotes the intersection of the family 8 of all subgroups of G which contain K. This

3

1. Fundamental Group Concepts

intersection is a subgroup called the subgroup generated by K. The integral powers bk of an element b can be usefully defined for all integers k and the usual rules of exponents shown to hold, that is, the equalities bo = 1,

b"b" = b"+",

(b")" =bmn

are valid for integral m and n. As a consequence, the subgroup (b) generated by a single element b of G proves to coincide with the powers of b. Whenever [(b): 11 is finite, it is the order of the element b, the smallest positive power of b equal to 1. Otherwise, b is said to be of injinite order. A group G is said to be cyclic iff G coincides with (b)for a suitable choice of the element b. A group G is called abelian or commutative iff x y equals y x for every subset ( x , y } of G. It follows immediately that a cyclic group is abelian. Among many of the elementary theorems of group theory, one of the most famous is that of Lagrange. The order [ H : I] of every subgroup H of a finite group G (1.2) THEOREM. is a divisor of the order [G : 1J of G . One of the simple ways of describing a finite group of low order is by means of its Cayley table. We give the Cayley table of a group of order sixteen. (See Table (1.3)).

(1 *3)

CAYLEY TABLE .G(1.3)

2 3 4 5 6 7 8 910111213141516 3 4 5 6 7 8 110111213141516 9 4 5 6 7 8 1 2111213141516 910 5 6 7 8 1 2 31213141516 91011 6 7 8 1 2 3 413141516 9101112 7 8 1 2 3 4 5141516 910111213 7 8 1 2 3 4 5 61516 91011121314 8 1 2 3 4 5 6 716 9101112131415 912151013161114 1 4 7 2 5 8 3 6 1013161114 91215 2 5 8 3 6 1 4 7 1114 91215101316 3 6 1 4 7 2 5 8 12151013161114 9 4 7 2 5 8 3 6 1 13161114 9121510 5 8 3 6 1 4 7 2 14 9121510131611 6 1 4 7 2 5 8 3 151013161114 912 7 2 5 8 3 6 1 4 161114 912151013 8 3 6 1 4 7 2 5 1 2 3 4 5 6

The elements of the group G(1.3) defined by Table (1.3) are denoted by the integers 1 through 16. The product inn is found in the mth row and nth column of the Cayley table, for instance, 9(7) equals 1I . This table enables us to give specific examples of some of the concepts we have introduced.

4

1. Groups and Modules

The set (1) is a subgroup of every group. The other subgroups of the above group are Subgroups of Order Two

Hzi

5>, Hzz ={I, 9>, Hz3 ={I, 11}, H24 = { l , 13}, Hz5 ={I, 15).

Subgroups of Order Four

={I, 3, 5, 7), H4z 5, 9, 131, H43 ={l, 5, 10, 14}, H44 ={l, 5, 11, 15}, H45 ={l, 5, 12, 16). ff41

Subgroups of Order Eight

2, 3, 4, 5,6, 7, 81, Hs2 ={I, 3, 5, 7, 9, 11, 13, 15}, H83 ={I, 3, 5, 7, 10, 12,14, 16). ffsi

A cyclic group of order n is denoted by the symbol C , . Each group of order two is cyclic and denoted by C , . The subgroups, H41, and H 4 5 , are cyclic of order four, with generators, 3, 10, and 12, respectively; that is, H41 = (3), H43 =(lo), and H 4 5 =(12). The subgroup HB1is cyclic with generator 2, among others. One notes that a cyclic group of order exceeding two has more than one generator. The groups, H , , and H S 3 ,are known as the dihedral and octahedral groups of order eight, respectively. Orders of the Elements of G(1.3)

O(1) = 1, O(2) = 8, o(5) = 2, o(6) = 8, o(9) =2, 0(10) = 4 , 0(13) = 2 , o(14) = 4 ,

o(3) = 4, O(4) = 8, o(7) = 4, o(8) = 8, ~ ( l l =) 2 , 0(12) = 4 , 0(15) = 2 , o(16) = 4 .

This example illustrates Lagrange’s theorem that the order of a subgroup (consequently, of an element) is a divisor of the order of any containing finite group. The Cayley table of each subgroup H of G can be read directly from the Cayley table of G. For example, the subgroup H42 has the representation as shown in Table (1.4). (1.4)

CAYLEY TABLE Hd2 1 5 913 5 113 9 913 1 5 1 3 9 5 1

5

1. Fundamental Group Concepts

The reduction of problems about complicated groups to related problems about simpler ones is a standard method of group theory. Such reductions are sometimes achieved by means of group homomorphisms. A mapping h with domain a group G and range a group G' is called a group homomorphism iff it preserves the binary operation of G in the sense that

(1.5)

h(gg') = h(g)h(g'),

($7, s'>= G.

An injective homomorphism of G into G' is called a monomorphism; a surjective homomorphism of G onto G' is called an epimorphism; and a bijective homomorphism of G onto G' is called an isomorphism. An endomorphism of G is a mapping of G into G satisfying Eq. (1.5), while an automorphism of G is a bijective mapping of G onto G satisfying this equation. Each g E G determines a mapping ie by means of the definition

i,(x) = g x g - ' ,

x

E

G.

It can be shown that is is an automorphism, called the inner automorphism generated by g. There are many examples of irreducible representations given in the sequel. Each of these irreducible representations is a homomorphism of a group G into a group of matrices. The mapping h with domain G(1.3) and range H42 given by

(1.6)

h(1) = h(3) = h(5) = h(7) = 1 , h(2) = h(4) = h(6) = h(8) = 5, h(9) = h(l1) = h(13) = h(15) = 9, h(10) = h(12) = h(14) = h(16) = 13,

is a homomorphism of G(1.3) onto H42. Although it is easy to verify directly that h is a homomorphism, we omit this in favor of an argument given below which makes h an example of a general type of homomorphism. We recall a number of significant facts and definitions about group homomorphisms. The mapping h denotes a homomorphism of the group G into the group G' in the following discussion. The group G' is a homomorphic image of G iff there exists a homomorphism of G onto G'. The image h(1) of the identity 1 of G is the identity 1' of G'. The image h(x-') is the inverse of the image h(x), that is, (1.7)

h(x-') = (h(x))-',

x E G.

The kernel K of a homomorphism h is a subgroup

(1.8)

K

= (g : g E G, h(g) = l'}

consisting of all elements of G which map into the identity 1' of G'. The = kernel K of the homomorphism h defined in (1.6) is the subgroup (1, 3, 5, 71.

6

1. Groups and Modules

As a matter of fact, the kernel K of a homomorphism h is a special kind of subgroup called normal or invariant. This concept can be conveniently discussed after introducing a notational convention of group theory. Let H and K be two complexes of a group G. Their product, written HK, is the set

(1.9)

HK = { g : g

= hk,

h E H, k

E

K)

of all group products whose left factor belongs to H and right factor to K. The notation is especially used in the case where H consists of a single element g and K is a subgroup of G. The sets (1.10)

gK

= (x : x

Kg

={ X

=gk, k

EK}

and (1.11)

: x = kg, k E K }

are called the left and right cosets of K in G, respectively, determined by the representative element g. An important group theoretical fact is that any two left cosets, gK and g'K, either coincide, or else have no elements in common. Consequently, each subgroup K of a group G determines a partition of G into the left cosets of K in G. Analogous results hold for right cosets. The product specified by (1.9) I S an associative binary operation on the collection 8 of all nonempty complexes of the group G. A normal or invariant subgroup K of a group G is a subgroup such that (1.12)

K

= gKg-',

g

E

G.

This equation can be written in the equivalent form (1.13)

Kg=gK,

gEG,

which asserts that the right coset of K determined by the representative element g coincides with the left coset determined by it. It may be shown that the relation K 3 gKg-', g E G, is equivalent to (1.12), so that it is a sufficient condition for normality. The subgroup gKg-' is a conjugate subgroup of the subgroup K. A subgroup K is normal iff it is self-conjugate. The element gxg-' is a conjugate of the element x of G. The conjugacy class of an element x is the set of all elements of G which are conjugates of x. An element x is self-conjugate iff it coincides with all of its conjugates. (1.14) THEOREM. The ensemble, written G/K, of all left cosets of a normal subgroup K of a group G, constitutes a group called the factor group of G modulo K. The binary operation of the factor group GjK is complex multiplication.

7

1. Fundamental Group Concepts

Proof. The associative law for complex multiplication implies that (gKXg’K) = g(K(g‘K1) = g((Kg’)K). From which it follows by (1.13) and further applications of the associative law that g((Kg‘)K) = g((s‘K)K) = g(g’(KK)) = (gg’)(KK)= (gg’)K. Consequently, complex multiplication is a binary operation on G/K since the complex product (gK)(g‘K)of any two left cosets gK and g‘K is the left coset (gg’)K. Since (gK)K = g(KK) = gK

and

K(gK) = (Kg)K = (gK)K = gK,

it follows that the coset K is the identity element of this binary operation. Furthermore, (gK)(g-’K)

= (gg-l)K = K

and

(g-’K)(gK) = (g-’g)K

= K,

so that gK has g-’K for an inverse. G/K is a group since the associative law is valid for complex multiplication. There exists a homomorphism v of G onto its factor group G/K for every normal subgroup K. This homomorphism, called the natural homomorphism, is defined by v(g) = gK,

9 E G.

To see that v is a homomorphism, note that v(gg’) = 99’K

= (gK)(g’K) = v(g)v(g’)

whenever { g , g’} is a subset of G. It can be verified that the subgroup H , , = {I, 3, 5, 7) of the group G(1.3) is normal. The left cosets (1, 3, 5, 7), (2, 4,6, S}, (9, 11, 13, 15}, and (10, 12, 14, 16) of provide an example of the above considerations. We prove the fundamental theorem as follows. (1.15) THEOREM. Let G‘ be a homomorphic image of G under the homomorphism h whose kernel is K . Then G’ is isomorphic to the factor group GIK of G modulo the kernel K.

Proof. Observe that gK

= g‘K

implies that g

= g’k,

k E K, so that

h(g) = h(g’k) = h(g’)h(k) = h(g’). Consequently, one can define a mapping h‘ of GIK into G‘ by h’(gK) =

8

1. Groups and Modules

Note that (1.16)

h”(gK)(g’K)I = h’(gg’K) = h(gg’) = h(g)h(g’)= h’(gK)h’(g’K).

Thus the mapping h’ is a homomorphism. Every element g’ of G‘ is an image Iz(g). It follows that every g’ is the image h’(gK) of some gK under h‘, so that h’ is subjective. The equalities h(g) = h’(gK) = h’(g’K) = h(g’) imply that 1’ = h(g-’g)

= h(g-l)h(g) = h(g-l)h(g’) = h(g-’g’),

which shows that g-lg‘ is an element of K. Consequently, g‘K = g K and h’ is a monomorphism. Therefore, h‘ is an isomorphism. Note that the homomorphism h defined in (1.6) is an application of these ideas. It can also be shown that if G’ is the homomorphic image of G under the homomorphism h, then h establishes a one-to-one correspondence f between the subgroups of G’ and those subgroups of G which contain the kernel K of / I . In particular, there exists such a correspondence between the normal subgroups of G‘ and the normal subgroups of G which contain the kernel K . We wish to discuss an important method for constructing new groups from old ones, or of analyzing a group G in terms of two of its subgroups H and K. Let H and K be any two groups and consider the Cartesian product G = H x K consisting of all ordered pairs (h, k), h E H , k E K. (1.17) DEFINITION. The external direct product of H and K is the set G together with the binary operation o defined for all pairs, (h, k ) and (A’, k‘), of G by the formula (1.18)

~ [ ( hk ), , (A’. k’)] = (hh’, kk‘).

It can be shown that (G, o)so defined is a group for every choice of the groups H and K. The external direct product of H and K is denoted by HOK. Let H and K be subgroups of a group G. In general, the product H K is not a subgroup of G. For example, consider the subgroups, H = (1,9} and K = { 1, 15}, of C( 1.3). The product H K is the complex {1(1), ~ 5 19(1),9(15)) ,

= {1,15,9,31,

which does not contain 7, the inverse of 3, so that it is not a subgroup of C( 1.3). The following lemma clarifies the situation.

9

1. Fundamental Group Concepts

(1.19) LEMMA. The product H K of two subgroups H and K of a group G is a subgroup of G if and only if HK coincides with KH. The condition of Lemma (1.19) is met whenever at least one of the factors is a normal subgroup of G. (1.20) DEFINITION. The group G is the product of its subgroups H and K if and only if G equals HK. (1.21) DEFINITION. The group G is the internal direct product of its two normal subgroups H and K iff G = HK; (ii) H n K = (1).

(i)

As an example (see the problems at the end of Chapter 1) G(3) is the product of its subgroups H = (1, 2, 3,4} and K = { I , 5}, but it is not the direct product since K is not a normal subgroup of G(3). These two concepts of external and internal direct products can be extended to a finite number of factors in a direct manner. Additional details for important special cases are given in Definitions (3.69) and (3.71) of this chapter. When H a n d K are subgroups of G with K normal in G, there is an interesting theorem somewhat related to the above considerations.

(1.22) THEOREM. Let G be the product H K of the subgroups H and K with K normal in G. Then H n K is normal in H and GIK is isomorphic to H/(H nK). Proof: To see that H n K is normal in H , note that hence (1.23)

hHK'=H

and

hKh-'=K,

h(H n K)h-' c hHh-' n hKh-'

The hypothesis implies that each g that (1.24)

E

hEH;

= H n K.

G is of the form hk, h E H , k E K, so

gK = hkK

= hK

and every left coset of K in G arises from a representative element I? belonging to H . Denoting the natural homomorphism of G onto G/K by v, let f be the restriction of v to H . According to (1.24), f is an epimorphism. An element h of H is in the kernel K' offiff it is in the kernel of v, that is, K'

(1.25)

= H n K.

It follows from Theorem (1.15) that H/K

=H/(H nK ) z

G/K = HKIK.

10

1. Groups and Modules

There is another useful theorem as follows which has the same appearance as the law of cancellation of common fractions. (1.26) THEOREM. Let H and K be normal subgroups of a group C with K contained in H . Then the factor group G / H is isomorphic to the factor group (GIK)/(H/K).

Proof. The natural homomorphism v of G onto G / K maps the normal subgroup H of G onto a normal subgroup H’ of G/K. Under v, each element h of H maps into the left coset hK of G / K , which shows that H‘ is none other than H / K . Let v‘ be the natural homomorphism of G / K into its factor group ( G / K ) / ( H / K ) The . map v’ 0 v is a homomorphism of G onto ( G / K ) / ( H / K ) whose kernel is the normal subgroup H . It follows that G/H = ( G / K ) / ( H / K ) .

(1.27)

2. RINGS AND FIELDS

This section is a very brief introduction to the concepts of rings and fields. The primary purpose is to report some standard definitions and results. (2.1) DEFINIIION. A ring is an algebraic system (A, w , w ’ ) consisting of a set A together with two binary operations w and w’ called addition and multiplication, respectively. Let {x,y} be contained in A. Then w(x, y) is denoted by x + y and w’(x, y) by xy. Addition is a commutative binary operation. The following rules hold for addition:

+ +

+

{x, Y, z} c A. (i) x (y z) = (x y) + z, (ii) There exists a unique OEA, called the additive identity, such that x+O=O+X=X,

XEA.

(iii) For every x E A there exists a unique y E A, called the negative of x and written -x, such that

x +y

=y

+ x = 0.

The following rules hold for multiplication : ( 9 X(YZ) = ( X Y k (ii) x(y + z) = xy + xz, (iii) (y + z)x = yx + zx,

{x, Y, 4 A. {x,y, z} c A. {x,y, z} c A.

The most familiar example of a ring is the set Z of the integers with the usual interpretation of addition and multiplication. The logical development of the required properties from a small set of axioms requires a rather long mathematical argument. We ask the reader to take it for granted that the

2. Rings and Fields

I1

integers satisfy the customary rules. In addition to these general properties, the integers enjoy various other properties that rings in general do not. Some of these are found in the following definitions. A ring A has a multiplicative identity, denoted by 1, iff A contains a unique element 1 such that l x = x l = x for every x E A. A ring A is a commutative ring iff xy = yx for every {x, y} c A. Let A be a ring containing more than one element with a multiplicative identity 1. Then A is said to be a division ring iff x # 0 implies that there exists a unique y E A such that xy = yx = 1. A commutative division ring is known as a.field. A ring A contains divisors of zero iff there exists {x, y} c A with xy = 0, but x # 0 and y # 0. An integral domain is a commutative ring with identity which contains no divisors of zero. The even integers form a ring with no identity and no divisors of zero. The rational numbers, the real numbers, and the complex numbers are examples of fields. Most of our work will be concerned with these fields, but we will give a short discussion at the end of this section concerning less common fields. We introduce two general families of rings in order to discuss other ring concepts. For each positive integer n, the first of these families consists of the three rings Q,, R , , and K, of all n x n matrices over the rational, the real, and the complex fields, respectively. The second of these families conR[x],and Q[x]of polynomials in a single insists of the three rings K[x], determinant x over these same three fields. The ring Q2 of all 2 x 2 matrices over the rational field provides an example of a noncommutative ring containing divisors of zero. Given the two elements

of Q, , note that the products

and

are distinct and that the last is the zero matrix even though neither of the factors is zero. The last equality implies that

/I 1

is a left

and

1y y1

is a right

12

1. Groups and Modules

divisor of zero. The ring R [ x ] of polynomials in a single indeterminant x with coefficients from the real field R is an example of a ring without divisors of zero. Both Q2 and R [ x ] have a multiplicative identity. Neither Q , nor R [ x ] are division rings, since neither of the nonzero elements

of Q2 and R [ x ] ,respectively, have inverses. Let D denote the set of all 2 x 2 complex matrices of the form

/ -%

where 5 denotes the conjugate of the complex number z. It can be shown that D is a ring under the binary operations of matrix addition and multiplication. Furthermore, D is a noncommutative ring in which every nonzero element has an inverse. Consequently, D is a division ring, called the ring of (Hamilton’s) quarternions, which is not a field. In the theory of rings, the concepts of subring and ideal are analogous to those of subgroup and normal subgroup in the theory of groups. A subset J of a ring A is a subring of A iff J is a ring under the restriction of the binary operations of A to the subset J. A subring J of A is a left ideal iff the product aj belongs to J for every a E A and j E J. A subring J of A is a right ideal iff the product ja belongs to J for every j E J and a E A. A subring J of A is a rwo-sided ideal iff J is both a right and a left ideal. In a commutative ring, the concepts of left, right, and two-sided ideals coincide, of course, and are denoted merely by the term ideal. The subring J of all even integers of the ring Z of integers is an ideal of 2. The subring J of all polynomials p ( x ) with zero constant term is an ideal of the ring R [ x ] of all polynomiak in an indeterminant x with real coefficients. The subset J of all matrices of the form

I: Ell

is a commutative subring, which is not an ideal, of the ring Q2 of all 2 x 2 matrices with rational components. An ideal J of a ring A is called a proper ideal iff it does not coincide with A. Let J and J’ denote’tleft, right, two-sided) ideals of the ring A. The proper ideal J is a maximal ideal of its kind iff it is not properly contained in any other ideal J’ of its kind. The proper, nonzero ideal J is a minimal ideal of its kind iff J properly contains no ideal J’ of its kind other than the zero ideal (0). Observe that a maximal, two-sided ideal J can sometimes be properly contained in a proper left ideal J’ with no contradictions in terms. I n the same way, a minimal, two-sided ideal J can

13

2. Rings and Fields

properly contain a nonzero right ideal J'. The ideal J is a prime ideal iff ab E J, {a, b} c A, implies that either a E J o r b E J . The ring R [ x ] contains a maximal ideal J consisting of all polynomials p ( x ) in R [ x ] such that p(c,,) = O for a fixed real number c,,. The ring Q2 contains a minimal left ideal J consisting of all 2 x 2 rational matrices

with uI2 = a22 = 0. The ring 2 of integers contains a prime ideal J consisting of all integral multiples of the number 2. The verification of these facts is left as a problem. The concept of a group homomorphism has a natural extension to that of a ring homomorphism. A mapping h with domain a ring A and range a ring A' is a ring homomorphism iff (2.4)

h(x

+ y ) = h(x) + h(y)

and

h(xy) = h(x)h(y)

for x, y contained in A. All of the concepts, homomorphic image, epimorphism, monomorphism, isomorphism, endomorphism, and automorpliism have their natural extensions from groups to rings which we leave to the reader. The homomorphism theorems on groups also have their analogs in the case of rings. We turn t o an investigation of the basic ideas involved. First, note that every ring A is an abelian group ( A , w ) where w denotes the binary operation of addition on A . Since each subring or ideal J of A is a normal subgroup of the group A, one can consider the factor group A/J of A modulo its ideal or subring J . The elements of A / J are the left cosets of J in A which are denoted by the symbols x + J , rather than x J , as in the case of groups. The abelian group A / J can be made into a ring, meaningfully related to the original ring, with the product of two left cosets x J and y + J defined by

+

(2.5)

(X

+ J)(y + J ) = X Y + J .

This rule is satisfactory only f o r the case of an ideal J, not merely a subring. The resulting ring is called the factor ring or residue class ring of A modulo its ideal J. The verification that A/J is a ring with this definition of the binary operation of multiplication, a task more characterized by its length than its difficulty, is omitted. The definition of the multiplication for A/J guarantees that the natural mapping v of the abelian group (A, o)onto its factor group A/J is, in fact, a ring homomorphism of the ring A onto the factor ring A/J. Thc extension of the basic homomorphism theorems from a group G to a ring A is straightforward. In fact, most of the work has been done, since one

14

1. Groups and Modules

starts with a knowledge of the results for the abelian group (A, w ) of the ring A . The kernel K of a homomorphism I7 of the ring A into the ring A’ is the set K = {X : x E A, h(x) = 0}, (2.6) where 0’ is the additive identity in A’. It is easy to show that K is an ideal of the ring A . The fundamental result is the following theorem. (2.7) THEOREM. Let I1 be a homomorphism of the ring A onto the ring A‘. Then A‘ is isomorphic to the factor ring A/K of A modulo the kernel K of the homomorphism h . Proof. The result for groups implies that the abelian group ( A , w’) is isomorphic to the factor group A/K under the homomorphism h’ defined from Iz by means of

h’(x + K)

(2.8)

=

4~).

It remains merely to show that h’ preserves the binary operation of multiplication. However,

h‘[(x

+ K)(y + K)] = /I’(XY + K) = ~(xY)= k(x)h(y) = h’(x + K)h’(y + K)

and the result follows. There are also ring analogs of Theorems (1.22) and (1.26) where H and K are subrings of the ring A . (2.9) THEOREM. Let A be the sum H + K of the subrings H and K with K an ideal of A . Then H n K is an ideal in H and A/K is isomorphic to H / ( H n K). Proof: lt is easy to see that H n K is an ideal in H. By Theorem (1.22), there exists a group isomorphismfof A/K onto H / ( H n K). Recall that each coset of A/K can be written in the form h K, h E H, and thatfis defined by

+

f(h

(2. lo)

+ K) = h + (H n K).

Consequently, the problem reduces to showing thatfalso preserves the binary operation of multiplication in A/K. However, (2.1 1) f[(h

+ K)(h’ + K)] =f(hh’ + K) = hh’ + H n K = (h + H n K)(h’ + H n K) =f(h + K),f(h’ + K),

as was to be shown.

The analog of Theorem (1.26) is the following.

(2.12) THEOREM. Let a ring A contain ideals H and K with K belonging t o H. Then the factor ring A / H is isomorphic to the factor ring (A/K)/(H/K). The proof is left to the reader.

15

2. Rings and Fields

The ring Z of integers provides a convenient example of some of the above considerations. The subset V consisting of all integral multiples (4) of 4 is easily seen to be an ideal of Z. The residue classes of Z modulo V consist of the sets { ...) -8, - 4 , 0 , 4 , 8, ...} = [O], { ...) -7, -3, 1,5, 9, ...} = [l], {... , -6, - 2 , 2 , 6, 10,. . .} = [2], { ..., -5, - 1 , 3, 7, 1 1 , ...} = [3]. We list in Tables (2.13) and (2.14) the binary operations of addition and multiplication for the residue class ring Z / V. (2.13)

(2.14)

The abelian group ( Z / V , a) is a cyclic group of order four. The factor ring Z/V has [2] for a divisor of zero. In any ring with a multiplicative identity 1, the elements with multiplicative inverses are called units. The set {[l], [3]} is the set of units of ZjV. The set T of all integral multiples (3) of 3 is a prime ideal in the ring Z of integers. The addition and multiplication tables for the three residue classes, [O], [I], [2], of Z/T are as follows in Tables (2.15) and (2.16). (2.15)

(2.16)

ADDITION

MULTIPLICATION

16

1. Groups and Modules

One verifies from Table (2.16) that ZIT is a commutative ring in which every nonzero element has an inverse. Consequently, ZIT is a field of only 3 elements. Similarly, it can be shown that if ( p ) denotes the set P of all integral multiples of a prime number p , then P is a prime ideal of the ring Z of integers and the factor ring is a field containing p elements. Furthermore, every field of p elements is isomorphic to the field Z / ( p ) . Finite fields are called Galoisfields in honor of their discoverer Galois. For every choice of the prime p and positive integer n, there exists exactly one class, denoted by the symbol GF(p"), of isomorphic fields of order p". The orders of all the elements of the abelian group (A, w) of a ring A may be bounded by some positive number k . Such rings are said to be ofJinite characteristic. The smallest positive integer n which bounds the additive orders of the elements of a ring A of finite characteristic is called the characteristic of A. For example, the characteristic of Z/(4)is 4 and that of Z / ( 3 ) is 3. It may be shown that any field of finite characteristic is of characteristic p where p is a prime. One should not conclude from this that a field of finite characteristic p is necessarily a finite field. We turn to an important class of algebraic systems for which this section introduced some of the necessary facts and terminology. 3. ABELIAN GROUPS, MODULES, AND VECTOR SPACES

This section is a brief review of the theories of abelian groups, modules, and vector spaces. These algebraic systems are similar in that each of them has an abelian group M as its basic structure. Each of them is associated with a set S of operators and a mapping 0,called scalar multiplication, from the Cartesian product S x M into M. The image a(a, in), a E S, m E M, is denoted by am and called the scalar product of m by a. They differ in that S is the ring Z of integers, an arbitrary ring A with multiplicative identity 1, and a field K in the cases of an abelian group, a module, and a vector space, respectively. We begin by recalling some customary notational changes in passing from the theory of a general group G to that of an abelian one M. First, the image o(m,m'), {m, m'} c M, is denoted by m + m', rather than mm'. The identity element of M is given the symbol 0 rather than 1 while the inverse of m E M is given the symbol -m. The product of two subgroups, H and K, of M is indicated by H + K ; the direct product by H OK. Each subgroup K of M is normal; its left (right) cosets are denoted by m + K, m E M. These conventions agree with those introduced for the additive group of a ring A.

17

3. Abelian Groups, Modules, and Vector Spaces

The basic definitions of the powers of an element m of M agree with those previously given, but the expressions for them assume a different form in the new notation, to wit,

(3.1)

mk = km,

k E Z, m E M .

The concept of multiplication by an integer replaces that of exponentiation. The rules of exponents assume the following altered form: Om = 0, lm = m, a(a’m) = (aa’)m, (a + a’)m = am + a’m, a(m + m’) = am + am’. where (0, 1, a, a’] c 2 and (0, m, m‘j c M. These rules remain essentially unchanged in passing from the case of an abelian group to that of a module. Although the theory of noncommutative finite groups is still active, the theory of finite abelian groups is well known. I n fact, the theory of many classes of infinite abelian groups is complete. We wish to introduce sufficient terminology to describe the results for an important class of abelian groups. A subset K of a group M is said t o be a set of generators of Miff the subgroup ( K ) coincides with M . A group M is said to befinitely generated iff it contains a finite set {ml, . . . , mk} of generators. In the case of an abelian group M, this means that each m E M can be written in the form m

(3.3)

= nlml

+ ... + nkmk,

where { n l , . . . , n k ) c 2. The set T of all elements

(3.4)

{m: m E M , nm = O E M , some ~ E Zn # , 0],

some nonzero multiple of which equals the additive identity 0 of M, is a subgroup called the torsion subgroup of M . An abelian group M is called torsion free iff its torsion subgroup consists only of (0). A set {m, : 71 E n} of generators of M is said to be free iff the equality

(3.5)

n,lm,l

+ . . . + n,, mXk= 0,

nn,E Z , 1 5 i I k ,

implies that

(3.6)

n,, = - . * = rink

= 0.

A free set of generators of M is sometimes called a basis of M. A free abelian group M with a basis {m} consists of all multiples {nm}, n E 2, of the element m.It is easy to see that M is isomorphic to the additive

18

I . Groups and Modules

group of integers. A free abelian group M with a basis {ml, . . . , mk} consists of all integral linear combinations

+ . + nk mk

nlml

(3.7)

* ’

of the basis elements. Thus M is isomorphic to the direct sum

(3.8)

Z 0 . e . @Z

of k copies of the additive group 2 of integers. In other words, M can be identified with the group G of all k-tuples (nl, . . . , nk) of integers where the operation of addition is defined by

(3.9)

(ni,

. . . nk) + (El’, . . . 2

9

a,’)

= (n1 f nl’, . .., nk $-

nk’).

A subgroup N of a free abelian group M is free. However, a more important fact is true. (3.10) THEOREM. Let N be a subgroup of the free group M with a finite basis. Then there exists a basis {ml, . . . , m,} of M and a set of integers {dl, . . . , d j } ,j _
m=m, +.--+m,,

miEMi.

(3.1 1) THEOREM. Let {g,’, . . . ,g,‘} be a set of generators of the abelian group M. Then there exist integers, 0 5 r r j _< k, such that M is the internal direct sum

c, @

” *

@ cj @ cj+l

of cyclic subgroups where each C i , r d i \ d i + l ,r 5 i 5 j - 1, and where

cj+l@ - . .

8Ck

i 5 j , is of finite order d i , with

OCk

is a free subgroup of M. Proof: Let F be the free abelian group with a basis {g,, . . .,gk).The mapping f’of F into M defined by f(nlgl

+ + nkgk) = n,g,’ + + nkgk’ ’’’

’’*

3. Abelian Groups, Modules, and Vector Spaces

19

is a homomorphism of F onto M whose kernel N is a free subgroup of F. Let {q,..., mk} and {dl, ..., dj} be the basis and set of integers, respectively, whose existence is asserted in Theorem (3.10) so that the set (dim,, . . . ,djmj}is a basis of N. The set of images {q’, . . . , mk’},where 1I i 5 k, mi’ =f(mi), is a set of generators of M. Let r be the smallest integer such that d, exceeds 1. Then mi belongs to N and mi’ is the identity of M for i less than r. Consequently, {mr’, . . . , mk’} is a set of generators of M. Suppose that

n,rn,’+...+n,m,’=O;

(3.12)

then n , m , + . . - + n k m k belongs to N and dilni,r < i < j , while ni=O whenever j < i k by choice of the basis. It follows that each summand of (3.12) is the identity and that the sum

(mi) 0 . * 0 (m,?

(3.13)

is direct. It is easy to see that mi’ has order di,r 5 i order for j < i < k. The result follows.

<j , and mi’ has infinite

This theorem states that every finitely generated abelian group M is isomorphic to an abelian group M‘ whose elements are n-tuples, with components either from Z/(m),for suitable choices of m, or from 2. Thus one obtains a very specific representative element from each class of isomorphic finitely generated abelian groups. We have developed these facts about abelian groups to exemplify and to motivate the introduction of modules. A module is a triple {M, A, T C }consisting of an abelian (3.14) DEFINITION. group M, a ring A with multiplicative identity 1, and a mapping n from the Cartesian product A x M into M. The image n(a, m), a E A , m E M , is denoted by the juxtaposition am, called the product or scalar product of m by a. The following rules are required to hold:

(3. 5,

(i) (ii) (iii) (iv)

l m = m, a(a’m) = (aa’)m, (a + a’)m = am + a’m, a(m m’) = am + am’,

+

where (1, a, a’} c A and {m, m’} c M. Naturally, one adopts the notation M for both the module and the underlying abelian group. One speaks of the module M over the ring A or the A-module M. In this terminology, an abelian group M is a module over the ring 2 of integers or a 2-module. The definitions introduced for abelian

20

1. Groups and Modules

groups carry over to A-modules. A set {m, : 7t E I’I} is a set of generators of M iff every element m E M can be written in the form (3.16)

m

= a,,mnl

+ . . + a,* mnk, *

where (a,,, . . . , ankfis contained in the ring A. The set of generators is an A-basis iff every element m of M can be expressed uniquely is the form of (3.16). An A-module M is called A-torsion free iff am = 0, a E A, m E M, implies that either a = 0 or m = 0. (3.17) DEFINITION. A subgroup N of the A-module M is called a submodule or an A-submodule if and only if an, a E A, n E N, is always an element of N. Since every submodule N of an A-module M is a normal subgroup of the abelian group M, the set {m + N), m E M, of left cosets of N in M constitutes an abelian group. Note that (3.18)

m +N

= m’

+N

implies that m - m’ belongs to N. This means that (3.19)

am - am’ = a(m - m’)

is an element of N. Thus, the left cosets, am + N and am’ + N, coincide. Therefore a scalar multiplication of the elements of the factor group M/N by elements of A can be defined by (3.20)

a(m + N)

= am

+ N.

The abelian group M/N together with this definition of scalar multiplication is an A-module called the factor module of M modulo N and denoted by M/N also. A proper submodule N of the A-module M is maximal iff N is not properly contained in any proper submodule N’ of M. A submodule N of M is an irreducible or minimal submodule iff N properly contains no submodule N’ of M other than the submodule (0). Let the sequence (3.21)

N, c N, c - * * c N,,,

be an ascending chain of submodules of the A-module M. The factors of the chain (3.21) are the factor modules, N,/N,, . . . , N k + , / N k . (3.22) DEFINITION. An A-module M is said to satisfy the ascending chain condition (A.C.C.) on A-submodules if and only if every properly ascending chain

21

3. Abeliun Groups, Modules, and Vector Spaces

contains only a finite number of elements. An A-module M is said to satisfy the descending chain condition (D.C.C.) on A-submodules if and only if every properly descending chain

(3.24)

N,

2

N,

3

*..

3

N,

3

*..

contains only a finite number of elements. One of the basic results on A-modules is the following theorem.

(3.25) THEOREM. An A-module M satisfies the ascending chain condition if and only if every submodule N of M is finitely generated. (3.26) DEFINITION. A finite ascending chain (3.27)

(0) = N , c N,

c

... c N k + J= M

of A-submodules of the A-module M which begins with (0) and ends with M, is called a composition series of M if and only if each submodule N, is a maximal submodule of its successor N , + l , I I i 5 k . The factor modules, N,/N,, . . . , N,+,/N,, are called the composition factors of the series; their number k is the length of the composition series. The ascending series of A-modules (3.28) and

(3.29)

N, c N, c ... c Nk+l

s, c s, c

..*

c

sj+,

of an A-module M are said to be equivalent iff k = j and there exists a bijection n on the set (1, . . ., k } such that the factors N , + l / N iand Sn(i+t)/Sn(i) are isomorphic, 1 s i 5 k. Following is an important theorem concerning composition series.

(3.30) THEOREM (Jordan-Hiilder). Any two composition series (3.31) and

(3.32)

of an A-module M are equivalent. The reader is referred to one of the standard treatises for a proof.

(3.33) DEFINITION. A mapping h with domain and range the A-modules M and N, respectively, is called an A-homomorphism if and only if 11 is a

22

1. Groups and Modules

homomorphism of the abelian group M into the abelian group N which preserves the scalar multiplication. In symbols, h satisfies the two conditions (3.34)

/z(m

+ m’) = h(m) + /7(m’),

(m, m’} c M ,

/?(am) = a/7(m),

aEA, mEM.

and (3.35)

The set of all A-homomorphisms of an A-module M into an A-module N is given the special symbol Hom,(M, N), that is, (3.36)

Hom,(M, N) = {/7: 11 is an A-homomorphism of M into N}.

Let S ( M , N) denote the set of all functions with common domain M and common range N where M and N are abelian groups. If {A h} c S(M, N); then the sum f + g is that element of %(M,N) whose value is f(m) + k(m) for each element m of M. Let z be the element of %(M,N) whose value is the identity 0 of N for each m of M. Given an elementfof %(M, N), let -f be the element of S(M, N) whose value is -f(m) for each m of M, that is, -f is the negative off. It can be shown that the set S(M, N) is an abelian group with these definitions. Let Hom(M, N) denote the set of all homomorphisms of the abelian group M into the abelian group N . Then it can be shown that Hom(M,N) is a subgroup of S(M, N). Now let M and N be A-modules. It can be shown that Hom,(M, N) is a subgroup of Hom(M, N), which can be made into an Amodule when A is commutative. Define af, a E A, f E Hom,(M, N) by (3.37)

[afJ(m) = a ( f (m)),

mE M

If both the domain and the range of the set Hom,(M, N) are taken t o be the A-module M , then even more algebraic structure is obtained. In such an instance. Hom,(M. M) is a ring when the product of {A h) c Hom,(M, M) is defined t o be their composition, that is, (3.38)

f/l

=fo

h.

These remarks complete our introduction t o the basic definitions and theorems in the theory of A-modules. A more detailed investigation of Amodules is intimately connected with the determination of the properties and ideal structure of the ring A acting on the module M. We shall consider a special case of this in the following chapter, but now turn to a more detailed examination of the situation in which A is taken t o be the field K of complex numbers. Such an examination will reveal a number of useful facts for our development of the theory of group representations and, at the same time, give us meaningful examples of the rather general ideas we have introduced concerning A-modules. We repeat an earlier definition.

23

3. Abelian Groups, Modules, and Vector Spaces

(3.39) DEFINITION. A complex vector space is a triple {M, K, R) consisting of an abelian group M, the field K of complex numbers, and a mapping rc from the Cartesian product K x M into M. The image n(a, m), ci E K , m E M, is denoted by the juxtaposition cm, called the product or scalar product of m by a. The following rules are required to hold :

(i) (ii) (3.40) (iii) (iv) where (1,

ci,

l m = m, a(a’m) = (ctci’)m, (a + a’)m = am + a h , a(m + m’) = am + am’, a’} c K and

{m, m’} c M.

(3.41) EXAMPLE. The universal example of a vector space M over the field K of complex numbers is the set M of all functions with domain a nonempty set 6 and range the field K of complex numbers. The group operation of addition is defined by taking the sum f + h of two elements f and 11 of M to be the map from G to K whose value is given by (3.42)

[f

+ I?](S) =f ( s ) +

sE

I?(S),

6.

The module operation of scalar multiplication is defined by taking the product af,ci E K , , ~ M, E to be the function from G to K whose value is given by

(3.43)

[afl(s) = r(f(s)),

sE G.

The zero element of M is that function z from 6 to K which maps every s E G into the zero complex number. If h E M, then the additive inverse -12 of I? is that function whose value is -h(s) for every s E G. Whenever 6 is a finite set, say

(3.44)

G = { I , . . . , n},

functions f and g from 6 to K can be denoted by row vectors

(3.45)

f =(q,.. . , c,)

and

(3.46)

g = (4,. . . , d;),

respectively, where ci = f ( i ) and di= g(i), 1 5 i 5 n. In this case, the rules of addition and scalar multiplication assume the familiar forms

(3.47) and

(3.48)

f + $7 = (c,, . . ., c,,) + (4,. . . > 4) = (c1 + d,, . . . , c, + a,) ~ f (q, = . . . , w,),

a E K,

24

1. Groups and Modules

respectively. The detailed verifications of all the axioms is left to the reader. The symbol C,, is sometimes used to denote vector spaces whose elements are n-tuples of complex numbers displayed in a row. For some purposes, it is convenient to display the components in a vertical column. Such vectors are called column vectors. The space of n-component column vectors is sometimes denoted by the symbol C". Recall that the symbol K,, is also used to denote the set of all n x n matrices with entries from the field K. There are a number of rules and lemmas needed for basic computational purposes which are minor extensions of the axioms in one direction or another. In the following equations, letters at the beginning of the alphabet such as a, j?, ai, pi,and so on, denote complex numbers. Letters such as x, y, x i ,y i , and so forth, denote vectors from the space M. The associative law has an important extension which implies that the symbol m, + . * + m k ,mi E M, has a well-defined meaning without the insertion of parentheses. In addition, one has

(3.49)

Ox = 0

and

(3.50)

-(ax) = (-.)x

(3.51)

.(xi + - - - + x , ) = a x ,

(3.52) (3.53)

a0 =0, =a(-x),

+..*+ax,,

+ + a,)x = alx + - . + a, x, (a1x1 + + a,x,) + (PIX, + ... + &X") (a1

* * *

* * *

= (a1

+ PI)XI +

* *

+ (an + B 3 X n ,

(3.54) The proofs of these are left as problems. A large part of the theory of modules over arbitrary fields is identical with the theory of modules over the complex numbers. As a matter of fact, none of the theorems of this section depend upon the nature of the field. Nevertheless, we largely restrict our considerations to the complex case and almost always use the term vector space to mean a module, denoted by M, over the complex numbers. To emphasize this aspect of M, we sometimes refer to it as a K-module or K-space. One of the important properties of vector spaces which distinguish them from more general types of A-modules is that every vector space is free (torsion free). Let the product am, U E K, m E M, be the zero vector of M. If a is not zero, then

(3.55) by (3.49).

m

= Im = (a-la)m = a-'(crm) = a-'O = O

25

3. Abelian Groups, Modules, and Vector Spaces

(3.56) DEFINITION. A nonempty subset {m,: .n E II} of M is free or linearly independent if and only if the equation (3.57) (a,,,

a,lm,l

+ . . + a,, *

...,aZk)t Kand {m,,, . . .,m,,}

m,, = 0,

c M, implies that

(3.58) a,, = * * . = a,, = 0. Note, in particular, that any set {m}, m E M, with m different from zero is linearly independent. Note the set {0} is not linearly independent. The concept of linearly independent coincides with that of free previously introduced for general A-modules. A set S of M is linearly dependent iff it is not linearly independent. A vector m of M is said to depend linearly or to be a linear combination of the set {ml,. . . , mk}of vectors of M iff there exists a set {ml, . . . , ak}of complex numbers such that

(3.59)

m=a,m, + * - . + a , m , .

A vector space M is nontrivial iff it contains a nonzero vector. A subspace S of M is a subgroup of M which satisfies the conditions of Definition (3.17). We can now prove an important theorem for vector spaces. (3.60) THEOREM. Every nontrivial vector space M contains a basis. Proof. Let 5 denote the ensemble of free subsets of M. Partially order 8 by set inclusion, that is, if ( S , S'} c 8, then S < S' if and only if S c S'. Let Q be a linearly ordered subset of 5. It is easy to see that the subset (3.61)

L = U S SEE

is a least upper bound of Q. By Zorn's lemma, 5 contains a maximal element B. If M contains an element m which is not a linear combination of elements in B, then B u {m}is a free subset B' properly containing B, which is a contradiction. Therefore B is a basis of M.

(3.62) COROLLARY. Every free subset S of a vector space M is contained in a basis B. Proof. Apply the above argument to the ensemble 8 of all free subsets of M which contain S. A nontrivial vector space M contains many distinct bases; however, the cardinality of any two bases B and B' of M is the same. We demonstrate this fact for finitely generated spaces by means of the following lemma.

(3.63) LEMMA. Let the subset {gl,. . . ,g k }be a set of generators of the vector space M. Every subset S of M containing more than k elements is linearly dependent.

26

I . Groups and Modules

Proof. It is to be noted that a subset S of M is linearly dependent whenever i t contains a linearly dependent subset. There is nothing to prove when M

is the trivial space. Otherwise, proceed by induction on k. When the generating set (gl} consists of a single element, let S be any subset of M containing distinct elements zgl and Pg,, different from zero. Then the equation (-P/a>(a91)

+ P91

=0

implies that {agl,pgl} is linearly dependent and consequently so is S. Assume the result for all positive integers k not exceeding n where 1 I n. Let {gl, . . . , gfl+l}be a generating subset of a vector space M and let S be a subset containing the set N ={mi, . . . , m f l + d

of distinct vectors. If N is contained in the space M’ generated by {gl,. . . ,gn}, the result follows from the induction hypothesis. Otherwise some element, say m l , is of the form m1

=a191

+

* * .

+ angn + an+lgn+li

where a,+ is different from zero. Each of the remaining elements of N has an expansion mi = pi,lgl ... + P i , n + l g n + l , 2 5 i < n 2.

+

+

Denote by D the set {z2, . . . , z , + ~ }of n

+ 1 vectors defined by

z i = m i - (Bi,.+l/afl+l)ml,

2I is n

+ 2.

Observe that D is contained in the space M’ and is linearly dependent by the induction hypothesis. Let { x 2 , . . . , x , , + ~ be } a set of complex numbers, not all zero, such that ~

2

+~... 2+ ~ n + 2 ~ n +=2O .

Then the equation,

+ . * .+ ~ n + 2 P n + 2 , n + l h l

(-1/%,+1)(~282,n+l

+X2mZ+...+Xn+zmn+2=0,

shows that N and consequently S is a linearly dependent set. This completes the induction and the proof. (3.64) THEOREM. Let the subsets { m l , . . ., mj) and inl,.. ., nk) be two distinct bases B and B’ respectively, of the vector space M. Then the integersj and k are equal. ProoJ Since each of these bases is a set of generators for M, it follows, by two applications of Lemma (3.63), that k cannot exceed j and j cannot exceed k . Consequently, j must equal k .

3. Abelian Groups, Modules, and Vector Spaces

27

This equality makes possible the following definition.

(3.65) DEFINITION. The dimension of the trivial vector space is zero. Otherwise, the dimension of a finitely generated vector space M is the number of elements in any basis of M. (3.66)

EXAMPLE. Let M be the vector space of all r-tuples, M

= {m:m = (cl,

. .., c,),

ciE K } ,

of complex numbers with the operations of addition and scalar multiplication of Example (3.41). It is easy t o see that the set {m,,. . . , m,},each element mi being the r-tuple with ith component 1 and all the rest zero, is a basis, called the standard basis, of M. Therefore, one notes that there exists an r-dimensional vector space M for each integer r. We call this space M the canonical r-dimensional vector space.

(3.67) THEOREM. Each r-dimensional vector space N is isomorphic to the canonical r-dimensional vector space M and, consequently, to any other r-dimensional vector space N’. Proof Let {n,, . . . , n,} be a basis B’ of N. Let n and ii be any two elements of N with the expansions n

= clnl

+

9 . .

+ c,n,

and

ii

= dlnl

+

*..

+ d,n,,

respectively, in terms of the basis B‘. Let h be the mapping from N to M whose value for the typical element n is given by

h(n) = h(c,nl Note that

h(n + 6 ) = h[(c, + dl)n,

+ . * . + c,n,) = (c,, . . . , c,).

+ . . + (c, -t- dr)n,] = (cl + d,, . . . , c, + d,) = (cl, . . . , c,) + ( d l , . . . , d,.) = h(n) + h(ii). *

Also, if M. belongs to K , then h(cm) = h(crc,n,

+ - + q n , ) = (ac,, . . . ,ac,) = a(c,,

. . . , c,)

= ah(n).

Thus h is a homomorphism of N into M. It is easy to see that I? is a bijection so that it is an isomorphism. Furthermore, if h’ is an isomorphism of any r-dimensional vector space N’ onto M, then h-‘ h’ is an isomorphism of N’ onto N. 0

(3.68) REMARK. One should avoid the mistaken impression that any two A-modules M, and M, over the fields A , and A , are isomorphic whenever they have the same dimensions over their respective fields. Vector spaces over nonisomorphic fields are never isomorphic. However, if A , is a finite

28

1. Groups and Modules

extension of the field A,, then each r-dimensional A,-module M can be extended to an r-dimensional A,-module M‘ in such a manner that M’ is an A,-module. However, the A,-dimension of M‘ is larger than r except in the case that A , is a trivial extension of A,. Consequently, in more general situations, one is compelled to speak of the A,-dimension or the A,-dimension of an abelian group M which is simultaneously an A,-module and an A,-module. Let 8 be a set {M,, . . . , M,} of subspaces of the vector space M. A useful concept is determined by means of the following definition. (3.69)

DEFINITION.

The vector space M is the sum, written M =Mi

+

* * .

+ Mk,

of the subspaces {Mli . . . , M,} if and only if each m of M can be written in at least one way as the sum

+ + mk,

m = m,

(3.70)

*.

mi E M i , 1 I i I k .

A special case of (3.68), more frequently encountered, is given in terms of the next definition. (3.71)

DEFINITION.

The vector space M is the internal direct sum, written M=M,@*..@M,,

of the subspaces [Mf, . . . , M,) if and only if each m of M can be written in exactly one way as the sum m

= m,

+ * . .+ mk,

m iE M i , 1 5 i 5 k.

These two definitions are the natural extensions, in the case of vector spaces, of Definitions (1.20) and (1.21) to the case of more than two subgroups. We also have an extension of the concept of external direct product appropriate to vector spaces as follows.

(3.72) DEFINITION. The external direct sum of the set { M I , .. .)M,} of K-spaces, also written M=M,@..*@M,, is the vector space M consisting of all k-tuples

M

= {m : m = (m,,

. . . ,mk), mi E Mi}

with addition defined by the rule (3.73)

(m,,

. . . , mk) + (m,’, . . . , mk’) = (m,

+ ml’, . . . , m, + m,’)

and scalar multiplication defined by the rule (3.74)

a@,, ..., m,)

= (am,, . . . ,am,),

u E K.

29

3. Abelian Groups, Modules, and Vector Spaces

These ideas come into play in the definition of a complementary subspace N’ of a subspace N of a vector space M. The subspace N’ is a complementary subspace of the subspace N iff M is the internal direct sum N O N ’ of N and

”.

(3.75) LEMMA.Let the vector space M properly contain the subspace N which properly contains the trivial subspace (0). Then there exists a complimentary subspace N’ such that M is the internal direct sum N ON’. Proof. Let {m,}, 71 E n, be a basis C of N and let B be the basis, existing by Corollary (3.62), of M which contains C. Let {m,,}, 71’ E Il’,be the nonempty set C‘ of those elements of B not belonging to C. Let N‘ be the subspace (C)of M generated by C‘. Since every element m E M is a linear combination (3.76)

m = a,;m,,

+

* *

. + affkm,,

+ a,,, m,, + . . . + I

ctffj,

mffj,,

where { a f f sa,,.} , c K , {mffS} c C, {m,,.} c C’, 1 5 s 5 k, 1 I t s j , it follows that M is the sum N N’. Let n and n’ be elements of N and N’ whose expansions with respect to C and C’ are the linear combinations

+

~,,m,, + ’ * * and

a,,,rn,,,

+

mnr>

+ . + a,,,

a,, E K ,

m,,, ,

E

mffzE

c,

K , m,,. E C’,

respectively. The equality n =n’ implies that a linear combination of the basis elements is zero. Therefore, all the coefficients of n and n’ are zero which means that both n and n‘ are zero. Consequently, M is the direct sum N 0 N’.

(3.77) REMARK. We wish to show that the preceding lemma is not valid for general A-modules. Let (2) denote the submodule of all multiples of 2 in the Z-module of the integers. Suppose that the module Z of the integers is the direct sum Z = (2) @ N’,

where N‘ is a submodule of Z containing only one even integer, namely, the number 0. If n‘ is a nonzero integer of N ‘ , then 2n’ is a nonzero even integer in N’. Therefore N’ is the submodule (0). This observation contradicts the assumption that Z is the direct sum (2) @ N ’ . Consequently, we see that even as well-behaved a Z-module as the integers Z themselves does not satisfy Lemma (3.75).

30

1. Groups and Modules

The concept of factor module has been introduced. We give an additional treatment for the special case of a factor space. Let N be a subspace of the vector space M. Then N is a normal subgroup of the abelian group M, and the factor group M/N is well defined. The equality of the cosets [m] and [m’] implies that m - m’ belongs to N and consequently, that am -am‘ belongs to N. Therefore, the cosets am + N and am’ N are the same. We can now give the standard definition.

+

(3.78) DEFINITION. Let N be a subspace of the vector space M. The factor space M/N is the factor group M/N together with the scalar multiplication given by a(m+N)=am+N,

(3.79)

E E K , mEM.

Results (1.19, (1.22), and (1.26) have natural extensions to modules and, in particular, to vector spaces. As an example, we consider the case of (1.22) which leads to the following lemma. (3.80) LEMMA. Let M be the sum N + N’ of two subspaces, N and N’. Then the factor space M/N’ is isomorphic to the factor space N/(N n N’).

Proof. Let v be the natural mapping of the abelian group M onto the factor group M/N’. Letf’ be the restriction of v to the subgroup N. Then, exactly as in the proof of (1.22),ff is a homomorphism of the subgroup N onto the factor group M/N’. The homomorphism ,f’ defines an isomorphism f of N/(N n N’) onto M/N’, where the mappingfis given by (3.81)

f(n

+ N n N’) =f’(n>,

where n + N n N’ is any element of N/(N n N’). We wish to show thatf is a vector space isomorphism, that is, that (3.82)

f[a(n

+ N n N’)] = gf(n + N n N’),

a E K.

However, (3.83)

f[a(n

+ N n N’)] = f ( m + N n N’) =f’(an) = v(an) = an + N’ = cx(n + N’) = av(n) = a.’(n) = af(n + N n N’).

An important special case of (3.80) is the following corollary. (3.84) COROLLARY. Let the vector space M be the direct sum N @N’ of its subspaces N and N’. Then M/N‘ is isomorphic to N. Proof. The intersection N n N’ is the zero space (0) and the factor space N/(O) is isomorphic to N.

To illustrate these ideas, we give an example.

31

3. Abelian Groups, Modules, and Vector Spaces

(3.85) EXAMPLE. Let {m,, . . ., m,} be a basis of the vector space M. Let N and N’ be the subspaces generated by {m,, m2} and { m 3 , m,}, respectively. As in the proof of Lemma (3.75), M is the direct sum N @ N’. Any coset of M/N’ is of the form (3.86)

Elml

+ ... + a4m4 + N’ = Elml + a2m2 + N’,

where {al, . . . , a,} c K. Further, if (3.87)

alml

+ a2m2 + N

then (3.88)

alml

= al’ml

+ a2’m, + N’,

+ a , m, = al’ml + a2’m2.

Consequently, the mapping f from N onto M/N‘ such that (3.89)

f(alml

+ a, m,)

= Elml

+ a, m, + N’

is a bijection which preserves the algebraic operations, that is, f is an isomorphism of N onto M/N‘.

The relationships among vector spaces satisfying the chain conditions and those which are finitely generated are stronger than in the case of general A-modules. The next theorem indicates the special implications in the case of vector spaces. (3.90) THEOREM. Let M be a vector space. The three following conditions on M are equivalent: (i) M is finitely generated ; (ii) the subspaces of M satisfy the ascending chain condition (A.C.C.); and (iii) the subspaces of M satisfy the descending chain condition (D.C.C.).

ProoJ First, observe that condition (i) implies by Lemma (3.63) that M and all of its subspaces have bases containing not more than k elements for some integer k . Theorem (3.25), unproved in this textbook, then implies the equivalence of (i) and (ii). Given (i) and the existence of the integer k mentioned above, let (3.91)

N, c N, c ..* c N,

c

...

be any properly ascending sequence of subspaces. Since the inclusion is proper, the dimension of N i + , must exceed that of Ni by at least 1. Since the dimension of no subspace exceeds k , no properly ascending series of subspaces contains more than k + 1 members. Given (ii), let (3.92)

N,

2

N,

3

3

N, 2 . * *

32

1. Groups and Modules

be a properly descending sequence of subspaces. Let Ni’ be a subspace complementary to Ni with Ni’ c Nf + ; then the series

,

(3.93)

N,’ c N,’ c .

c N,’ c

...

is a properly ascending sequence. It follows that (3.92) contains only a finite number of members. Given (iii), let { m l , . . . , m,, . . .} be a free subset S of M. Let Bidenote the S-complement of {ml, . . . , mi} and let Ni be the subspace generated by Bi . Then one obtains the descending chain (3.94)

N,

3

N,

2

... 3 N, 3

.

a

*

of subspaces. Since this chain must be finite, it follows that any basis B of M contains only a finite number of elements. Thus M is finitely generated. Such relations as those of Theorem (3.90) are unusual as shown by considering the following example. (3.95) EXAMPLE. The ring Z of the integers, themselves a 2-module, does not satisfy the descending chain condition (D.C.C.) on submodules. To see this, consider the family {.Zi), 0 5 i, of submodules defined by Zi = (2i).Then (3.96)

z,3z1 3 * . . 3 z r 3 . . .

is a properly descending chain of infinite length. On the other hand, let (3.97)

z, cz,c . * - cz,c . . ’

be an ascending chain of nontrivial submodules of 2. The union (3.98)

hJ=

t) Z i ,

i= 1

is a module of 2.If s is the smallest positive integer contained in N , then N = (s). However, s must be contained in some first submodule of the union (3.98), say Z , , from which it follows that 2, = Z r + l=... = N ,

that is, there are only a finite number of distinct submodules in the ascending chain (3.97). Nevertheless, one should avoid the misconception that there is some fixed number k such that no ascending chain contains more than k elements. It is easy to see that there exists ascending chains of submodules of any finite length in the Z-module of the integers. The question of the existence of a composition series for the subspaces of a vector space M is easy to settle. One has the following straightforward result.

33

4, Linear Transformationson Vector Spaces

(3.99) THEOREM. A vector space M has a composition series of subspaces if and only if M is finitely generated. Proof. Suppose that M is finitely generated and let the subset {m,,. . . , m,} be a basis of M. Let N, be the subspace of M generated by the set {m,,. . . , mi}, 1 i 5 r. The ascending chain of subspaces (3.100)

(0) = No c N, c * . . c N,

=M

is a composition series for M. The fact that the dimension of N i , 0 _< i < r, is only one less than the dimension of N i + l implies that Ni is a maximal subspace of N i + l . Suppose, conversely, that (3.100) is a composition series of M. Since the dimension of N i , 0 < i 5 r, diminishes by one on passing from Ni to Ni-,, it follows that M has dimension r, in particular, M is finitely generated. The proof of the Jordan-Holder theorem is simple in the case of vector spaces where the theorem assumes the following form. (3.101) THEOREM (Jordan-Hiilder). Any two composition series (3.102) and (3.103)

N, c N, c . . * c N k + l

s, c s, c

* * *

c

sj+l

of the finitely generated K-space M are equivalent. Proof. It is easy to see that the subspace N of the vector space M is maximal and the factor space M/N is minimal or irreducible if and only if the dimension of M exceeds that of N by one. It follows that the number of elements in a composition series of an r-dimensional vector space M is r + 1. Consequently, k a n d j must be equal in the series (3.102) and (3.103). Each of the factors occurring in either of the composition series is a one-dimensional K-space. Therefore, the identity permutation 71 is the required correspondence. We turn now to the fundamental problem of determining the nature of the K-homomorphisms of one K-space M into another K-space N. 4. LINEAR TRANSFORMATIONS ON VECTOR SPACES

The study of the homomorphisms of a given class of algebraic systems is one of the classical problems of algebra. Usually, the successful pursuit of such an investigation requires not only a deep knowledge of the particular algebraic systems involved but also a good deal of algebraic sophistication. Fortunately, finite-dimensional vector spaces constitute an important class of algebraic systems for which reasonably complete results can be obtained at an

34

I . Groups and Modules

elementary level. This section is devoted to an explicit determination of the nature of the elements of Hom,(M, N) where M and N are finitely generated K-modules or, in other words, finite dimensional complex vector spaces. An element h of Hom,(M, N) is, of course, a K-homomorphism of the Kmodule M into the K-module N. However, we adopt the usual custom and speak of h as a linear transformation from M to N. To be specific, we repeat a definition. (4.1) DEFINITION. A linear transformation from the complex vector space M to the complex vector space N is a function Twith domain M and range N for which

+

(i) T(m m‘) = T(m) (ii) T(crm) = crT(m),

+ T(m’),

{m, m’} c M, U E K , mEM.

The words linear operator and linear mapping are frequently used synonyms for linear transformation. (4.2.) EXAMPLE.Let {ml, m 2 , m,} be a basis B of the three-dimensional vector space M. Let h be a mapping from M into K such that

+ cr2 m2 + u3 m,) = crl for each m whose expansion is ulml + cr2 m2 + u3 m3 in terms of the basis B. Note that, if m’ is an element with the expansion Plml + /I2 m2 + P3 m, , then h(a,m,

(4.3)

Mm

+ m’) = h[(Ximi + a2m2 + u3m3) + ( B I ~ +I Pzmz + P3m3)I =~[(UI = cr,

+

+ B&I + ( x 2 + BzImz + (u3 + P d m J = h(m)

+ h(m’).

By a similar argument, (4.4)

h(cm) = ccr,

= ch(m),

c E K, m E M.

Thus h is an element of Hom,(M, K ) . This is not only an example of a linear transformation from the vector space M to the vector space K, but it also i5 an example of a special kind of linear transformation which is sufficiently important to give rise to the following definition. (4.5) DEFINITION. An element h of Hom,(M, K), that is, a linear transformation from the K-space M to the complex numbers Kis called a linearfunctional on M. According to earlier remarks, which are briefly discussed below in the special caSe of vector spaces, the set Hom,(M, N) of A-homomorphisms of an A-module M into an A-module N is also an A-module for a commutative ring A. Consequently, Hom,(M, K), the set of all linear functionals on M is a vector space called the dual space of M and denoted by the symbol M* (since K is a commutative ring).

4. Linear Transformations on Vector Spaces

35

(4.6) EXAMPLE. This example is so general that every linear transformation from a finite dimensional K-space M to a K-space N is a special instance of it. Therefore we term it the canonical linear transformation. Let the set {m,, . . . , m,} be any basis B of the r-dimensional vector space M and let the set {n,, .. . , n,) be any collection I of r vectors, distinct or not, in the vector space N. Let x be any element of M whose linear expansion in terms of the basis B is (4.7)

x = Elml

+ ... + arm,,

C ~ K, ~ E1

5 i 5 r.

Let T be the function with domain M and range N whose value for x is given by (4.8)

T(x) = T(a,m,

+ ... + arm,) = alnl + * . . + arn,.

The proof of the fact that T is an element of Hom,(M, N), that is, a linear transformation from M to N is so easy and so fundamental that we leave it to the reader. The argument goes exactly as in Example (4.2). However, we give a theorem which shows the central importance of the canonical linear transformation. Let the set {ml, .. . , mr>be a basis B of the K-space M and (4.9) THEOREM. let the set {n,, . . . , n,} be any set Z of r vectors in the K-space N. Then there exists exactly one linear transformation T of Hom,(M, N) such that T(mi) = n i ,

(4.10)

1 5iI r.

Proof. The linear transformation defined in (4.8) is one such linear transformation. Thus the theorem is valid if T is the only linear transformation with such properties. Let T' be any linear transformation of Hom,(M, N) with

(4.I 1)

T'(mi) = n i ,

1

< i 5 r.

Then the value of T' for a general element x such as that of (4.7) is given by

T'(x)= T'(a,m, = a,n,

+ . . * + arm, = a,T'(m,) + ... + arT'(m,)

+ ... + a,n,.

Thus we see that T'(x) and T(x) are equal for every x of M, which shows that the functions T' and T ar e equal. (4.12) REMARK. One notes that we have given a sort of answer to the nature of the elements of Hom,(M, N), namely, each pair of sets (4.13)

B

= {ml,

. . . , m,}

and (4.14)

Z = {n, * . ., nrly 9

36

1. Groups and Modules

where B is a basis of M, determines a unique linear transformation by (4.10). Conversely, given a basis B and a linear transformation T, the set Z is uniquely determined by (4.10) as well. Nevertheless, this answer is not entirely satisfactory since two different pairs {B, I} and {B', Z'} may actually define the same linear transformation T by means of (4.10). A fully satisfactory description of the situation depends upon the introduction of the concept of the matrix of a linear transformation Twith respect to a pair {B, C } of bases of the finite-dimensional vector spaces M and N, respectively. (4.15) DEFINITION. Let T be an element of Hom,(M, N) where both M and N are finite-dimensional K-spaces. Let {m,, . . . , m,} be a basis B of M and { n l , . . . , n,} be a basis C of N. Then there exists an s x r matrix of complex numbers a:, 1 I u I s, 1 I v Ir, called the matrix of T with respect to the pair {B, C } . The elements of the array are determined by the following sets of equations, (4.16)

T ( m i ) = a i ' n l + . . . + a:n,,

I
The matrix {a,"} is ordinarily referred to merely as the matrix of T unless there is some doubt about which pair {B, C} of bases is understood. Theorem (4.9) and Remark (4.12) assure us that there exists a unique matrix corresponding to every element T of Hom,(M, N) for every choice of the pair (B, C). A linear transformation T ordinarily has different matrices with respect to different choices of the pair { B , C}. Consequently, one must find the relationship between the matrices {a;} and {'au"}of Twith respect to the pairs {B, C} and {B', C'}, respectively. (4.17) THEOREM. Let the bases B and B' of the K-space M consist of the sets {m,, . . ., mr} and {ml', .. ., q'},respectively. There exist uniquely determined matrices {p,"} and {'p,"}, 1 I u, u r, such that (4.18)

mi' = pilml + * . . Pirmr >

and (4.19)

mk =

'pklml'

+ . + ):Inr', * '

1l i < r , 1 5 k 5 r.

Furthermore, the matrices {p,"} and {'p,"} satisfy the relationship (4.20) Proof. The existence of the unique matrix {p,"} of coefficients that satisfy (4.18) follows from the fact that B is a basis; that of {'p,"} follows similarly. Inserting (4.18) into (4.19), one finds that (4.21)

37

4. Linear Transformationson Vector Spaces

which implies by the linear independence of B that

(4.22)

6,'

=c

6,'

=

In a similar fashion,

(4.23)

p ? 'pki.

1)jpki.

Thus the matrices {p,"} and {'p,"} are inverses of each other. We say that the matrix {p,"} relates the basis {m,'} to the basis {mu}. The desired relationship between the matrices of a linear transformation T with respect to two different pairs of bases can now be expressed by the following theorem.

(4.24) THEOREM. Let the linear transformation T of Hom,(M, N) have the matrices {a,'} and {'av"}with respect to the two pairs of bases {B, C} and {B', C'} of the K-spaces M and N, respectively. Let {p,"} relate the basis B' to B and (4,") relate the basis C' to C. Then one has the equations, t

t

a, =

2 'q,'a$p:,

iI r. 1I tI s, 1 I

Proof. We use the notation introduced above. We find that

(4.25)

T(m,')

=

T(pilml + * * .

= pilT(m,) = pil(a,'nl = =

+ p;mr)

+ . . + p;T(mr) *

+ . . + ulsns) + . . . + pir(urlnL+ . . . + a:ns)

1PiYC auvnv) = CPi" (a,.(C

c (C 'qUtauvP:)n''*

'qutnt'))

It follows from the definition of {'a,'}, 1 5 t 5 s, 1 I i 5 r, that (4.26) If we denote each of these matrices by its corresponding upper case letter, then Eq. (4.26) assumes the form

(4.27)

A'

=

Q'AP

=

Q-lAP,

where A is the matrix of Twith respect to the pair {B, C} and A' is its matrix with respect t o the pair {B', C'}. Two matrices A and A' are said to be in the relation R iff there exists invertible matrices P and Q' such that Eq. (4.27) is satisfied. The two matrices A and A' represent the same linear transformation T only if they are in the relation R. Conversely, if the matrix A' is in the relation R with the matrix A of a linear transformation T with respect to the pair { B , C}, then A' is the matrix of T with respect to a suitable selected pair {B',C"}. As one easily

38

1. Groups and Modules

verifies, the relation R is an equivalence relation on the set of all s x r complex matrices. Each linear transformation T is associated with an equivalence class of matrices, each of which defines Tfor some pair { B , C } of bases of M and N. The rzull linear transformation T is the element of Hom,(M, N) which maps each element of M into the zero element of N. It is clear that the matrix {a,") of the null linear transformation has all of its components equal to zero. In order to select a particularly suitable representative element from each equivalence class of matrices for nonnull linear transformations, it is convenient to introduce the following definitions. (4.28) DEFINITION. Let f be a function with domain D and range R. The image o f f , written Imf, is the set of all images of elements of D underf, that is, (4.29)

Im f

=

{r: r E R , r = f ( d ) , d E D}.

Whenever Tis an element of Hom,(M, N), the image of Tis a subspace of the range N and the kernel of Tis a subspace of the domain M . The dimensions of these subspaces are sufficiently important to deserve the following treatment. (4.30) DEFINITION. Let T be an element o f Hom,(M, N) where M is an r-dimensional K-space. If T is the null linear transformation, then its rank is zero. If Tis a monomorphism, then its nullity is zero. Otherwise, the nuNity of Tis the dimension of the kernel of Tand the rank of Tis the dimension of the image of T. (4.31) THEOREM. Let T be an element of rank t of Hom,(M, N) where M is an r-dimensional space and N I S an s-dimensional one. Let {a,'}, 1 5 j I s, I ii I r, be an s Y r matrix such that and all the remaining entries are zero. Then there exists a pair {B, C} of bases of M and N such that (a,'} is the matrix of T with respect to this pair. Proof. If T is the null homomorphism the result is true. Otherwise, the space M is the direct sum H @ Ker T where the sets (hl, ..., h,) and { k , , . . . , k,} are bases D and E for H and Ker T, respectively. The set of images {T(h,), . . . , T(h,)} is linearly independent. For suppose that {el, . . ., c,} is a set of complex numbers such that

(4.32)

+ +

Then the element c,h, .. c,h, is common to H and Ker T, that is, it is the zero vector. Since D is linearly independent, it follows that ci=O,

I
39

4. Linear Transformations on Vector Spaces

Any element m of M can be written in the form m = clh,

+ ... + c,h, + d,k, + ... + d,k,,

from which it follows that T(m) = clT(h,)

+ . . . + c, T(h,).

Thus we see that {T(h,), . . . , T(h,)} is a basis for the subspace the Im T and consequently that the rank of T is u. If {T(h,), . . . , T(h,)} is not a basis for N, then we supplement it with sufficiently many vectors to form a basis C of N. We take for the basis B of M the union D u E. We have the results,

(4.33)

T(hi) = T(hi),

1

< i < u = t,

and

(4.34)

T(ki) = 0,

l
Consequently, the matrix of Twith respect to the pair { B , C} has the required form. Two matrices A and A' are usually called equivalent iff they are in the relation R of (4.27). We have shown that every equivalence class of R contains a canonical form of the above type. The following corollaries can be read off from this result.

(4.35) COROLLARY. Let T be an element of Hom,(M, N) where M is an r-dimensional K-space. Then the nullity of Tplus the rank of Tis equal to the dimension of M. (4.36) COROLLARY. Let T be an element of Hom,(M, N) where both M and N are r-dimensional K-spaces. Then T is a monomorphism if and only if T is an epimorphism. We turn now to a discussion of linear transformations on a single vector space M. When discussing an element T of Hom,(M, M), it is customary to define the matrix of T with respect to a single basis B of M. Consequently, Eqs. (4.16) assume the special form

(4.37)

T(m,) = a:ml

+ + airm,,

1 5iI r,

where {m,, . . .,m,} is the given basis. Let the matrix {p,"} relate the basis {m,'} to the basis {m,}. Then the matrix {'a,'} of T with respect to the basis {m,'} is given by

(4.38)

'air=

c'puia~p~,

40

1. Groups and Modules

where the matrix {'pu'}is the inverse of {p,"}. Two matrices A and A' are said to be similar iff there exists a matrix P such that

(4.39)

A'

= P-'AP.

The result is that A and A' are matrices of T with respect to different bases only if A and A' are similar. Conversely, if A and A' are similar and A is the matrix of T with respect to some basis B, then A' is the matrix of T with respect to some other basis B'. Unfortunately, the concept of similarity does not lend itself to so simple an analysis as that of equivalence. A careful treatment involves more space than we are able to give to the topic. Before passing to the next section where these matters are partially discussed, we make one additional observation about the nature of the set Hom,(M, M).

(4.40) THEOREM. Let Tand T' be elements of Hom,(M, M) and c an element of K . Then the sum of T and T', defined by (4.41)

[T+ T'](m)

=

T(m)

+ T'(m),

m E M,

and the scalar product of T b y c, defined by

(4.42)

[cTl(m) = c(T(m)),

m

E

M,

are linear transformations. Furthermore, the product TT',defined by

(4.43)

[TT'](m)

=

m E M,

T(T'(m)),

is a linear transformation. In addition, Hom,(M, M ) is a vector space under the sum and scalar product definitions as well as a ring under the sum and product definitions. We leave the proof of these facts as a useful exercise for the reader. 5. INVARIANTS OF LINEAR TRANSFORMATIONS

The present section is devoted to a brief sketch of the various types of invariants and canonical forms which arise in the theory of linear transformations on an r-dimensional K-space M. The symbol T always denotes an element of Hom,(M, M). DEFINITION. A nonzero element m of M is an eigenvector of T corresponding to the eigenvalue n of K if and only if (5.1)

(5.2)

Tm = nm.

(5.3) DEFINITION. A nontrivial subspace N of M is an eigenspace of T corresponding to the eigenvalue n E K if and only if (5.4)

Tm=xm,

mEN.

41

5. Invariants of Linear Transformations

DEFINITION. A subspace N of M is an invariant subspace of T if and only if Tm is an element of N whenever m is an element of N.

(5.5)

In order to discuss the existence of these invariants for a particular linear transformation T, we introduce a few definitions and results from the theory of linear equations and determinants. The determinant of the matrix A is denoted by the symbol ( A 1 and that of the matrix {a;} by the symbol Juvu 1 whenever A and (a:} are square matrices. The row rank of a matrix is the maximal number of linearly independent rows which it contains and the column rank is the maximal number of linearly independent columns. The row rank and the column rank of a matrix are equal and are called its rank. We assume the following fundamental theorem on the solutions of homogeneous linear equations. (5.6) THEOREM. Let A denote the r x r complex matrix whose elements are the set {a:), 1 2 u, u 2 r. Then the dimension of the space of solutions of the set of homogeneous equations (5.7)

equals r minus the rank of A . In particular, the system above has a nontrivial solution if and only if the determinant I A I is zero. Theorem (5.6) enables one to reduce the problem of the existence of eigenvectors and eigenvalues of a linear transformation T on the r-dimensional K-space M to the study of its matrix with respect to any basis B of M. Let {a:} be the matrix of T with respect to the set (m,, . . . , m,} of elements of B. Let m be an element of M having the expansion m = tlml

+ ... + trm,

in terms of B. Then Eq. (5.2) leads to the following relations between the matrix {a,'} of Tand the components {ti},1 2 i 5 r, of an eigenvector m with eigenvalue IK, namely, By Theorem (5.6), this system of equations has a nontrivial solution if and only if )nd"u- a;1 = 0 , that is, if and only if, (5.9)

IIrZ-Al

=o,

where A denotes the matrix { a t } and Z the r x r identity matrix. Let A' be the matrix of T with respect to a basis B' related to B by a matrix P. Then the necessary and sufficient condition assumes the form (5.10)

17CI-A'l

=o.

42

1. Groups and Modules

Letf(t) be the rth degree polynomial in t, which is given by

I

f ( t ) = tz - A

(5.11)

1.

It follows from (4.39) that

f(r>=

ItZ-Al

=

ItPP-'-PA'P-'/

=

/P(tZ-A')P-'/

=

ItI-A'I,

so thatf(t) is independent of the particular matrix of Twhich is employed in its calculation. (5.12) DEFINITION. The rth degree polynomialf(t) of Eq. (5.11) is called the characteristic polynomial of the linear transformation T. Our previous considerations have led to the following theorem.

(5.13) THEOREM. The complex number n is an eigenvalue of the linear transformation T i f and only if n is a root of the characteristic polynomial f(t) of T. This theorem is fairly useful for the calculation of eigenvalues and eigenvectors of linear transformations on spaces of low dimension, but less so for higher-dimensional ones. A famous result on linear transformations is given by the following theorem. (5.14) THEOREM (Cayley-Hamilton). Letf(t) be the characteristic polynomial of the linear transformation Ton the r-dimensional K-space M. Then one has

T'+ arPlTr-'+ ... + a,T+ aoZM= 0,

where 0 denotes the null linear transformation.

A monic polynomial g ( t ) is one whose leading coefficient is 1. Theorem (5.14) asserts that a linear transformation T on an r-dimensional space is a root of a monic polynomial of degree r. Not uncommonly, such a linear transformation Tis a root of a polynomial of lower degree. (5.15) DEFINITION. Let T be an element of Hom,(M, M) where M is an r-dimensional K-space. The nonconstant monk polynomial f o ( t ) of lowest degree which has T a s a root is called the minimalpolynomial of T. There are a number of facts about the minimal polynomialf,(t) of Twhich we quote and do not prove. The minimal polynomial of T is unique and is a divisor of every polynomial g(t) which has T as a root. In particular, the minimal polynomial is a divisor of the characteristic polynomialf(t) of T. However, every linear factor off(t) is also a factor offo(t).

43

5. Invariants of Linear Transformations

One of the reasons for limiting our considerations to K-spaces over the field K of complex numbers is that complex polynomials have simpler factorization properties than do polynomials over more general fields. If h(t) is an element of K [ t ] ,then h(t) has a factorization (5.16)

h(t) = a,(t - p l y . . . (t - &)ak

unique up to the order of the factors, as the product of its leading coefficient

a, times the powers of distinct monic first degree polynomials. In particular,

the characteristicf(t) and minimalfo(t) polynomials of a linear transformation T have the factorizations (5.17)

f ( t )= (t -

‘‘‘

( t - pk)ak

and (5.18)

fO(t)

= ( t - PI)”

‘

.‘ ( t - pk)”,

respectively, where 1 I p i 5 c l i , 1 5 i I k . The set {pi, .. . , p k } of distinct roots off(t) occurring in the factorization (5.17) is the complete set of eigenvalues of the linear transformation T. It is clear that there exists no nonconstant common divisor of the set = {pl(t>,. *

(5.19)

. ,P k ( t ) )

of polynomials defined by (5.20)

1 Ii 5 k .

p i ( t ) = f ( t ) / ( t - pi)(li,

There is an interesting result from the theory of polynomials over a field which asserts that given any set such as $‘3 of relatively prime polynomials there exists a second set (5.21)

= {q1(0,

. .. q k ( a 9

of polynomials such that the equation (5.22)

1 = Pl(t)ql(t)

+ ... +Pk(t)Clk(O

is a polynomial identity. This identity implies that (5.23)

I M =pl(T)ql(T)

+

* ’‘

+ pk(T)qk(T).

Consequently, if x is any element of M, then (5.24) Let Mi denote the image of the linear transformation p , ( T ) , 1 _< i 5 k . Then (5.24) implies that M is the sum of these subspaces, that is, (5.25)

M = M , +*.*+Mk.

44

1. Groups and Modules

Define the set {d,(t), . . . , dk(t)}of polynomials by (5.26)

1 I i I k,

di(t) = ( t - pi)'[,

and observe that the pairs {di(t),pi(t)} of polynomials are relatively prime, 1 I i k . As above, there exist pairs {ri(t),si(t)>of polynomials such that (5.27)

1

+ d,(t)si(t),

1 I i 2 k.

= pi(t)ri(t)

These identities give rise to the linear transformation equations, (5.28)

ZM =pj(T)ri(T) + di(T)si(T),

1 5 i I k.

A substantial number of the following results depend upon the commutativity of the linear transformations defined by any two polynomials p ( T ) and q(T) in the linear transformation T of Hom,(M, M). Since every element x i of the subspace M i , defined above is of the form p i ( T ) y for a suitable choice of y, it follows that

(5.29)

di(T)Xi = d,(T)p,(T)y= f ( T ) y = 0.

Consequently, Mi is contained in the kernel of the linear transformation d i ( T ) .Conversely, if x is an element of the kernel of di(T),then (5.30)

x = pi(T)ri(T)x + di(T)si(T)x= p i ( T ) r i ( T ) x+ s,(T)d,(T)x = pi(T)ri(T)x,

so that x belongs to M i . Therefore, M i is the kernel of di(T).Furthermore, p,(T)ri(T)xi equals xi for every xi of Mi and p i ( T ) x j= 0, j # i. Assume that (5.31) Then (5.32)

0 = x1 + ... + x k ,

0 = pi(T)ri(T)(x, + * .

xi E M i .

- + xk)= x i ,

so that each of the summands of (5.31) is zero and the sum (5.25) is direct. (5.33) DEFINITION. Let T be an element of Hom,(M, M) where M is an r-dimensional K-space. The characteristic subspace of T corresponding to the eigenvalue p i is the subspace M i , the kernel of the linear transformation Our previous considerations lead to the following theorem. (5.34) THEOREM. Let T be an element of Hom,(M, M) where M is an r-dimensional K-space. Then M is the direct sum (5.35)

M

= Mi @

of the characteristic subspaces of T.

... @ Mk

45

5. Invariants of Linear Transformations

One notes that each characteristic subspace M i of Tis an invariant subspace

of T since x E Mi implies that (5.36)

di(T)Tx = Tdi(T)x = 0.

Let Mi be an invariant subspace of the linear transformation T on the r-dimensional K-space M. The restriction Ti of T to Mi is the mapping T i on M i , which is defined by

Timi= Tm,,

(5.37)

miEMi,

The mapping Ti is an element of Hom,(M, , Mi). Let the r-dimensional K-space M be the direct sum (5.38) LEMMA.

M = MI @ . . . @ Mk

(5.39)

of invariant subspaces M i , 1 I i I k , of the linear transformation T of Hom,(M, M). Let fi(t) be the characteristic polynomial of the restriction Ti of T t o M i . Then the characteristic polynomialf(t) of Tis the product of the polynomialsfi(t), 1 I i I k. Proof. Let the ensemble { B l , .. . , Bk}be a collection of linearly independent subsets of M where each Bi ,1 5 i 5 k,is a basis of the invariant subspace M i . The union

B=UB,

is a basis for the space M. The matrix of T with respect to the basis B consists

of a series of blocks along the main diagonal, each corresponding to the matrix of one of the restrictions Tiwith respect to the basis B i . It follows that the characteristic polynomial f ( t ) is the product of the polynomials fi(t), 1 4 i I k. Let the decomposition (5.39) be the direct decomposition of M into the characteristic subspaces of the linear transformation T. It follows from the definition of the characteristic subspace Mi corresponding to the eigenvalue p i that

(5.40)

[(Ti - piZM,)"']mi = 0,

m iE M i ,

so that the minimal polynomial of Ti has the form (t - pJal where 1 I p i I a , , 1 Ii I k. The characteristic polynomial of Ti must be of the form (t - pi)', from which it follows by Lemma (5.38) that the characteristic polynomial of Ti is (5.41)

fi(t)

= (t

- pi)ai,

1 I i I k.

This observation implies that the dimension of the characteristic space Mi must equal the exponent ai . We require two additional definitions to describe the situation more fully.

1. Groups and Modules

46

(5.42) DEFINITION. The linear transformation T on the K-space M is nilpotent if and only if there exists an integer n such that T" is the null transformation. (5.43) DEFINITION. The linear transformation T on the r-dimensional K-space M is nilcyclic iff there exists a basis {m., ... , m.} of M such that Isist-I,

(5.44) and (5.45)

Tm,=O.

We refer the reader to standard works on linear algebra for the proof of the following key theorem. (5.46) THEOREM. Let T be a ni I potent linear transformation on the r-dirnensional K-space M. Then M is the direct sum

M

(5.47)

=

M t EB ... EB M,

of invariant subspaces M;, lsi s k , of T such that the restriction T, of T to the invariant subspace 1\1 i is a nilcyc1ic linear transformation. Our previous results show that T; - P;!M, is a nilpotent linear transformation on the characteristic subspace M;. Theorem (5.46) implies that each of the characteristic subspaces M; is a direct sum (5.48) of invariant subs paces Mij, 1 s j s n;, on each of which T; - Pi1M , is nilcyclic. Let {m., ... , m.} be the special basis, which exists by Definition (5.43), of Mij. The behavior of T; and consequently of T on this particular basis is described by the set of equations (5.49)

Tmj=pimj+mZ' ... '

Tm,_t=Pim'_l+m.,

Tm,=pim,.

Consequently, the matrix of T (or its restriction T i ) on the invariant subspace Mij is a txt matrix of the general form

(5.50)

n., =

0

0 0 Pi 0 1 P;

0

0

Pi I

0

0 0 0

0 0 0 Pi

The matrix of Twith respect to a properly chosen basis for the original space M is in quasi-diagonal form, that is, it consists of zeros except for a succession of blocks of the form Bij down the main diagonal. This particular

47

5. Invariants of Linear Transformations

result is known as the Jordan canonicalform of the matrix of T. One obtains a useful special case from these considerations. THEOREM. Let T be a linear transformation on the r-dimensional K-space M whose characteristic polynomial factors into the product of r distinct, linear factors. Then there exists a basis {ml, . . . , m,} of the space M consisting of eigenvectors of T. Proof. The subspaces M i , 1 5 i Ir, in the decomposition (5.35) are all onedimensional. Since T - p i IM,is nilpotent on M i , it follows that every nonzero vector of Mi is an eigenvector of T. Furthermore, the set (5.51)

h, f.. where each m i , 1 the space.

3

m,),

i 5 r, is an element of Mi different from zero, is a basis of

Theorem (5.51) is too restrictive. Given an r-dimensional K-space M, r > 1, there exist many linear transformations T on M with the property that M has a basis of eigenvectors of T even though T itself has repeated eigenvalues. An element T of Hom,(M, M) is called semisimple iff M has a basis of eigenvectors of T. We introduce several new concepts in order to discuss some special cases where the proof that certain linear transformations are semisimple is straightforward.

(5.52) DEFINITION. A mappingfwith domain the set M x M , M a K-space, and range the field K (any field of characteristic zero) is said to be a bilinear form on M if and only if the following conditions hold: 6 ) f(x + Y, z> =f(x, z) + f ( Y , z), (ii) f ( x , Y + z) =f(X, Y) +f(x, z), ( i 4 f (cx, Y) =f(x, CY) = cf (x, Y),

{x, Y, z l = M. {x, Y> z> c M. c E K, {x, Y> c M.

If 0 is the zero vector in M, then f(m, 0) =f(m, 00) = Of(m, 0) = 0,

m E M.

Similarly, f ( 0 , m) vanishes for every element m of M. (5.53) DEFINITION. A bilinear form f on a K-space M is said to be nondegenerate if and only if given any nonzero element x of M there exists a y of M such that f(x, y) is different from zero.

Let M be any K-space. The set Hom,(M, K ) is a vector space according to prior remarks. Recall that this vector space is called the dual space of M and its elements are called linear functionals. The linear functionals on a space M with a nondegenerate bilinear form have a special representation according to the following lemma.

48

1. Groups and Modules

(5.54) LEMMA. Let f be a nondegenerate bilinear form on the r-dimensional K-space M. Let h be any linear functional on M. Then there exists a unique element n of M such that h(m) =f(m, n),

(5.55)

m E M.

Pro06 First, we establish the existence of the required element n. Let h be the null linear functional. Then the zero element of M satisfies (5.55). Otherwise, the kernel of h I S an ( v - 1)-dimensional subspace H of M by Corollary (4.35). Let { m , , .. . , m,- ,} be a basis B' of H and m, an element of M not contained in H. The set B' u {m,} is a basis of M so that every element m of M has the expansion

m = <'ml + ... + {"m,.

(5.56)

Furthermore, the set of equations (5.57)

0 = f ( m , , m) = C f ( m , ,m,)tj,

1 I i I Y - 1,

has a nontrivial solution since the number of variables exceeds the number of equations. Let n' be some nonzero element of M corresponding to a nontrivial solution of the system of Equations (5.57). Iff(m,, n') is zero, then f ( m , n') vanishes for every m of M, which is an impossibility. We define the element n by

n = [h(mr>if(mr n'>ln'. 9

It follows that h(m,) equalsf(m,, n). From (5.56), we have that (5.58) h(m)

= h(t'm,

+ . . . + t'm,)

= < ' f ( r n , , n)

= ('h(m,)

+ ... -t- t'f(m,,

n)

+ . . . + t'h(m,) + ... + t'm,,

=f(tlml

n) =f(m, n)

for every m E M, as was to be shown. To show uniqueness, let ii be an element such that /7(m) equalsf(m, ii) for every m of M. Consequently, f(m, n - ii) = f ( m , ii) -f(m, n) = 0,

which Implies that ii

-

m E M,

n is the zero vector. Thus n is unique.

A useful correspondence can be set up between any basis {inl, . . . , m,} of an r-dimensional K-space M and a basis of its dual space M*. Let the set {m,, . . . , m,} be any basis B of M. By means of Theorem (4.9), define B* to be the subset {m,*, . . . , m,*} of M* determined by

(5.59)

m,*(m,) = 6,*,

1 5 i I r.

We leave to the reader the argument that M* has B* for a basis called the dual basis of B.

49

5. Invariants of Linear Transformations

Let f be an element of M*, that is, an element of Hom,(M, K ) and let T be an element of Hom,(M, M). The composition f 0 T is also an element of Hom,(M, K ) . Let T* be the mapping with domain and range M* which makes each element f of M* correspond tofo T, that is T * ( f )=f

(5.60)

0

f

T,

E

M*.

The mapping T* is a linear transformation on M*, that is, an element of Hom,(M*, M*). For, if {f, g } c MY,then (5.61) [T*(f + s)l(m) = [ ( f +s) Tl(m) O

= [ f + sl(T(m>)= f (T(m))+ g(T(m)) = [ f 0 TI(@ =

[T*f

+ [9

O

+ T*g](m)

TI(@

=

[T*fl(m)+ [T*gI(m)

for every m belonging to M which implies that (5.62)

T*(f+ g)

=

T*f+ T*g.

In a similar manner. one shows that (5.63) (5.64)

T*(af) = aT*f, DEFIN~TION.The

a E K,

f

E

M*.

element T* of Hom,(M*, M*), defined by T*f=foT,

fEM*,

is called the adjoint of T. A very important class of spaces have associated with them a special kind of form which is almost a bilinear form. (5.65) DEFINITION. A mappingfwith domain the set M x M, M a K-space, and range the complex numbers K is said to be a positive definite, hermitian symmetric form on M if and only if the following conditions hold for (x, y, z> c M, CI E K : (i) f(x, x) 2 0, f(x, x) = 0 iff x = 0, (ii) f ( x , Y) = f ( Y , x), ( W f (ax, Y) =f. (x,Y), (iv) f ( x , Y + z) = f ( x , Y) + f ( % 2).

The mapping f of this definition is commonly called an innerproduct on M and the image f ( x , y) is usually denoted merely by (x,y) when there is no reason for confusion. A complex vector space W with an inner product f is frequently called a n inner product space.

50

I . Groups and Modules

(5.66) EXAMPLE. There are many ways of introducing an inner product into the canonical space C, of r-tuplets. One possibility is to define the inner product of the vectors

x

= (a,,

. .., a,)

and

by means of the formula

(5.67)

(x,y)

= a16,

+

y

*

..

=

(bl, .. . , b,)

+ arb,.

The verification that this definition satisfies all the required properties is left to the reader. The existence of an inner product on a complex vector space M permits the introduction of a number of useful concepts related to those of length and angle in a Euclidean space. The length or norm of a vector m of M is defined to be the nonnegative square root of (m, m) and written IlmIl. Two vectors m and m’ are said to be orthogonal if and only if (m, m’) vanishes. An indication of the relationship between these ideas and the metric properties of an inner product space is given in the next two results. (5.68) THEOREM (Schwarz’s inequality). Let m and m’ be vectors of the inner product space M. Then (5.69)

I(m, m?I

llmll llm’ll.

Proof. If m’ is the zero vector, then m’ equals Om’ so that I(m, m’)I

=

I(m, 0m’)I

= Ol(m,

m’)I

=0 5

llmll llm’ll.

Now suppose that m’ is different from zero. For any number c, note that (5.70) 0I (m - cm’, m - cm’) = (m, m) - c(m, m’) - c(m’, m)

+ I c I2(m‘, m’).

The substitution c = (m, m’)/(m’, m’) gives rise to the particular result that (5.71) 0 5 (m, m) - I (m, m’) 12/(m’,m’) - I (m’, m) I2/(m’, m’) + I (m, m’) 12/(m’, m’). This inequality implies that I(m, m y 2 5 llmllZ llm’/I2

or

I(m, m?I 5 llmll llm’ll.

(5.72) DEFINITION. A subset {m,, . . . , mk}of the r-dimensional inner product space M is called orthogonal if the inner product (m,, m,) vanishes for i different from J . It is called normal if llmLll= 1 for 1 < i 5 k. It is called orthonormal if and only I f (5.73)

(m,, mj)

= dji,

1

< i, j < k .

51

5. Invariants of Linear Transformations

A unitary basis {ml, . . . , m,} of the r-dimensional inner (5.74) DEFINITION. product space M is a basis which is an orthonormal set.

It is easy to see that every orthonormal set is linearly independent. In order to prove that every r-dimensional inner product space M has a unitary base, it is convenient to note the following lemma.

(5.75) LEMMA. Let {ml, . . . , mk} be an orthonormal set N of the r-dimensional inner product space M. Let m be any element of M. Then the vector m - (m, ml)ml - ... - (m, mklmk is orthogonal to every element of N. Consequently, it is orthogonal to the subspace generated by N. We leave the proof to the reader.

(5.76) THEOREM(Gram-Schmidt). Let {m,, ..., m,} be a basis of the r-dimensional inner product space M. Then there exists an orthonormal basis {nl, . . . , n,} such that the subspace generated by {ml, . . . , mk} coincides with that generated by {n,, . . . , nk} for 1 5 k < r. Proof. The proof is by induction on r. When M is one-dimensional, let n, be the vector ml/llml 11. Assume the result for 1 5 r 5 k and let {ml, ..., mk+l} be a basis of the ( k 1)-dimensional inner product space M. By the induction hypothesis, there exists an orthonormal basis {nl, . . . , nk} of the subspace M' generated by {m,, . . ., mk} with the properties required by the theorem. The vector

+

nk+l'

= mk+l

- (mk+l, nl>nl

- ". - (mk+l, nk)nk

is orthogonal to the set {nl, . . . , nk} and to the subspace M'. Since M' is a proper subspace of M, nk+l' is not the zero vector. We define the vector nk+, to be the vector nk+l'/llnk+l'lland note that the set {n,, . .. , nk+,} is a unitary basis of M with the required properties. This completes the induction and the proof. Letfbe a bilinear form on the K-space M, with K a field of characteristic zero. Let the symbol Hom,(M, M) stand for the set of all K-homomorphisms T of M into itself such that

(5.77)

f(Tm, Tm') =f(m, m'),

{m, m'} c M.

Clearly the identity element ZM of Hom,(M, M) is in this set so that it is not empty. Let T and T' be elements of Hom,(M, M). Then note that

(5.78)

f ( [ T o T'lm, [To T'lm') =f(T(T'(m)),T(T'(m'))) =f(T'm, T'm') =f(m, m'),

52

1. Groups and Modules

which implies that T T' is also an element of Homf(M, M). Consequently, Hom,(M, M) is a multiplicative semigroup ofihe ring Hom,(M, M). Suppose that f i s nondegenerate; then T i n Homf(M, M) and Tm equal to zero imply that 0

(5.79)

f ( m , m') =f(Tm, Tm') = f ( O , Tm') = 0

for every m' of M. It follows from (5.79) and the nondegeneracy of fthat m is zero, that is, the kernel of T is (0). Therefore T is a monomorphism. When M is finite-dimensional, but not in general, T is also an epimorphism and consequently an isomorphism. Thus in the finite-dimensional case, T has an inverse T' which belongs to Hom,(M, M). Note that (5.80) f(m, m') =f((n")m,

(TT')m') =f(T(T'm), T(T'm')) =f(T'm, T'm')

for (m, m') c M, which implies that the inverse T' of an element T of Homf(M, M) is also in Hom,(M, M). Thus Homf(M, M) is a group whenever f i s a nondegenerate bilinear form on a finite-dimensional K-space. (5.81) EXAMPLE. Let M denote the set of all four-tuples of real numbers and use the definitions of Example (3.41) to define an R-space, that is, a vector space over the field of real numbers. This space M together with the bilinear form f defined for the elements

m

=

(a,, a,, a,, a,),

ai E R ,

by (5.82)

f(m, m') = a,b,

and

m' = ( b l ,b,, b,, b,),

6, E R,

+ a, b, + a , b, - a46,

is sometimes referred to as Minkowski space. It is easy to see that f is a nondegenerate bilinear form. The group Homf(M, M) is called the homogeneous Lorentz group in this case. The previous discussion can be repeated with almost no change for the case of a positive definite, hermitian symmetric form f o n a finite-dimensional inner product space. The group Homf(M, M) is called the r-dimensional unitary group when M is an r-dimensional space with inner product f. The invertible elements of Homf(M, M) are the elements of the unitary group in the case of an infinite-dimensional Hilbert space M. (5.83) DEFINITION. An element {au"},1 I u, v I r, of K, is called a unitary matrix if and only if the following equations are satisfied by its components:

(5.84) and (5.85)

C a/iivw= S,",

1 I u, w I r,

C a/iiwu= S,",

1 5 v , w I r,

5. Invariants of Linear Transformations

53

It is easy to show that (5.84) and (5.85) are equivalent so that either is a suitable definition of unitary matrix. The proofs of the following two theorems are left to the reader. (5.86) THEOREM. The matrix of a linear transformation Twith respect to the unitary basis {m,, .. . ,m,} of the inner product space M is unitary if and only if T i s a unitary transformation on M. (5.87) THEOREM. Every unitary matrix is the matrix of a unitary transformation T with respect to the basis B if and only if B is an orthogonal set of vectors, all of which have the same length. The proof of the following lemma is essentially the same as that of Lemma (5.54) for finite-dimensional spaces. Let h be a (continuous) linear functional on the Hilbert space (5.88) LEMMA. M. Then there exists a unique element n of M such that

h(m) = (m, n),

(5.89)

m

E

M.

Let T E Hom,(M, M) and let ( , ) denote a nondegenerate bilinear form on M.The map h from M to K defined by

h(m) = (Tm,n ) for m E M and fixed n E M is a linear functional on M. Hence there exists a unique n* E M such that h(m) = (m, n*) by Lemma (5.54). Given any pair {u, v} c M, one has (5.90)

for every m, which implies that (u + v)* that (cn)* = cn*. (5.91)

+

+

(m, (u + v)*) = (Tm, u v) = (Tm, u) (Tm, v) = (m,u*) + (m,v*) = (m,u* + v*)

DEFINITION.

= u*

+ v*.

Similarly, one can show

The mapping T* defined by T*n=n*,

nEM,

is an element of Hom,(M, M) which is called the adjoint of T with respect to the nondegenerate form ( , ). Similarly, if ( , ) is an inner product on M , the corresponding T* is called the Hilbert space adjoint of T. One should avoid the misconception that the adjoint as defined here coincides with that of Definition (5.64) although it is customary to use the same notation in either case. The previous adjoint is an element of Hom,(M*, M*); that defined in (5.91) is an element of Hom,(M, M).

54

1. Groups and Modules

(5.92) DEFINITION. A n element Tof Hom,(M, M ) where M is equipped with a nondegenerate form ( , ) is self-adjoint with respect to ( , ) iff (5.93)

(Tm, m’) = (m, Tm’),

{m, m‘} c M.

(5.94) THEOREM. Let U be a linear transformation on the r-dimensional inner product space M. Then U is a unitary transformation on M if and only if U-’ coincides with U * , the Hilbert space adjoint of U. Proof. Suppose that U* is the inverse of U. Then

(m, m’) = (m, U*Um’) = (Um, Um),

{m, m’} c M.

Conversely, suppose that U is a unitary transformation on M. Then 0 = (Um, Um’) - (m, m’) = (m, U*Um’) - (m, m’) = (m, U*Um’ - m’), {m, m’} c M. From which it follows that U*Um‘ = m’.

m’ E M.

Therefore U * is a left inverse of U and, by previous discussions, the inverse of u. (5.95) THEOREM. Let T be a linear transformation on a finite-dimensional K-space M such that every proper invariant subspace N of T has a complementary invariant subspace N’, that is, M is the direct sum N @ N’. Then Tis a semisimple linear transformation on M . Proof The proof is by induction on the dimension of M. The result is true for one-dimensional spaces. Assume its validity for spaces of dimension r with 1 2 r ~ n Let . T be a linear transformation with the properties of the theorem on an ( n + 1)-dimensional space. Let p1 be a root of the characteristic polynomial f ( t ) of T. Let (m) be the invariant subspace generated by an eigenvector corresponding to the root p,. By hypothesis, there exists a complementary invariant subspace W such that M = (m) @ W. Let : M + M/(m) denote the natural map of M onto its factor space. Note T induces a linear transformation T’ on M/(m) defined by T’[x (m)] = Tx + (m). Let U’ be a proper invariant subspace of T‘ and let U = v-’(U’). The space U is invariant under T so that there exists a complementary invariant subspace V with M = U @ V. It follows that M’ = M/(m) is the direct sum of the invariant subspaces U‘ and V’ of T‘. Thus T‘ is a linear transformation on an n-dimensional space which satisfies the hypothesis of the theorem. Consequently, M’ has a basis {m,’, . . . , m,,’} consisting of eigenvectors of T‘ 11

+

55

Problems

by the induction hypothesis. Select wi E W so that mi’ = wi + (m), 1 I i 2 n. Hence

T’(wi

+ (m)) = Twi + (m) = lli wi + (m)

where Ai is the eigenvalue of T‘ corresponding to an eigenvector mi’. Thus Tw, - Ai wi E (m) n W = (0),that is, TWi

= lli wi

.

Therefore, the set B = {m, wl, . . . , wn} is a basis of M consisting of eigenvectors of T. This establishes the validity of the theorem for n + 1 and obtains the general result. For convenience, we use the word form to denote a map ( ,) from M x M to M which satisfies conditions (i) and (ii) of Definition (5.52) and either (iii) of (5.52) or (iii)’ (cx, y)

= c(x,

y)

and

(x, cy) = T(x, y).

(5.96) COROLLARY. Let M be a finite-dimensional vector space over a field K of characteristic zero. Let ( , ) be a form on M such that (x,x) = 0 implies x = 0 and let T be a linear transformation on A4 such that (Tm, m’) = (m,Tm’) for {m, m’} c M. Then T is a semisimple linear transformation. Proof. One shows that T has the property of Theorem (5.95). PROBLEMS

Let f : D -+ R be a function with domain D and range R. Injection : The map f is an injection if, for {d, d’} c D, f ( d ) =f (d’) implies d = d‘. Surjection: The map f is a surjection if, for r E R , there exists d E D, such that f ( d ) = r. Bljection: The map f is a bijection if it is both an injection and a surjection. Let g : R -+ W be a function with domain R and range W. Composition: The function (g o f ) : D-,Wis defined ford E D by (g f ) ( d )= g ( f ( d ) ) .It is called the composition of g and.6 I-Function: For any nonempty set S , the map 1, : S -+ S is defined by I,(s) = s for s E S. Inverse: Let f : D -+ R and g : R -+ D be two functions. The function f is a left-inverse of g ( f is a right-inverse of g ) if 0

f o g = 1, (9

o f = ID).

1. Groups and Modules

56

1. Let f : A + B, g:B + C, and h : C + D be functions with the indicated domains and ranges. Show that the binary operation of composition is associative, that is, (h g) 0 f = h ( g f ) . 0

0

0

2. Show that the composition of two injections is an injection.

3. Show that the composition of two surjections is a surjection. 4. Show that f :D such that g 0 f = 1., 5. Show that f :D g': R + D such that f

+R

is an injection implies that f has a left-inverse g

R is a surjection implies that f has a right inverse g' = I,.

-+ 0

6. Show that f has a left-inverse g and a right-inverse g' only if g = g'. The unique map g determined in this case is called the inverse off. Note that 4, 5 , and 6 imply that a bijection f :D + R has a unique inverse g: R + D such that gof=l,andfog=l,. Let A = { 1, . . . , n) denote the set of the first n positive integers. Let S, = {f:A + A :f is a bijection}. The first six problems show that S, is a group under the operation of composition. It is called the symmetric group of dcJgreen.

7. Let A

= (1,2,

3}, and write out the elements of S, .

Look up the words relation and equivalence relation in a modern algebra book if they are unfamiliar. 8. Let H be a subgroup of the group G. Two elements a, b of G are said t o be in the relation R, that is, aRb iff a-'b E H . (a) Show that R is an equivalence relation on the set G. (b) Show that the equivalence classes of R consist of the left-cosets of H in G.

9. Let G be a finite group containing H as a proper subgroup. Show that the distinct left-cosets of H in G give a decomposition of G into disjoint subsets. 10. Let a belong to the group G. The map aL : G -+ G with domain and range G whose value at x E G is given by aL(x)= ax is called the left-multiplication dejned by a. Show that aL is a bijection on G.

11. Determine the right and left cosets of the subgroup H = { I , 2, 3) of the Group ( I ) . See page 57. Do the same for subgroup K= {I, 5, 7, 11). You should find that the right and left cosets coincide for H but not for K . Work out the right cosets for the subgroup N = {1,2, 3, 4) of Group (2).

Problems

57 CAYLEYTABLE OF GROUP(1)

1 2 3 4 5 6 7 8 9101112 2 3 1 5 6 4 9 7 8121011 3 1 2 6 4 5 8 9 7111210 4 5 6 2 3 1121011 7 8 9 5 6 4 3 1 2111210 9 7 8 6 4 5 1 2 3101112 8 9 7 7 8 9101112 1 2 3 4 5 6 8 9 7111210 3 1 2 6 4 5 9 7 8121011 2 3 1 5 6 4 101112 8 9 7 6 4 5 1 2 3 111210 9 7 8 5 6 4 3 1 2 121011 7 8 9 4 5 6 2 3 1

IRREDUCIBLE REPRESENTATIONS OF GROUP(I)" T(1) T(2) T(3) T(4) T(5)ll 12 21 22 T(6)ll 12 21 22

1 1 1 1 1 1 0 0

1

1 0 0 1

3 4 5 6 7 8 9101112 1 1 1 1 1 1 1 1 1 1 1 - 1 -1 -1 1 1 1 - 1 -1 -1 1 1 1 1 - 1 -1 - 1 -1 -1 -1 1 - 1 -1 -1 -1 -1 -1 1 1 1 p4p2 p5 p3 p 0 0 0 0 0 0 0 0 0 0 0 1 p z p 4 p p3p5 0 0 0 0 0 1 p 4 p z p 5 p 3 p pzp4 p p3 p5 0 0 0 0 0 0 pzp4 p4 1 pz 0 0 0 0 0 0 0 0 0 0 0 1 p 4 p Z p Z l p4 0 0 0 0 0 1 p z p 4 p 4 1 pz p 4 p z pz 1 p4 0 0 0 0 0 0 2 1 1 1 1

a The symbol p in the above table denotes the complex number cos(60) i sin(60). The symbols T(6)-denote the components of matrices. More conventionally, the data of this table would be presented in the form:

+

I/

T(6)11 T(6)12 T(6)21 T(6)22

/I

'

A similar notation is used throughout the book for listing irreducible representations. For example, the values T(6)(4) and T(6)(7) are

The table of irreducible representations of Group (1) supplies six homomorphisms of Group (1) either into the group of nonzero complex numbers (under multiplication) or into the group GL,(K) of invertible 2 x 2 complex matrices.

1. Groups and Modules

58 CAYLEY TABLEOF GROUP(2) ~

1 2 3 4 5 6 7 8 910111213141516 2 1 4 3 6 5 8 710 9121114131615 3 4 1 2 7 8 5 61112 91015161314 4 3 2 1 8 7 6 5121110 916151413 5 6 7 8 2 1 4 314131615 9101112 6 5 8 7 1 2 3 41314151610 91211 7 8 5 6 4 3 2 1161514131112 910 8 7 6 5 3 4 1215161314121110 9 9 1 0 1 1 1 2 1 3 141516 2 1 4 3 6 5 8 7 10 9 1 2 1 1 1 4 1 3 1 6 1 5 1 2 3 4 5 6 7 8 1112 9 1 0 1 5 1 6 1 3 1 4 4 3 2 1 8 7 6 5 121110 9 1 6 1 5 1 4 1 3 3 4 1 2 7 8 5 6 1314151610 91211 5 6 7 8 2 1 4 3 14131615 9 1 0 1 1 1 2 6 5 8 7 1 2 3 4 15161314121110 9 7 8 5 6 4 3 2 1 161514131112 910 8 7 6 5 3 4 1 2 ~~

~~~~

CAYLEY TABLEOF GROUP(3) ~~

12345678 21436587 34217865 43128756 56871243 65782134 78563412 87654321

12. The results of Problem 11 and the text suggest that the left cosets of a subgroup H of a finite group G determine a partition of G into disjoint subsets, each of which contains the same number of elements as H . This is true and is the basis of Theorem (1.2) (Lagrange’s rheorem) that the number of elements of a subgroup H of a finite group G is a divisor of the number of elements of G. Using the ideas of Problems 9 and 10, prove Lagrange’s theorem. 13. Show that the product of the left cosets 4H = (4, 5, 61 and 7H = (7, 8, 9} of the subgroup H = (1.2, 3) of Group ( I ) is the left coset 10H = { 10, 1 1, 12). Determine the Cayley table G / H . 14. Find two left cosets of the subgroup K whose product is not a left coset of K .

=(I,

5, 7, 11) in Group (1)

59

Problems

15. Verify that Group (1) is the product H K of its subgroups H and K

= { 1,7},

= {I, 2, 3,4,

5, 6)

but that it is not the internal direct product.

16. Let g belong to the finite group G. Let ig: G -+ G, be the inner automorphism determined by g, that is, ig(x)=gxg-' for x E G. Show that ig is a permutation (bijection) on G. Prove that a subgroup H of G is normal if and only if H is mapped onto itself by every such inner automorphism. 17. Find the kernels of each of the six homomorphisms of Group (1). mentioned above. Verify that each of them is a normal subgroup of Group (1). 18. Prove that the element in the (i,j)th position of a Cayley table is a conjugate of the element in the ( j , i)th position. For example, the element in position ( 5 , 14) of Group (2) is 10 while that in position (14, 5) is 9 so that 9 is a conjugate of 10. Work out the classes K , of conjugate elements of Group (2). Your answer should be Kl = (l), K , = {2}, K3 = {3}, K4 = {4}, K , = ( 5 , 6}, K6 = (7, S}, K , = (9, lo}, K , = {11, 12}, K9 = (13, 14}, and K,, = (15, 16). 19. Prove that the product H K = {hk:h E H , k E K } of two subgroups H and K of a group G is a subgroup of G iff H K = K H . 20. Letf: G + G' be a homomorphism from the group G onto the group G'. Show (a) The imagef(H) of a subgroup H of G is a subgroup H' of G'. (b) Show that the counterimage f -'(IT)= { x E G :f(x) E H ' ) of a subgroup H' of G' is a subgroup H of G. (c) Show that the image of a normal subgroup H of G is a normal subgroup H' of G'. (d) Show that the counterimage f -'(H') of a normal subgroup H' of G' is a normal subgroup H of G. (e) Show that if H , and H , are subgroups of G containing the kernel N off, thenf(H,) = f ( H 2 ) implies H I = H , . (f) Conclude from the above that there exists a one-to-one correspondence between the subgroups of G containing the kernel N off and the subgroups of G'. 21. Let R denote the set of all continuous, real-valued functions defined on the interval 0 2 t < 1. Given ( 9 ,h} c R, let g + h be defined by (g + h)(f) = g(t) + h(t) and gh by (gh)(t)= g(t)h(t).Verify that R is a commutative ring for these assumptions. 22. Let to be a real number with 0 I to I 1. For the ring R of Problem 21, let J denote the subset consisting of all functions vanishing at t o . Prove J a maximal ideal.

60

1. Groups and Modules

23. The set R 2 of all real 2 x 2 matrices is a ring under the standard matrix operations. Show that R 2 contains no proper two-sided ideals. Show the set of all n x n complex matrices is also a ring containing no nontrivial two-sided ideals. 24.

Consider the sets R, T, and L of all 4 x 4 real matrices of the form all

a,2

a21

an

0 0

0 0

0 0

0 0

a33

a34

a43

a44

0 0 0 0

R

0 0 0 0 0 a33 0 a43

0 0 a34 a44

T

all a21

0 0

0 0 0 0

0 0

0 0

a33

a34

a43

a44

L

Show that R is a ring under the usual matrix operations, that T is a maximal two-sided ideal ofR, and that L is a maximal left ideal. Note that the maximal two-sided ideal T is properly contained in the maximal left ideal L. 25. The set J of all diagonal 2 x 2 real matrices constitutes a subring of the set R of all 2 x 2 real matrices. Show that J is not an ideal. Nevertheless, the set J is a subgroup of R regarded only as an abelian group under addition. Thus the factor group R/J can be defined in the usual manner. Show this abelian factor group can not be made into a ring by the" obvious definition" of product (x + J)(y + J) = xy + J. 26.

The set of 2 x 2 real matrices

are the elements of a cyclic group generated by

under matrix multiplication. Consider the set R consisting of all 2 x 2 real matrices which are real linear combinations of the form r o ! + rIA + r2A2 + r 3 A 3 . Show that R is a ring. 27. The complete analog of Problem 20 is valid for rings. Formulate and prove the parallel results. Hint: One uses the results for groups applied to the additive structure of the ring R. Then there remains only to verify that multiplication goes right. 28. Let U be a proper subspace of the vector space V. Suppose that B = {u., ... , urn} is a basis of U and that C = {VI' ... , vn } is a basis of V. Prove there exists k = n - m vectors VI" ... , vk ' in the base of C such that the set D = {u,, ... , urn' VI', .. ·, vk ' } is a basis ofV.

Problems

61

29. Let each vector x belonging to V be specified by its coordinates with respect to the basis D of Problem 28. It is clear that x E U if and only if ti= 0, m < i (we continue the enumeration of D by u,+~ = v,’, u , + ~= v2’, etc.). Let E be any basis of V and let {ql, . . . , q n } denote the coordinates of y with respect to E. Show that it takes k linear equations in the coordinates {ql, . . . , qn} to specify that y belongs to U. 30. Let M and N be abelian groups. Let F(M, N) denote the set of all functions with domain M and range N. Complete the proof that F(M, N) is an abelian group under the operation defined in Section 3. Let A denote either a ring or an algebra in the following problems.

31. Let M and N be left A-modules. An elementfof Hom,(M, N) is a group homomorphism of the abelian group M into the abelian group N such that f(am) = af(m) for a E A, m E M. Show that Hom,(M, N) is a subgroup of the group F(M, N) defined in Problem 30. 32. A can be regarded as an abelian group under its additive structure. There exists a natural scalar product of A by itself under which the pair (b, a) goes into ba. Show that A is a left A-module over A with respect to this scalar multiplication. 33. When A is commutative, show that the scalar product defined for a E A, f~ Hom,(M, N) by (af)(m) = a(f(m)) makes Hom,(M, N) into a left A-module.

34. Given a E A, there exists a map a R , called the righi r n ~ ~ t i p ~ ~ ~ a r ~ ~ i i determined by a, defined by a,(b) = ba for b E A. Show that aR belongs to Hom,(A, A) when A is regarded as a left A-module over itself. 35. Suppose that A is not commutative, that is, there exists {a, b} c A with ab # ba. Show that Hom,(A, A) is not a left A-module under the multiplication defined in Problem 33. 36. Use mathematical induction to establish the computational rules: (3.49), (3.50), (3.51), (3.52), (3.53), and (3.54). 37. Complete the proof that the map T defined in Example (4.6) is a linear transformation.

38. Theorem (3.1 1) determines the nature of finitely generated abelian groups: One fundamental type M consists of the set {(q,. . . , nk): n iE Z } of k-tuplets of integers with the sum defined by coordinatewise addition. These groups have many properties in common with k-dimensional vector spaces. In particular, if mi denotes the element of M with 1 in the ith coordinate and 0

62

1. Groups and Modules

.

elsewhere. then {m,, . . . . m,} is a basis B of M. If {ml’, .. . mk’)isany set of k elements of the abelian group M’, prove there exists a unique homomorphism T of M into M’ whose values on the basis B are determined by T ( m i )= mi’, I
39. Other kinds M of finitely generated abelian groups are those consisting of k-tuplets (nl. . . . , nk), where n, E Z r r , the group of integers modulo ri. Again the sum is defined by coordinatewise addition. The set of m i with 1 E ZrLin the ith position and zero elsewhere determines a particularly useful set of generators. Given any set {m,‘, . . , , mk’)of elements in an abelian group M’. prove there exists a unique homomorphism T of M into M’ defined by T(m,) = mi’, if the order of mi’ is a divisor of ri. 40.

State and prove the analog of Problem 20 for vector spaces.

41. Let C be a finite group and S , the group of all permutations on G. It is a famous theorem of Cayley’s that there exists a monomorphismf: G+SG defined byf(cr) = crL, where uL is the left multiplication defined in Problem 10. Prove Cayley’s theorem.

A variation of Cayley’s theorem is of importance in representation theory. Let G be a finite group { g l , . . . ,g,,}. Let V be the n-dimensional vector space whose elements consist of all formal linear combinations

{k,g,

+ ... + k,g,: kj€ K }

of elements of G. The vector space operations are coordinatewise sum and scalar product. There exists a monomorphism T : C -+ Hom,(V, V) such that T(g ) is the linear transformation defined by T ( g ) g , = ggi , g E G . This is a satisfactory definition of T ( g ) by Example (4.6) and Problem 37. Show that the correspondence T defined above is a monomorphism of G into CL(V), the general linear group on V. 42.

43. Apply the considerations of Problems 41 and 42 to the group H , , whose Cayley table is presented in (1.4). Determine the matrices of the linear transformations T(5)and T(9). The most familiar n-dimensional real vector space is the set V of all mtuples {(xl.. . . , q,):xiE R}. This set V can be considered as a subset of the n-dimensional complex space U consisting of all n-tuples

44.

{(z,, . . . , 2”): zi E K ) .

63

Problems

The subset V is not a subspace of U. Why? However, the set U can be regarded as a 2n-dimensional real space W with basis consisting of all vectors of the form (1, 0, ..., 0), (i, 0, ..., 0),

..., (O,O, ..., l), (O,O, ..., i).

Try to formulate these facts precisely. Show that V is a subspace of W. Make a precise statement about complex and real n x n matrices analogous to that of Problem 44 for vector spaces. 45.

Chapter 2

The Representution The09 of Finite Groaps

A fundamental tool in abstract algebra is the analysis of an abstract algebraic object A by means of a homomorphism h of A into a more concrete algebraic object B . The term representation is applied in the case where h is a homomorphism of A into the algebra of all linear transformations on a vector space V. The theory of group representations concerns itself with the classification of the homomorphisms of an abstract group G into groups of linear transformations or matrices. This chapter develops the representation theory of finite groups over finite-dimensional vector spaces. The field K of the vector spaces is frequently (not always) taken to be the complex numbers. The mathematical treatment is at a level between that of representation theory for physicists as presented by Wigner (1959) and that of representation theory for mathematical specialists such as given by Curtis and Reiner (1962). Since it is hoped that this book will serve as a point of departure for rhose who plan to read Curtis and Reiner, an effort has been made to conform to the definitions and terminology introduced by these authors. However, the basic analysis is done in terms of the group aIgebra of a finite group rather than in terms of a semisimple algebra in order to present the material at a somewhat lower level of mathematical sophistication. In the first section of this chapter we introduce the basic definitions of the theory o f group representations and give examples to illustrate them. The representation theory of finite abelian groups is worked out completely. It is 64

1. Basic Concepts and Defnitions

65

shown that every finite-dimensional representation of a finite group over the field K of complex numbers is completely reducible. In the second section we introduce the concept of the group algebra KG of a finite group G over the complex numbers K along with examples and fundamental definitions. The decomposition of the group algebra KG into the direct sum of minimal ideals is obtained. In the third section we develop the concepts and terminology of the theory of semisimple algebras in terms of the group algebra KG. A more detailed examination is made of the decomposition of the group algebra KG into the direct sums of minimal left- or minimal two-sided ideals. Section 4 contains a full account of the Pierce decomposition of a simple algebra in terms of a minimal two-sided ideal in KG. The number of classes of equivalent irreducible representations of a finite group G is shown to be equal to the number of distinct classes of conjugate elements in G. In the fifth and last section we define the character of a representation of the group G and establish the theorem that two representations of G are equivalent if and only if they have the same characters. The more important facts are established about character tables and illustrations provided in the case of certain groups. We summarize some of our notational conventions for the convenience of the reader. Most of the groups to be discussed are finite and commonly denoted by G and G'. Exceptions to this occur primarily in the cases of the full linear group GL(V) o f all invertible linear transformations of the vector space V and the full matrix group GL,(K) of invertible n x n matrices over the field K . Elements of groups are frequently denoted by either g and g' but many exceptions occur. Subgroups are sometimes denoted by H and K so that the symbol K is used both for subgroups and for fields. Vector spaces are denoted by U, V, and W. Elements of these spaces by boldfaced letters such as x, y, and z. Elements of fields are frequently denoted by lower case Greek letters such as a, j?, and y. Homomorphisms are denoted by I? and f as well as by T i n the case of linear representations. Linear transformations are also denoted by T . 1. BASIC CONCEPTS AND DEFINITIONS IN THE REPRESENTATION THEORY OF FINITE GROUPS

Let 6denote a subset of Hom,(V, V), the set of all endomorphisms of a finite-dimensional vector space over K. To say that 6 is reducible means that V contains a nontrivial subspace U which is a n invariant subspace of every linear transformation T belonging to 6. 6 is irreducible means that no nontrivial subspace U of V is an invariant subspace of each element T of 6. 6 is decomposable means there exist nontrivial, invariant subspaces U and W of V such that V is the direct sum U 0 V. 6is indecomposable means that

66

2. The Representation Theory of Finite Groups

there exists no nontrivial, invariant subspace U with a complementary, nontrivial invariant subspace W. G is completely reducible means that whenever U is a nontrivial invariant subspace of each T in 6, then there exists a second, nontrivial invariant subspace W such that V is the direct sum U 0W. The preceding definitions are meaningful for any family 6 of linear transformations belonging t o Hom,(V, V). They are used mainly in the case of a homomorphism T of a finite group G into GL(V). Here 6 is taken to be the Im T , the set of all elements of GL(V) of the form T(g) with g in G. Note that these definitions apply equally well t o the case of a family 6 of matrices contained in GL,(K). Here the vector space V is taken to be the coordinate space K". We now turn t o the most important definition in this chapter. Let G be a finite group and let V be a vector space over the complex numbers K. (1.1) DEFINITION. A linear representation of G with representation space V is a homomorphism T with domain the group G and range the full linear group GL(V).Two representations T and T' of G with representation spaces V and V' are said t o be equivalent if there exists an isomorphism Iz of V onto V' such that

T'(g)/z = /zT(g),

g

E

G.

The dimension (V : K ) of V over the field K of complex numbers is called the degree of T and denoted by deg T. The Im T is a subgroup G' of GL(V) which is contained in Hom,(V, V). The representation T is said t o be reducible, irreducible, decomposable, indecomposable, o r corripletely reducible if and only if G' is a reducible, irreducible, decomposable, indecomposable, o r completely reducible set of linear transformations in Hom,(V, V).

(1.2) DEFINITION. A matrix representation of G of degree n is a homomorphism T of the group G into the full matrix group GL,(K). The matrix representations T and T' are equizalent if they both have the same degree n and there exists a fixed matrix A in GL,(K) such that T'(g) = AT(g)A-',

LJ E

G.

The concepts of reducible. irreducible, decomposable, indecomposable, and completely reducible are immediately applicable to matrix representations as well as linear representations. The concept of equivalence for representations is an equivalence relation 91 on the set 5 of all linear o r all matrix representations. This statement means that (1) T i s equivalent to T for

67

1. Basic Concepts and Definitions

every representation T ; (2) T is equivalent t o T' if and only if T'is equivalent to T ; and (3) T is equivalent to T' and T' is equivalent to T" imply that T is equivalent to T". Thereby, the classification problem for representations is largely reduced to the characterization of all distinct equivalence classes of representations of a given group rather than to the determination of all of its individual representations. It evolves that it is sufficient to characterize the distinct classes of equivalent irreducible complex representations of which there are only a finite number in the case of a finite group. Therefore, our principal efforts in the following pages are to determine methods of finding all of the distinct classes of equivalent complex irreducible representations of a given finite group G. Each representation T of a finite group G is a homomorphism of G into G', a subgroup of the full linear group GL(V)over the representation space V of T. The adjectives reducible, irreducible, decomposable, indecomposable, and completely reducibte are extended from the representation T to the representation space V so that one speaks, for example, of an irreducible representation space V meaning, of course, the representation space of an irreducible representation T . The following facts are true of a representation T, either linear or matrix, merely because T is a homomorphism of a group G into a group G':

and

T(1)

=I,

T w o results about linear transformations and homomorphisms are particularly useful in giving specific examples of representations of a finite group G. First, there exists a unique linear transformation T in Horn#, V) for any specific choice of the set {Tv,, . . . , Tv,} of images of a basis {v,, . . . v.} of V. Second, it is sufficient to show that a mapping T with domain a group G and range Hom,(V, V) has the following properties: (1.3)

(i) T(gg')v, = T(g)T(g')v, for g,g' in G, and vi any element of { V I , . . ., v,>, (ii) T ( l )is the identity 1, in Hom,(V, V),

in order to show that T is a representation of G. We give several examples of representations of groups before continuing with the general discussion.

68

2. The Representation Theory of Finite Groups

For our first example, let G denote the cyclic group C, of (1.4) EXAMPLE. order four whose Cayley table is as shown in Table (1.4').

(I .4')

CAYLEY TABLEOF C, 1234 11234 22341 33412 44123

Let V denote a four-dimensional vector space over K with a basis consisting of the set {vl, . . . , v4). For each i, 1 < i < 4, define T(i)by the formula T(i)vj= v i +j

,

where the subscript i +.j is to be computed from the Cayley table of C4.The fact that the correspondence i 4 T(i)is a representation of C, can be shown by verifying that conditions (i) and (ii) of (1.3) are satisfied. (i) T(i+j)v, = V c i + j ) + k = v i + ( j + k ) = 7'(WWk1 = CWT(j)lvk; (ii) T(l)v, = vL+, = vk so that T(1) =I,. The family of linear transformations {T(i)}can also be described by means of their matrices { M ( i ) } .These are =

1 0 0 0 O O O 0 0 1 0 ' 0 0 0 1

M(3) =

0 0 1 0 0 0 0 1 1 o o 0 ' 0 1 0 0

M(1)

0 1 M(2)= 0 0

0 0 1 0

0 0 0 1

1 0 0 0

0 1 0 0 w4)=

0 0 1 0

o o o

1 1 0 0 0

It is simple, although tedious, to verify from their matrices that the set {T(1). T(2), T(3),T(4)) of linear transformations satisfy the same multiplication table as the group C,. Again we find that the mapping T of C, into Hom,(V, V) is a linear representation of C., This linear representation T gives rise to a matrix representation M which makes the matrix M(i) correspond to each element i of G. The linear representation T is said to uflord the matrix representation M . More generally, if T is a linear representation of a group G with representation space V and the set {vl, . . . , v,} is a basis B of V. then each linear transformation T(g),g E G, has a matrix M ( g ) with respect to the basis B. The correspondence M which makes correspond to each g of

69

1. Basic Concepts and Definitions

G the matrix M ( g ) is a matrix representation of G which is called the matrix representation afforded by T .

We consider a second example of broad significance in both representation theory proper and in its application to other disciplines. (1.5) EXAMPLE. Recall that a transformation group G on a nonempty set S is a group, each element g of which induces a bijection on the set S, such that if g, g’ are elements of G and x belong to S, then (gg’)x = g(g’x).

In greater generality, a group G is said t o act as a transformation group on the nonempty set S if there exists a homomorphism h of G into the group P ( S ) of all permutations on the set S. Familiar examples of transformation groups are GL(V), the group of all invertible linear transformations on a vector space V, and U,, the group of all unitary transformations on an n-dimensional inner product space. We illustrate a standard procedure for determining a representation of a transformation group G acting on a nonempty set S. Let V(S) denote the vector space of all functions with domain S and range the field K of complex numbers. We recall that V(S) is a vector space over K with (i) [ f + g](x) defined to be f ( x ) + g ( x ) and (ii) [ufJ(x) defined to be uf(x). We define a linear transformation T(g) belonging to Hom,(V(S), V(S)) for each g in G and show that this correspondence T is a representation of G. I f f E V(S) and g E G, we define T ( g ) by its action on f. that is, T(df=fog-1. It can be shown that T ( g ) is a linear transformation on V(S). Let g,g’ be elements of G. Then T(gg’)f = f o (gg’)-I

so that (1.6) Furthermore,

so that (1.6‘)

=~

=f 3 [(g’)-l 0 g-

‘ 1 = [ S o ( g ’ ) - l ] 09- ’

m T ( g ’ ) f= l LWT(g’)lJ;

Th’)= T(g)T(g’). T(l)f=fo1-1

=fo

T(1) = 1”(S)

1 =f, ’

It follows from (1.6) and (1.6‘) that T is a homomorphism of G into GL(V(S)) and, consequently, is a representation of G. This argument is entirely independent of the number of elements in either G or S.

70

2. The Representation Theory of Finire Groups

A simple generalization of this example can be obtained by letting V(S) denote the vector space of all functions with domain S and range a vector space U over the field K of complex nur,ibers. The above arguments extend to this case with virtually no modification and demonstrate that there exists a representation T of G with representation space V(S) in this instance as well. To be more specific. let S consist of the letters {a, b, c} and V be a three-dimensional vector space with the set (va. v b . vc} as a basis. Let the symbol S 3 denote the symmetric group on the set {a, 6, c} and denote its elements by g1 = (a),g z = ( a b c), g 3 = ( a c b), g4 = (a b), gs = (a c), and .96 = ( b c). A functionfE V(S) can be defined by means of a triplet of vectors (u", u,,. u,) where f ( a ) = u,, f ( b ) = ub. and f ( c ) = u,. A basis of V(S) consists of the nine functions,f,, . . . . , , f 3 3 which are defined by f 1 1 = (vu

3

O, O),

fi2

= (vb

O).

f22

= ( O , vb

.f32

= (O,

f21

= (O.

vo

f3l

= (O.

O , v',).

3

1

O, O ) ,

0, vb).

O, O)?

f13

= (vc

>

f23

= (O.

vc >

f33

= (O,

0, Vc).

O).

>

O),

The linear transformation T(gl) corresponding to g1 is clearly the identity. We look at the image T(g,) under the foregoing scheme.

[T(g2)f1(a>= (J"9 2 -')(a> = f ( c > , [7-(92)fl(b)

= (.Po 9 2 - I P )

=f(a),

and [7-(g2>fl(c> =

Cf

0

9 2-

'>(c>= f ( b ) .

I t follows from these relations that 7-h72)f;* =.f*13

7-(g2)s,2 =.f2*.

T(g2)f21

=f31,

T(g2).f22 = f 3 2

[(.92).f31

=f;I.

T(g2).f32

-_

=flZ

7-(92)f,,

T(g2)f23 = f 3 3

7

T(g2)f33

1

=f23r 7

'f13.

The matrix M ( g , ) of T(q,) with respect to this basis arranged in dictionary order is given by 0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1

1 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

71

1. Basic Concepts and Definitions

The interested reader will be able to work out the matrices of any other of the linear transformations T ( g ) for himself. Before turning from this example, we wish to observe that this last representation is reducible. If

is the inverse of any element g of G, then the effect of T(g)on any f of V(S) of the form (u,, u b , u,) is given by (1.7)

T(g)(ua >

ub

>

uc)

= (ua’

> ub’ > uc’>.

If the three components off are equal to each other, then (1.8)

T(s)f=.L

S f G.

Thus, for example, the subspace U of V(S) spanned by all complex multiples of (v,, v,, v,) is an invariant or reducing subspace of the representation. We return to the development of the general theory. Consider two families L and L‘ of linear transformations belonging to the two sets Hom,(V, V) and Hom,(W, W), respectively. Assume that each element of each of the two sets can be uniquely labeled with an element from the index set IT. Thus L consists of the set {T(n): 7c E rI} and L‘ of the set {T’(n): n E n}.Let A be an element of Hom,(V, W) such that (1.9)

T’(n)A = AT(7c),

7c E

rI.

(1.10) LEMMA. The family L is reduced by the Ker A and the family L‘ is reduced by the Im A . Proof. Let T(n)denote any element of L and v denote any element of the Ker A . Then AT(n)v = T’(7c)Avwhich is zero since v belongs to the Ker A . It follows that T(n)(Ker A ) is contained in the Ker A for each 71, and the Ker A is an invariant (reducing) subspace of L. On the other hand, if T’(n) is any element ofL’ and Ax is any element of the Im A , then T’(n)Ax = AT(7c)x which is an element of the Im A . Therefore T’(n)(Im A ) is contained in Im A for each n, and the Im A is a reducing subspace of L‘. (1.11) LEMMA.Let the family L defined above be an irreducible family, then either the Ker A must be the null space of V or the Ker A must coincide with V. Proof. Since L is reduced by the Ker A , it must be a trivial subspace of V, that is, the Ker A must be either the zero subspace or else coincide with V. (1.12) LEMMA.Let the family L‘ defined above be an irreducible family, then either Im A must be the zero space of W or the Im A must coincide with W. Proof. The proof is analogous to that of Lemma (1.1 1).

72

2. The Representation Theory of Finite Groups

(1.13) LEMMA (Scliur). Let V and W be vector spaces of dimension greater than zero over the field K of complex numbers. Let both L and L' be irreducible subsets of Hom,(V, V) and Hom,(W, W), respectively, the members of each set being labeled with elements from the set FI. If there exists a linear transformation A in Hom,(V, W) such that

(1.14)

T'(n)A = AT(n),

71

E

rI,

then either A is an isomorphism of V onto W or A is the null homomorphism. Proof. By Lemma (1.1 I), the Ker A must either be (i) the zero space of V or (ii) the entire space V. In case (i), the mapping A is a monomorphism and the Im A is not the zero space of W. Therefore, the Im A coincides with the space W by Lemma (1.12) and A is an epimorphism. Thus, in case (i) the mapping A is an isomorphism of V onto W. In case (ii), A is the null homomorphism of V Into W since the Ker A coincides with all of V. (1.15) LEMMA (Scliuv). Let L be an irreducible family {T(n): n E I'I} of linear transformations in Hom,(V, V) where V is a vector space. of positive dimension over the field K of complex numbers. If there exists a linear transformation A in Hom,(V, V) such that AT(7t) = T(n)A,

n E rI,

then A is a multiple of the identity automorphism 1, on V. K

Proof: Let U, be the nonzero eigenspace corresponding to some eigenvalue of A . Let u be any element of U, and T(n) be any element of L. From AT(n)u = T(n)Au = IcT(n)u,

it follows that T(n)u belongs to U, so that the irreducible family L has a nonzero invariant subspace U, which must coincide with the entire space V. Therefore, the linear transformation A is equal to ~ 1 " . (1.16) LEMMA.Let U be a subspace of the vector space V over the field K of complex numbers. Let the set (PI , . . . , P,) be a family of projections of Hom,(V. V), each of which has the subspace U for its range. Then the linear transformation

(1.17)

P = ( l / n ) ( P , +..'+P")

is also a projection of U. v

Proof. It is sufficient to show that P is an idempotent with range U. Let V, then PivE U for 1 I i I n, so that the range of P is contained in U.

E

73

1. Basic Concepts and Definitions

If u E U, then P i u = u for 1 i 2 n, so that Pu = u and U is contained in the range of P, that is, the range of P coincides with U. Furthermore, (1.18)

P2x = P(Px)

= Px,

x Ev,

so that P is idempotent. We are now able to establish the most fundamental theorem of the representation theory of finite groups. (1.19) THEOREM (Muschke). Let T be a representation of the finite group G of order n on the m-dimensional space V over the field K of complex numbers. Let U be a reducing subspace of T. Then U has a complementary reducing subspace W. Proof. Let R be any projection on the invariant subspace U and note that T(g,)RT(gi-') is also a projection Pi on U for every gi E G. It follows from Lemma (1.16) that p

-'>+

= (l/W(g,)RT(g,

*.

. + ~hl)RT( gf l- ')l

is a projection on U. Moreover, for each choice of g E G, T(g)PT(g-') is equal t o P since it is merely a rearrangement of the summands of P. Consequently,

m p = PT(g),

(1.20)

9 E G.

If I, denotes the identity linear transformation on V, then (1.21)

m",

= [I,

-PI

- PlT(S),

9 E G.

Furthermore, 1, - Pisaprojection and i fx belongstoitsrangeW then (1.22)

T(g)X

= T(g)[lv

= [I, - P]V,

-PIX = [l, - P]T(g)x E W

for every g E G. This implies that W is a reducing subspace of the representation T. However, every x belonging to V can be written x

= Px

+ (1,

- P)x

= x,

+ x2,

where x, belongs to U and x2 belongs to W. Also, if x is common to U and W, then x =P x

= (1,

- P)x = P(1, - P)x = 0,

so that V is the direct sum U 0 W of the reducing subspaces U and W. Thus W is a complementary subspace of U as was to be shown. LetQ be a linear transformation on a vector space V, which is the direct sum U, @ @ U, of invariant subspaces of D. Then Q defines a linear

74

2. The Representation Theory of Finite Groups

transformationDi on each invariant subspace U i , such that for each ui € U i , (1.23)

DiUi =nui.

Suppose that T is a linear representation of a finite group G on the representation space V which is the direct sum U, @ . . . @ U, of reducing subspaces. Since each Ui is an invariant subspace of T , T ( g )defines a linear transformation T i ( g ) of Hom,(U,, Ui) for each g in G. If g and g’ are in G, then for any ui in U i Ti(gg’)ui= T(gg’)Ui= [T(g>T(g’>lui == T(g)[T(g’)uil (1.24) = T(g)[Ti(g’)uil = Ti(g)[Ti(g’)uil = ITi(g)Ti(~’>lui.

Consequently, we see that (1.25) Let Ti denote the mapping from G into Hom,(Ui, Ui) whose value at g of G is the linear transformation Ti(g)introduced above. According to (1.25), Ti preserves the algebraic operations. Furthermore,

( I .26)

T,(l)u,= T ( l ) U i

= ui

for every u i E Ui where 1 denotes the identity of G. It follows from (1.25) and (1.26) that Ti is a representation of G. The representation T is said to be the direct sum of the set {TI, . . . , Tk} of representations. We denote this by

(1.27)

T = T l @..*@Tk.

It is important to realize that the sum indicated here is of a special nature and that T(g> = Tl(g) @ ’ ’ ’ @ T k ( g ) is not a conventional sum of linear transformations. Each of the representations Ti is the restriction of T to the corresponding subspace Ui and can be denoted by the symbol TI U i . The terminology previously introduced can be applied to the summands occurring in (1.27). Thus we speak of Ui as being reducible, irreducible, and so forth, accordingly as the representation Ti is reducible, irreducible, and so forth. respectively. Now apply Maschke’s theorem t o obtain the following theorem. (1.28) THEOREM. Let T be a linear representation of the finite group G on the n-dimensional representation space V over the field Kof complex numbers. Then either V is irreducible or else V is the direct sum U @U’ of invariant subspaces where U is irreducible, that is, the restriction of T t o U is irreducible.

75

1. Basic Concepts and Definitions

Proof. The theorem is valid for the case of an irreducible space V. If V is reducible, then V contains a nontrivial reducing subspace W, on which T defines a representation T, . Either W, is irreducible or else W, contains a nontrivial reducing subspace W, . In this fashion, determine a properly decreasing sequence of subspaces

w, 3 w, 3 f . . )

(1.29)

each of which is an invariant subspace of T. Since the representation space V is finite dimensional, the sequence (1.29) must terminate in an irreducible subspace W,. Denote W, by U and use Maschke's theorem to write

V=U@U',

(1.30) where U is irreducible.

(1.31) THEOREM. Let T be a linear representation of the finite group G on the n-dimensional representation space V over the field Kof complex numbers. Then either V is irreducible or else V is the direct sum U, @ * 0 U, of irreducible subspaces.

-

Proof. The theorem is valid for the case of an irreducible space V. If V is reducible, then Theorem (1.28) asserts that V is the direct sum U, @ U,', where U, is irreducible. Whenever U,' is reducible, it can be decomposed as the direct sum U, 0 U,', where U, is also irreducible. In this way, one arrives at an ascending chain

u, c u, o u , c u, O U , ou, c * . * , where the sums are direct with each summand irreducible. The finite dimensionality of the vector space V implies that this chain terminates with V. Hence there exists an integer k such that V is the direct sum

v = u , @ "'Ouk, where each summand is irreducible. Such a decomposition of V gives a corresponding decomposition of T as (1.32)

T = T, @ ... @ Tk

where each summand is the restriction of T to the irreducible subspace U i . Thus one obtains the following corollary. (1.33) COROLLARY. Let T be a linear representation of the finite group G on the n-dimensional representation space V over the field K of complex numbers. Then either T is irreducible or T is the direct sum of irreducible representations.

76

2. The Representation Theory of Finite Groups

(1.34) THEOREM. Let T be a linear representation of the finite group G on the n-dimensional representation space V over the field K of complex numbers. If V is the direct sum U, 0 0U, of irreducible subspaces and W is any reducing subspace o f V, then either W coincides with V or V is the direct sum of W and some of the U i . Proof. First, note that if W' and U j are reducing subspaces of a representation T, then the intersection W' n Uj is a reducing subspace of T, common to both. This observation implies that if Ui is irreducible, then either W' n Uj is the zero space or W' n Ui coincides with U j. Second, we see that if W and V coincide, the theorem holds; otherwise, there exists some U i not contained in W. Relabel this subspace Uj, and note that the sum

w, = w OU;,

(1.35)

is direct since W n U j , consists o f the zero vector alone. Either W, coincides with V or else there exists a second irreducible subspace, say Ui, , such that W, n Uj, i s the zero space and the sum

w, = w, 0ui,

is direct. By repetition of the argument, one produces an ascending chain

w c w, c w, c

(1.36)

*

of subspaces in the finite-dimensional space V. It follows that (1.36) must terminate with some space W, which coincides with V From the nature of the construction, it follows that

v =w@ui,0.*.0Ui,,

(1.37) as was to be shown

Let T be a linear representation o f a finite group G with representation space V which has two direct sum decompositions, U @ W and U @ W'. A theorem from the general theory of vector spaces asserts that W and W' are each isomorphic to the factor space VjU and hence are isomorphic to each other. This theorem can be extended to the more general case in which U, W, and W' are invariant subspaces of the linear representation T. The conclusion now becomes that the representation defined by T on W is equivalent to the one defined by T o n W'. The argument in the case of a linear representation T is very similar t o the argument for vector spaces and necessitates the introduction of a factor representation. We recall that if 6 is an element of Hom,(V, V) with U a nontrivial invariant subspace of 6, then 6induces a linear transformation 6' on the factor space VjU. Consequently, each T ( g ) defines such a T'(g) by (1.38)

T'(g)(x

+ U) = T(g)x + u,

x

+ u E vju

77

1. Basic Concepts and Definitions

f o r g E G. Thus one can use the representation T of G to define a representation T' of G with representation space V/U. For each g of G, let T'(g) denote the linear transformation on the factor space V/U that is defined in Eq. (1.38). Note that if g, g' belong to G and x + U is any element of V/U, then T'(gg')(x+ U)

= T(gg')X

+ U = T(g)T(g')x+ U = T'(g)[T(g')x + U] = T'(g)[T'(g')(x+ U)l

= [T'~g)T'(g')l(x

+ U),

so that (9

T'(g9') = T'(dT'(9').

Furthermore,

T'(l)(x + U)

= T(l)X

+ U = x + U,

which implies that (ii) T'(1) = I",", Consequently, T' is a representation of G with representation space VjU. The representation defined above is called the factor (1.39) DEFINITION. representation of T o n the factor space V/U. (1.40) THEOREM. Let T be a linear representation of the finite group G on the vector space V over the field K of complex numbers. Suppose that V is the direct sum U @ W of the reducing subspaces U and W. Then the factor representation T' of G on V/U is equivalent to the representation T"of G on the reducing subspace W. Proof. Let v be the natural homomorphism of V onto V/U and observe that Y maps x of V into x U of V/U. Denote by A the isomorphism of W onto V/U which is the restriction of v to the subspace W. Then

+

(1.41)

ATn(g)W

= AT(g)w = T(g)w

+ U = T'(g)(w + U)

= T'(g)Aw,

w

E

W , g E G,

so that

(1.42)

AT"(g) = T'(g)A,

9 E G.

Thus T and T' are equivalent representations of G.

If T' and T" are two equivalent representations of a group G with representation spaces V' and V" respectively, we sometimes say that the two representation spaces V' and V" are equivalent, written V' z V".

78

2. The Representation Theory of Finite Groups

( I .43) COROLLARY. Let T be a linear representation of the finite group C on the vector space V over the field K of complex numbers. Suppose that V can be decomposed in two distinct ways. U 0 W, and U 0 W,, into the direct sum of reducing subspaces, with the first summand common to both decompositions. Then the second summands are equivalent. In symbols, if (1.44)

V=UQW, =UQW,,

then W, and W2 are equivalent, that is, the representations Tl and T2 defined by T o n the reducing subspaces W, and W, are equivalent.

ProoJ: The representations T , and T2 are each equivalent to the representation T' on the factor space VjU and, consequently, are equivalent to each other. We are now prepared to derive the fundamental theorem concerning the reduction of representation spaces into irreducible subspaces by means of the results obtained in (1.40) and 1.43). (1.45) THEOREM. Let T be a linear representation of the finite group G on the vector space V over the field K of complex numbers. Let U and W be equivalent subspaces of V. Suppose that U has the decomposition (1.46)

U = U l Q . * .QU,,

while W has the decomposition (1.47)

W

= W, @ . . .

QW,

into irreducible subspaces of T . Then (i) the number of summands in (1.46) and (1.47) are the same and (ii) there exists some permutation { j , , . . . ,j,} of { I , . . . . k ] such that the subspaces U i and Wj, are equivalent.

Proof. The proof is by induction on the integer k . If k is 1, then U is irreducible so that t must also be 1. Clearly U, and W, are equivalent in this case. Assume, for purposes of induction, that the theorem is valid for 1 Ik I m. Let U and W have the two decompositions (1.48)

U=U,

and

( I .49)

W

=

@~..@u,@u,+l w, Q . . . o w , ,

respectively. Since U and W are equivalent, there exists an isomorphism A of U onto W such that

79

1. Basic Concepts and Definitions

Let W,' be the image of U, under A . Then W,' is an irreducible subspace of W equivalent to U,. Theorem (1.34) implies that W is the direct sum

w = W,' 0wi, 0. . . 0W & . It follows that

u* = U 2 0 . - . @ U , + ,zu/u, M

W/W,'w w *

= w i l@ . . . O W i r .

By the induction hypothesis, there are rn summands in W*, each of which is equivalent to one of those occurring in the direct sum decomposition of U*. On the other hand,

u, w u/u*w w / w * ,

so that W* contains every summand of W except one. say W". which must be equivalent to U,. This completes the induction and establishes the theorem.

(1.50) THEOREM. Let T be an irreducible linear representation of a finite abelian group G on the representation space V over the field K of complex numbers. Then V is one-dimensional. ProoJ: Let T(go) correspond to any go of G. Then

It follows by Lemma (1.15) that T(go) is a multiplef(g,)lv of the identity on V. Any nontrivial subspace U of V reduces T. Thus V has no nontrivial subspaces, that is, V is one-dimensional. (1.51) THEOREM. Let Ckbe a cyclic group of order k. Then there is a one to one correspondence between the family { T } of irreducible representations of C, and the set (5) of kth roots of unity.

Proof. Let T be any irreducible representation of ck and a any generator o f C,. The representation space V of any irreducible representation T of G has

a basis {v} consisting of a single vector so that the image T(a) can be taken to be the linear transformation such that T(a)v = cv.

It follows that

v

=

I V = I,v

= T(1)v = T(ak)v = [T(U>],V= l k v ,

so that 5 is a kth root of unity. The kth root of unity icompletely determines the linear representation T. Thus, given a particular generator a of the group G, there corresponds to each irreducible linear representation T a unique kth root of unity i.

80

2. The Representation Theory of Finite Groups .

If j > j’ and a j = aj’, then a’-.’

.I

= 1,which

j =j’

(1.52)

implies that k divides j - j ‘ or

+ mk,

which means that [j

(1.53) whenever

=ij’+mk = Y

1;

5 is a kth root of unity.

The correspondence which pairs with each element a/ of G the complex number ij is a well defined mapping from G to the field K of complex numbers, since aj = ai’ implies that 5 j = [j’ according to (1.52) and (1.53). Thus we can define for each g = aj of G a linear transformation T(g) on the one-dimensional space V. Let (v} be a basis of V and define T(g) by T(g)v

= T(aj)v =

pv.

The correspondence T is a representation of G with representation space V. This follows from T(gg’)v = T(ajuj’)v = T(ajfj‘)v = ij+ j‘v = (iJ[j’)v = [j((j‘v) that is, and so that

= T(ai”(aj‘)vI

= %d[T(g’)vI = [n)T(g’)lv,

T(1)v = T(ak)v = Ikv = lv

= l,v,

T(1) = l v .

(ii)

Thus the irreducible representations of the cyclic group correspondence with the set of kth roots of unity.

c k

are in one-to-one

It is a well-known theorem that every finite abelian group G can be written as the direct sum (1.54)

G

= (ml) @

*

. . @(m,)

with each of the cyclic summands (mi) having order c ( ~such that ailcli+l, 1 5 i 5 r - 1 . Every irreducible representation T gives rise to a family { T I ,. . . , T,} of irreducible representations of the subgroups (mi), 1 5 i I r. Each representation Tiis the restriction of T to (mi),that is, (1.55)

Ti = TI (mi),

15 i I r.

1. Basic Concepts and Definitions

81

By Theorem (1.51), if the representation space V has a basis {v} consisting of the single vector v, then each of the representations Ti is characterized by a complex number C i , an aith root of unity, such that (1.56)

Ti(mi)v= Civ,

15 iI r.

Thus the value of the representation T for any element (1.57)

g =j,m,

of G,

+ +j r m , *

*

. Cj3v, is completely specified by the r-tuplet (Cl,. . . , [,) of a,th roots of unity. (1.58)

T(g)v

= (C1j1

*

Furthermore, if T‘is any representation of G with {v}‘ a basis of its representa tion space V‘ and (1.59)

T’(m,)v’= CiV’,

then T‘ and T a r e equivalent representations of G. Conversely, if (C1, ,. . , 5,) is an r-tuplet of complex numbers such that Ciai = 1 , 1 5 i 5 r, then the correspondence f such that (1.60)

f ( 9 ) = Cl” . . . [ A

where g is given by (1.57), is a well-defined mapping of G into the field K of complex numbers. This correspondence allows us to define a mapping T from G into Hom,(V, V) V a K-space with basis {v}. T ( g )is defined by (1.61)

T(g)v

=

([p.. . C))v.

The verification that T is an irreducible representation of G follows along the same lines as in the proof of Theorem (1.51). Thus we are lead to the following Theorem. (1.62) THEOREM. Let G be an abelian group with the direct sum decomposition G = (m,)@. (m,)

-

into cyclic subgroups (mi)of order m i , 1 5 i 5 r, where ail a i + l , 1 5 i r - 1. Then there is a one-to-one correspondence between the irreducible representations of G and the set of all r-tuples (it,. . . , [,), where Ci is an aith root of unity. If an irreducible representation T with representation space (v) corresponds to the r-tuple (Cl, . . . , C,) and g denotes j,m, + . . . + j , m,, then

T(g)v = ( C l j l . . . (‘3v. W e now turn to some specific instances of the representations theory of finite abelian groups.

82

2. The Representation Theory of Finite Groups

(1.63) EXAMPLE.We illustrate these general considerations by means of members of two of the three classes of isomorphic abelian groups of order eight. The Cayley table of the first of these two groups is as shown in Table ( 1.64). (1.64)

CAYLEY TABLEOF Cs 12345678 112345678 223456781 334567812 445678123 556781234 667812345 778123456 881234567

The group C , is a cyclic group of order eight which is generated by the element 2, for instance. From Theorem ( l . S l ) , it follows that the representations of C , are determined by the eighth roots of unity. Let o denote the complex number cos(ni4) + i sin(n/4). Then each of the numbers o,. . . , w 8 = 1, determines a distinct irreducible representation of C , . The complete collection of nonequivalent irreducible representations of C, is given by Table (1.65). (1.65)

IRREDUCIBLE REPRESENTATIONS OF Ce 1 2 3 4 5 6 7 8 T I 1 w w 2 w 3 w4 w 5 w 6 w7

T2

1 w 2 w4 w 6 w 8 w10 w I 2 w14

T3 1 w 3 w 6 w 9 w 1 2 w 1 5 w ' 8 w z 1 T~ I w4 w i z w i 6 w 2 0 w 2 4 w2a T5 I w 1 0 w 1 5 w Z O w Z 5 w30 w 3 5 T6 1 w 6 w 1 2 w 1 8 & 2 4 w 3 0 w 3 6 4,2,, T1 1 w 7 w14 w 2 1 w 2 8 w 3 5 w 4 2 w49 T a l l 1 1 1 1 1 1

The entries in the body of Table (1.65) constitute the matrices of the corresponding linear transformation with respect to any given basis of the one-dimensional representation space. (1.65') EXAMPLE.The next group G to be discussed is C, 0 C,. The members of C2 form the set (1, a} and those of C, the set (1, b, b2, b3}.The elements of C, @ C, are enumerated in the following order: 1 = (1, l), 2 = ( I , b), 3 = (1, bZ), 4 = (1, b3), 5 = ( a , I), 6 = ( a , b), 7 = ( a , b2), and 8 = ( a , b3).

83

I . Basic Concepts and Definitions

The Cayley table of G is shown in Table (1.66). The representations of C2 (1.66)

CAYLEY

TABLE OF

c z @ c4

12345678 112345678 223416785 334127856 441238567 556781234 667852341 778563412 885674123

are determined by the square roots of unity, 1 and -1, while those of C4 are determined by the fourth roots of unity, 1, i, - 1, and - i. If x and 6 are second and fourth roots of unity determining certain irreducible representations of C2 and C, ,then the pair {x,S} determine an irreducible representation T of C, 0 C, through the formula T[(u’,b ’ ) ] ~= xiS’v where {v} is a basis of a one-dimensional representation space V of T. These remarks lead to the family of eight irreducible representations of C, 0 C, which are listed in Table (1.67). The entries in the table are the matrices of the image Ti(gj)with respect to any basis of the representation space. (1.67)

IRREDUCIBLE REPRESENTATIONS OF Cz @ C4 1 2 3 4 5 6 7 8 1 i - 1 -i 1 i - 1 --i 1 i - 1 -i-1 -i 1 i 1-1 1-1 1-1 1-1 1-1 1 - 1 -1 1-1 1 T5 1 - i - 1 i 1 -i-1 i Ta 1 - i - 1 i-1 i 1 --i T ~ 1 1 1 1 1 1 1 T8 1 1 1 1 - 1 -1 - 1 -1

TI Tz T3 T4

1

These examples conclude the discussion of the representation theory of finite abelian groups. We turn now to the problems of the representation theory of nonabelian groups. It will appear that here the problems are considerably more complicated and require the introduction or a rather extensive mathematical apparatus.

84

2. The Representation Theory of Finite Groups 2. THE GROUP ALGEBRA KG OF A FINITE GROUP G

The results of Section 1 provide a powerful tool for the analysis of the representation space V of a linear representation T of a finite group. The principal purpose of this section is to develop a certain fundamental representation of an arbitrary finite group G. The immediate source of this representation is the group G itself. The usual representation obtained by means of G is called the regular representation and is denoted by the symbol 93 in this book. The representation space of the regular representation 93 is the group algebra KG of the finite group G. The group algebra KG is a vector space whose basis is identified with the elements of G and in which an operation of multiplication has been defined. Before entering into details, we summarize briefly the definition of the system known as an algebra in modern mathematics. Each algebra A is a vector space over a field K generally taken to be the field of complex numbers in this book. The symbol A is used to denote the set of vectors as well as the algebra itself. The connection between the algebra A and the field K is made explicit by saying that A is an algebra over K. In addition to the vector space operations of A, there is defined a binary operation called multiplication for the vectors. Thus there are three operations involved in the concept of an algebra: the operation of scalar multiplication, the operation of vector addition, and the operation of vector multiplication. In the following list of axioms, the symbols CI,P, and y denote elements of the field K of complex numbers, while the symbols x, y, and z are used for vectors of A. The symbol 1 denotes the multiplicative identity of K.

VECTORSUM (A Binary Operation on A) (1) x t (y 2 ) = (x Y)+ z. (2) There exists a unique 0 in A such that x 0 = 0 x = x. (3) Corresponding to x in A there is a unique y in A such that x y -tx = 0, where y is usually written -x. (4) x + z = z + x .

+

+

+

SCALARMULTIPLICATION (5) yx belongs to A. (6) 4 b x ) = ( a m . (7) a(x Y) = fix cry. (8) 1. P I X = ax Px. (9) Ix = x.

+ +

+ +

VECTORPRODUCT (A Binary Operation on A) (1) X(YZ = ( X Y N (2) 4XY) = ( 4 Y = X(UY>, (3) x(y 2 ) = xy -txz. (4) (Y z)x = yx zx.

+

+

+

+

+y =

2. The Group Algebra KG of a Finite Group G

85

The group algebra KG of a finite group G is an algebraic system that satisfies the above axioms and which is closely related to the group G. The initial step in its construction is to form a vector space associated with G. This may be done in the following way. Given any nonempty set S , the set F ( S ) of all complex valued functions with domain S and range K, the complex numbers, can be made into a vector space by means of the definitions: (i) Vector sum: Iff, g E F ( S ) , f + g is the element of F ( S ) whose value at s E S is given by [f+ gJ(s)= f ( s ) g(s). (ii) Scalar multiplication: I f f E F ( S ) and a E K, afis the element of F( S) whose value at s E S is given by [~fJ(s) = a(f(s)).

+

The argument that F(S) is a vector with these two operations is straightforward. In the case of a finite set S, each memberfof F ( S ) can be described by means of a table such as

where the first row of the table is an enumeration of the elements of S and the second is a tabulation of the images such that each image ci stands beneath its counter image s i . A second common mode of description is to write the element f of F ( S ) as a formal sum where the coefficients f ( s i ) of each element si is the value o f f at si. This notation can be abridged to (2.2) or

f=

n

1f(si)si

i= 1

These formal sums can be made into actual sums in F ( S ) by the identification of an element s of S with that function s* of F ( S ) whose value at s is one and whose value at s', different from s, is zero. With this notation, the above equations become (2.2')

or (2.3') We usually employ the notation of the unprimed equations, but think in terms of the primed equations. The vector space F(G), G a finite group, admits of a multiplication under which it becomes an algebra. The functions

86

2. The Representation Theory of Finite Groups

si* form a basis for the vector space F(G). A vector multiplication for F(G) can be defined in terms of these basis elements and extended to the other members of F(G) by linearity. Let si and sj be any two elements of G whose product s i s j is the element sk . We define

si*sj* = Sk*.

(2.4)

It follows that if h is a second element of F(G) with

h

=

c h(t)t,

isG

then the product f h should have the form

which becomes, after substituting q for st and rearranging,

This result justifies the definition: (iii) Vector product: Iff, h E F(G),fh i s the element of F(G) whose value at s E G is given by [fh](s) = ~,,Gf(st-1)17(t). The proof that F(G) i s an algebra over the field K of complex numbers with these definitions of vector sum, scalar multiplication, and vector product is a tedious argument which we advise the reader to skip. The algebra F(G) is called the group ulgebra of the group G and denoted in the sequel by KG. There is associated with each element b of an algebra A an element b, of the K-homomorphisms Hom,(A, A) of A into itself. The definition of bL is straightforward. If x E A, then bL(x) is defined by bL(x) = bx.

The fact that b, is linear follows from the observations that and

bL(x + y)

= b(x

b,(r*x)

+ y) = bx + by = bL(X) + bL(y),

= b(ctx) = a(bx) = a(bL(x)),

a

E

X,

y

E

A,

K, x E A.

The re~gulurrepresentation 3' 3 of the group G is the homomorphism of G into GL(KG) obtained by letting each image %(g) be the left-multiplication 9 , . a linear transformation on the vector space (algebra) KG. The symbol gL denotes the left-translation or left-multiplication defined by the element g* of KG. Recall our custom of denoting the element g* by the element g of G to which it corresponds and, consequently, of denoting its left-multiplication by g L .

2.

87

The Group Algebra KG of a Finite Group G

We wish to verify that the mapping % is a representation of the finite group G. Let g , g’ be elements of G and x be any element of KG, then W g g ’ ) x = (SS’)L(X) = (99% = g(g’x) = SL(SL’(X)) =W)(%’)X)

so that (2.7)

= Pw)%(g’)lx,

W g d ) = f#(g)Wg‘).

Furthermore, we see that f l ( l ) X = lL(X) = l(x) =x = lKG(X),

which implies that %(1) = l K G ,

(2.8)

Equations (2.7) and (2.8) show that the mapping % is a homomorphism of the finite group G into GL(KG) so that the term regular representation is justified, that is, % is a linear representation of G over the representation space KG. Before developing the structure theory of the group algebra KG of a finite group G, we consider a simple example in some detail. There are two classes of abstract groups of order six, one of which is abelian and the other is not. The permutation group S3 is a realization of the noncommutative class whose elements may be defined as follows: R1

a b c = ( a b c)’

R2 =

a b c ( c a b)’

a b c R 3 = ( b c a)’

a R4=(a

b c c b)’

a b c R 5 = ( b a c)’

a b c R 6 = ( c b a)‘

The multiplication table, group elements being designated by their subscripts, for the group S3 is shown in Table (2.10).

(2.10)

CAYLEY TABLE OF S3 123456 1123456 2231645 3312564 4456123 5564312 6645231

88

2. The Representation Theory of Finite Groups

The entry k at the intersection of the ith row and j t h column is the number of the product R, = Ri R, of the element R i by the element R, . The elements in KG which correspond to the group elements form the set &, R 2 , R 3 , R 4 , R 5 , R6)? where R2

R3

Rt

R2

R3

R4

R5

R6

Rt

R2

R3

R4

R5

R6

R2

R3

R4

R5

R6

Rt

R2

R3

R4

R5

R6

Rt

R2

R3

R4

R5

R6

1

0

0

(

R1

0 0

0

1

0

(2.1 1)

R, 0

R4 R5 0 0

Rl

0

0

0

0

1

0

0

0

0

0

0

0

0

1

0

0

0

0

0

=R2, =R3,

0

1

0

= Rt,

0

0

1

The notation for these elements can be conveniently shortened to (1 0 0 0 0 0) = R,, (2.12)

(0 1 0 0 0 1 0 0 (0 0 0 1 0 (0 0 0 0 1 (0 0 0 0 0

(0 0

0)=R2, 0)=R3, 0)=R4, 0)=R5, 1) = R 6 ,

where the first row in each tab-" is understood to be Rl

R2

R3

R4

RS

R6

and is suppressed. The form of the expressions in (2.12) show that the vector space KS, is isomorphic to K 6 . In particular, {Ri},1 5 i I 6, is a basis for KS3 called the standard or natural basis. It follows from the definition of vector product, that

[RiRj](s)=

rsG

Ri(st-l)R,(t).

89

2. The Group Algebra KG ofa Finite Group G

This value is different from zero for a given S if and only if at least one of the summands on the right-hand side is different from zero. The summand R;(st-1)R/t) differs from zero if and only if and

st-1=Sj

t=Sj'

Thus [RjR¥s) is different from zero if and only if

Since [R j Rj](sj Sj) (2.13)

=

1, these observations imply that where

Ri R, =R k,

SjSj =Sk'

Thus we see that the rules of (2.4) are satisfied and that the multiplication in the group algebra KS 3 is completely determined by the group multiplication Table (2.10). In particular, the effect of the left-multiplication R iL is determined from Eq. (2.13). Each left-multiplication R iL , being a linear transformation, has a matrix M(i) with respect to the standard basis. The elements of M(i) are denoted by the symbols M(i)jk' Thus we have 6

RiL(R j) =

I 1 M(i)kj R k·

k=

The explicit forms of the matrices are 1 0 0 0 0 o 1 000 o 00100 o M(I) = 00010 o o 0 0 0 1 o 00000 1

o

o

'

1 000 o 0 100 o 1 0 0 0 0 o M(3) = o 0 000 1 ' 00010 o o 0 0 0 1 o

o o

00001 o 0 000 00010 M(5) = 00100 1 0 0 0 0 o 1 000

o I

o o' o o

0 I 0 0 10000 o 1 000 M(2) = 00001 00000 00010 00010 o 0 0 0 1 o 0 0 0 0 M(4) = I 0 0 0 0 o 1 000 00100

o 0 0 0 0 000 I 0 o 0 0 0 I M(6) = o 1 000 o 0 1 0 0 1 0 0 0 0

o o o o' 1

o o o I

o' o o 1

o o o o o

90

2. The Representation Theory of Finite Groups

We return to the general considerations. The theory of algebras employs certain basic concepts which can be applied to a group algebra KG as a special case. (2.14) DEFINITION. A right ideal J of an algebra A is a subspace of A such that if x E J, y E A then xy is an element of J. A left ideal J of A is a subspace of A such that if x E J, y E A, then yx is an element of J. A two-sided ideal J of A is a subspace which is both a right and a left ideal.

In every algebra A, the subspace A and the zero space (0) are ideals referred to as fricial ideals. Ideals of A different from either of these are called nontricial ideals. A ready example of these concepts is had in the case of the algebra K, of all 2 x 2 matrices over the complex field. The argument that K2 is an algebra is essentially a review of the basic operations on matrices. The set of all matrices of the form

: :1 :I1 1 at 1

is a left ideal in K 2 , while the set of those of the form " ; ;

is a right ideal. It is an interesting fact that K2 contains no nontrivial twosided ideal. For let J be any two-sided ideal of K2 which contains a nonzero element

The products

/I2 2 1 . )I ::: ::: 1 / ; ;/ 1 ::: :::/ 1 0 I/ I( ::: :: /I / ::; 2: /I =

and

I ! )

=

belong to J. The first is the original matrix with its columns interchanged and the second with its rows interchanged. It follows from a sequence of such operations that J contains a matrix B with b,, different from zero whenever J contains a nonzero element of K 2 . On the other hand, it follows from

91

2. The Group Algebra KG of a Finite Group G

and the fact that J is a subspace, that J must contain the matrix

Continuing in this fashion, one finds that J must contain a basis of K2 consisting of the four matrices

Consequently, J coincides with the algebra K , .

(2.15) DEFINITION. A minimal ideal is an ideal which contains no nontrivial ideal of the same nature. A minimal left ideal contains no nontrivial left ideal, a minimal right ideal contains no nontrivial right ideal, and a minimal twosided ideal contains no nontrivial two-sided ideal. (2.16) LEMMA. A subspace L of the group algebra KG is a left ideal of KG if and only if L is an invariant or reducing subspace of the regular representation Y?. Proof. Let L be a reducing subspace of the regular representation 92, t be any element of G, and u be an element of L. Then Consequently, if h

=

ct

tu

= t,u = Y?(t)u E L.

h(t)t any element of KG, then

and L is a left ideal in KG. Conversely, if L is a left ideal of KG, g u E L, then %(g)u = gLu = gu E L

E

G, and

and L is a reducing subspace of Y?. We note that an irreducible subspace L of the regular representation % is a subspace of KG which contains no nontrivial reducing subspace. A minimal left ideal of KG is a left ideal containing no nontrivial left ideals. It follows as a corollary.

(2.17) COROLLARY. A subspace L of the group algebra KG is an irreducible subspace of the regular representation Y? if and only if L is a minimal left ideal of KG. We turn to the introduction of an important new idea. Since any left ideal L of the group algebra KG is also an invariant subspace of '93, the regular representation defines a representation sL whose representation space is L.

92

2. The Representation Theory of Finite Groups

Two left ideals L and L’ are said to be equivalent if their corresponding representations 91L and 91L, are equivalent. We denote this relation in symbols by L z L‘. (2. IS) THEOREM. Every left ideal L of the group algebra KG of a finite group G has a complementary left ideal L’ such that KG is the direct sum L @L’. Proqf: This is a special case of Theorem (1.19). (2.19) THEOREM. The group algebra KG of a finite group G is the direct sum of minimal left ideals:

KG = L , @ . . . @ I + .

(2.20) Furthermore, if

KG =L,‘ @... @L,‘

(2.21)

is another decomposition of KG as the direct sum of minimal left ideals, then k equals t and. possibly after rearrangement, the corresponding minimal left ideals are equivalent, that is,

Li M L j z ’ ,

(2.22)

1l ilk.

Proof. This is a particular instance of Theorem (1.45) for the case of the group algebra KG, where the term minimal left ideal is used rather than irreducible subspace. Observe that two minimal left ideals L and L’ of an algebra A either coincide or have only the zero vector in common. The sum L @ L’ is direct i n the second case. Furthermore. if L and L’ are distinct minimal left ideals of the group algebra KG, there exists a direct sum decomposition

KG = L @ L ’ @ * . . @ L k

(2.23)

into the sum of minimal left ideals. Our results and definitions have been concerned with representations of a finite group G up to the present. We enlarge our concepts to include the representations of an arbitrary algebra A. A linear representation of an algebra A is a mapping T* of A into the algebra Hom,(V, V) of all linear transformations of a vector space V over the field K of complex numbers. The mapping T* preserves the algebraic operations in the following sense. Let cx denote a complex number while x and y denote elements of the algebra A. Then

+

(i) T*(x y) = T*(x) (ii) T*(ax) = cxT*(x);

+ T*(y);

(iii) T*(xy) = T*(x)T*(y).

93

2. The Group AIgebra KG of a Finite Group G

Although the preceding definition applies to the concept of a representation for any algebra, we use it primarily in the case of the group algebra KC of a finite group. One notes that a representation T* of an algebra A is a linear transformation of the vector space A into a vector space Hom,(V, V). This contrasts with the case of a group representation T which is homomorphism of one group into another. The analysis of the representations of certain special types of algebras is more straightforward than the corresponding analysis of the representations of an arbitrary finite group. Consequently, some of our representation problems are simplified by noting that every linear representation T of a finite group G defines a corresponding representation T* of the group algebra KG. Let T be a linear representation of the finite group G on the finite dimensional vector space V over the field K of complex numbers. The representation T* is defined in the following manner. The set

G

= {gi, . . . , g,>

is a basis of the vector space KG. The set Im T = {T(gi),. . . , T(g,)) belongs to the vector space Horn#, V). By a well-known theorem of vector spaces, there exists a unique linear transformation T* with domain KG and range Hom,(V, V) such that T*(gJ = Tfgi),

(2.24)

9, E G .

In Eq. (2.24), the argument of T* on the left should be labeled g , * ; however, we eliminate the asterisk from such elements of KG by agreement. The argument of T on the right-hand side of (2.24) is an element of G. Since T* is known t o be linear, it suffices to show that

T*(xJJ) = T*(x)T*(y),

(2.25) Let f =

1,

f(g)g and Iz =

cr

f / f

from which it follows that

=

X,

y

E

KG

I7(t)t be any two elements of KG. Then

c c fw7(f)sf>

gsc

rsc

94

2. The Representation Theory of Finite Groups

We see that T* is a linear transformation of KG into Hom,(V, V) which satisfies Eq. (2.25) listed above. Therefore, T* is a linear representation of the group algebra KG on the representation space V. Furthermore, if T' any representation of KG such that (2.27)

T'(d =

m,

9

E

G,

then T' coincides with T*. On the other hand, let T* be any linear representation of the group algebra KG on the representation space V, with T * ( l ) the identity element 1, of Hom,(V. V). Then the mapping T of G into Hom,(V, V) defined by (2.28)

T(g) = T*(g),

9

E

G,

is a representation of the group G. Let T be a representation of the finite group G and T* a representation of the group algebra KG, related to each other according to one of the two schemes outlined above. Then T and T* have a common representation space V. Furthermore. the concepts of an invariant or reducing subspace U coincide for the two representations; that is, a subspace U is a reducing subspace of T if and only if it is a reducing subspace of T*. It follows immediately from this fact that the concepts of irreducible, decomposable, and completely reducible coincide for the two representations T and T*. The representation space of the regular representation % of the finite group G is none other than the group algebra KG. It follows that the corresponding representation %* of the group algebra KG is a representation of the group algebra KG with representation space also KG. We will show that the representation %* makes correspond to every f E KG the left-multiplication fL of Hom,(KG. KG). To see this, let f be any element of KG and v be some element of KG (considered as the representation space of %*). Since f can be expressed in the form

f=

c f(d%

SEG

we see that

In summary, (2.30)

!R*(f) =f L ,

f E KG.

95

2. The Group Algebra KG of a Finite Group G

The content of the preceding remarks is that the representation %* of the group algebra KG can be viewed consistently either as an extension of the regular representation % of the group G or as the regular representation of the algebra KG which makes correspond to each f of KG the left-translation f L. We now turn to an investigation of the representations of the group algebra KG of a finite group G over the field K of complex numbers.

(2.31) THEOREM. Every irreducible representation T of the group algebra KG occurs in the regular representation %* of KG. Proof. Denote by !RL* the representation of KG defined by %* on the minimal left ideal L of KG. Let T be any irreducible representation of the group algebra KG with representation space V. Define a linear transformation A of a minimal left ideal L of KG into the space V in the following manner. Select any nonzero vector v of V and denote by A the linear transformation of L into V defined for x E L by A X = T(x)v. Let f be any element of KG and note that whenever v E L [A%L*(f>lX= A[%L*(fBl = 4 f x I

= T(f"x1

=

[T(f)AIx.

It follows that

(2.32)

T ( f ) A = A91L*(f),

f

E

KG.

Since both 91L* and T are irreducible, Schur's lemma implies that either A is the null transformation or else A is an isomorphism so that sL* is equivalent to T. Suppose that for some irreducible representation T, A is the null transformation for each choice of minimal left ideal L of KG. Since KG is the direct sum KG=Ll@*..@Lk of minimal left ideals, it follows that T ( f ) v must be zero for every f E KG, which is a contradiction. Hence the group algebra KG must contain at least one minimal left ideal L such that %* is equivalent to T. In the sequel, the regular representation %* of the group algebra KG is denoted by the same symbol % as the regular representation of the finite group G unless there is serious danger of confusion about the meaning of the symbols.

(2.33) THEOREM. Let T be any linear representation of the group algebra KG over a finite-dimensional representation space V and let T have the decomposition T = T1 0 ' . @ Tk

96

2. The Representation Theory of Finite Groups

into irreducible components. Let x belong to a minimal left ideal L of KG such that 9IL is not equivalent to any of the summands Ti, 15iI k. Then it follows that T(x) must be the zero linear transformation. Proof. It was seen in the proof of Theorem (2.31) that if the irreducible representation T iis not equivalent to the representation !RL, then Ti(x) is zero whenever x E L. Since T(x) = T,(x) . . . Tk(x), it follows that T vanishes for x E L whenever !RLis not equivalent to any one ofthe T i ,1 5 i 5 k.

+ +

3. THE STRUCTURE OF THE GROUP ALGEBRA KG

The fundamental decomposition of a group algebra KG into the direct sum of minimal left ideals was obtained in Section 2 . I n Section 3, it is shown that the group algebra KG of a finite group G over the field K of complex numbers is the direct sum of minimal two-sided ideals. Every complex group algebra KG i s a sernisirnple algebra, a concept to be introduced in this section. The development of these results requires the introduction of a number of new algebraic ideas. A left ideal L of an algebra A is the principal left ideal generated by the element b of A whenever L coincides with the left ideal Ab. The element b is a generator of the ideal L. A nonzero element e of A such that ez equals e is an idernpotent. If the left ideal L has the idempotent e for a generator, then e is an idernpotent generator of L. Let 6 be any nonempty subset of the algebra A . The set L of all a E A such that as is zero when s E 6 is a left ideal called the left annihilator of 6 . The set R of all a E A such that sa is zero whenever s E 6 is a right ideal called the right annihilator of 6 . (3.1) THEOREM. Let the group algebra KG of the finite group G over the field K of complex numbers be the direct sum L @ L’ of the nontrivial left ideals L and L‘. Then there exists idempotents e and e‘ which are the generating idempotents of L and L’, respectively. Furthermore,

ee‘ = e‘e = 0.

(3.2)

Proof. Let 1 denote the multiplicative identity in KG. Then there exists the decomposition

1 = e + e‘,

If x is any element of L, then

x

= x l = xe

+ xe’,

e E L , e‘ EL’. xe E L, xe’ E L’.

From the uniqueness of the direct sum decomposition of x, we see that

x =xe,

0 =xe’

97

3. The Structure of the Group Algebra KG

In particular, it follows that e = e 2.

O=ee'.

In a similar fashion, one has for x' any element of L', x'

= x'e',

0

= x'e,

and

e'

= (e')',

0

= e'e.

Also one notes that every x E L can be written x = xe so that L c (KG)e. Since (KG)e is surely contained in L, it follows that L coincides with (KG)e. Thus e is an idempotent generator of L. In the same way, e' is an idempotent generator of L'. (3.3) COROLLARY. If L is a nontrivial left ideal of K C , then L has an idempotent generator e such that the right-multiplication eR, see (3.9), is a linear transformation of KG onto L. Moreover, xe and x coincide for every x in L. Proof: Theorem (2.18) asserts that KG is the direct sum L @L' where L' is an ideal complementary to L. Denote by e and e' the idempotents of Theorem (3.1). The element e is the required idempotent generator of L and eR is the required linear transformation. Let the group algebra KG be the direct sum (3.4) THEOREM. (3.5)

KG = L l @ . . . @ L k

of nontrivial left ideals {L,}, 1 5 i 5 k. Then there exists a family of idempotents { e , } ,1 I iI k , such that each e , I S a generating idempotent of the corresponding left ideal Li . Furthermore, (3.6)

e,e,

= 0,

i #.j.

Proof: Decompose the identity 1 of KG as the sum

(3.7)

1 = el

+ . . . + ek

and proceed as in Theorem (3.1). Two idempotents e' and e" are called orthogonal if e'e''

= e"e' = 0.

An idempotent e is called primitive if there exists no decomposition of e into the sum of orthogonal idempotents. (3.8) THEOREM. If e is a primitive idempotent of the group algebra KG, then the left ideal L generated by e is minimal. Conversely, if L is a minimal left ideal of KG, then any idempotent generator e of L is primitive.

98

2. The Representation Theory of Finite Groups

Proof Let e denote an idempotent generator of the left ideal L. Suppose that L’ is a nontrivial left ideal properly contained in L, then KG and L have the respective direct sum decompositions,

KG

= L’ @ L”

and

L = L’ @ L*,

where L* denotes the nontrivial left ideal L n L”. The idempotent e has the decomposition e = e‘

+ e*

with e’ and eh orthogonal idempotents generating L’ and L*, respectively. Thus e is not primitive. Let e be a nonprimitive idempotent with the decomposition e’ e“ into the sum of orthogonal idempotents. The left ideal generated by e has the decomposition

+

Le = L(e’ + e”) = Le’ + Le”. The ideals Le’ and Le” are contained in Le since

e’e = e’(e’ + e”) = (e’)’

= e‘ E

Le

and

e“e = e”(e’ + e”) = (e”I2= e“ E Le. However, Le’ does not coincide with Le since e“ belongs to Le but not to Le’. Thus L is not a minimal left ideal. (3.9) OBSERVATION. Every element b of an algebra A over the field K of complex numbers determines a right-multiplication (translation) b, belonging to Hom,(A, A). If x any element of A, then bR(x) is defined by

(3.10)

bR(x) = xb.

The argument that the right-multiplication b, is a h e a r transformation parallels the proof that the left-multiplication bL is a linear transformation. However, the homomorphism bR has an additional property which is not always valid for a general linear transformation. This property is as follows: If a E A. considered as an algebra, and x E A, considered as a vector space, then

b,(ax) = (ax)b

= a(xb) = a(b,(x));

that is, (3.11)

b,(ax)

= a(b,X).

99

3. The Structure of the Group Algebra KG

Let V and W be A-modules and let h be a group homomorphism of V into W. Suppose that h(ax) = a/z(x),

(3.12)

a

E

A, x

E

V.

Then h is called an A-homomorphism of V into W. The collection of all Ahomomorphisms of V into W is denoted by the symbol Hom,(V, W). Equation (3.11) asserts that bR is an element of Hom,(V, V) as well as an element of Horn#', V). We wish to emphasize the fact that the two left ideals L and L' of the group algebra KG are said to be equivalent if and only if the two representations '91L and 'illLf are equivalent. (3.13) THEOREM. If the left ideals L and L' are equivalent, then every equivalence mapping A from L into L' is given by a right-multiplication, that is, there exists b E KG such that A(X)

= bR(X)= xb,

X E L.

Proof. The statement that L and L' are equivalent implies that there exists an isomorphism A of L onto L' such that A%L(f)

= '9IL'(f)A,

fE

Let e be a generating idempotent of L. Then x

KG.

EL

implies that

xe = x .

It follows that A(x)

= A(xe) = A('91L(x)e)=

[A'illL(x)]e = ['91L.(x)A]e= %,.(x)(Ae) = %,.(x)b = xb = bR(X),

where b is the image Ae of e under A and b, is the right-multiplication defined by b. (3.14) REMARK. The element b determined above has the property that b equals eb. Furthermore, since b is an element of L', b is equal to be'. Thus the equation b = eb = ebe'

is valid. It is significant that any nonzero element y for which y

= eye'

defines a nonnull right-translation yR of L into L'.

100

2. The Representation Theory of Finite Groups

We turn to the consideration of certain general facts which can be conveniently expressed in terms of the concept of KG-homomorphism. (3.15) LEMMA.Let h be a KG-homomorphism of the minimal left ideal L into the minimal left ideal L’. Then h is either an isomorphism or the null homomorphism. Proof: This is Schur’s lemma (1.13) with a slight variation. Since h E Hom,,(L, L’), / I is also an element of Hom,(L, L’). Furthermore,

/?(ax) = a/7(x),

a

E

x

KG,

E

L,

so that (3.16)

PL(a)x) = sL,(a)(~W)

or (3.17)

[/?%L(a)l(x)= [ f l , W ~ l ( X ) .

Equation (3.17) implies that

’iNL3(a)/7= hSL(a),

a

E

KG.

Since both L and L’ are irreducible spaces for 91, it follows from Schur’s lemma that 17 must be either an isomorphism or the null homomorphism. The second form of Schur’s lemma has the following interesting consequence.

(3.18) LEMMA.Any KG-endomorphism h of a minimal left ideal L into itself is either an automorphism of L or the null endomorphism. ProoJ The proof consists in redefining the terms so as t o apply Lemma ( 1.1 5 ) . (3.19) THEOREM. Let L and L’ be minimal left ideals with generatingidempotents e and e‘ respectively. Then any element exe‘, different from zero, defines an equivalence mapping of L onto L’. Proof. The right-multiplication bR defined by a nonzero element b = exe‘ is not the nullhomomorphism since bR(e) = b. By Lemma (3.15), bR is a KG-isomorphism of L onto L’. It follows that if x E L then (3.20)

[bRflL(f)I~= bR[% ( f B 1 = bR(f~) = (fx)b =f(xb) = f ( b R ~ ) = sL’(f)(bRX) = [%L’(f)bRlxConsequently, bR%(f)

= fl~,(f)bR,

f e KG.

Since b, is not the null mapping, the representations 91, and SLr are equivalent, that is. the minimal left ideals L and L’ are equivalent.

3. The Structure of the Group Algebra KG

101

(3.21) REMARK.It is only asserted in (3.14) and (3.19) that the equivalence can be effected by an element of the form exe‘, not that it necessarily is effected by such. Nevertheless, if b is any element determining such a n equivalence by right-multiplication, then eb is a nonzero element of L’ so that an element of the prescribed form, namely, eb = ebe‘, also defines the same right-multiplication of L onto L‘. Thus we have the following theorem. (3.22) THEOREM. Two minimal left ideals (KG)e and (KG)e’ are equivalent if and only if there exists an element exe‘ different from zero. Every equivalence mapping from (KG)e onto (KG)e’ is given by a right-multiplication defined by such an element. We now present a fundamental theorem on primitive idempotents. (3.23) THEOREM. If e is a primitive idempotent of KG, then exe is a multiple of e for every x E KG. Conversely, if exe is a multiple of the idempotent e for every x in KG, then e is a primitive idempotent. Proof: Every primitive idempotent e is the generator of a minimal left ideal L of the form (KG)e. If exe is null, then it is of the form Oe. Any nonzero element b of the form exe determines a right-multiplication b, of L into L such that

f E KG.

‘ % , ( f ) b= ~ bR’%(f)-

(3.24)

Since !RL is irreducible, it follows from Schur’s lemma that b, must be a multiple of the identity transformation on L. Consequently, In particular,

x(exe) = ~ x , x E L , e(exe)

= exe = Xe,

xEK E

K.

Conversely, let e be an idempotent of KG such that Suppose that with

(e’)2 = e‘,

Then, it follows that Consequently,

exe = Xe, e = e‘

x

E KG.

+ e“,

(e”)’ = e”,

and

e’en= e”e’ = 0.

e’ = ee’e = ze. Xe‘ = X2e = (xe)’

= (e’)’ = e’.

Therefore, the number x must be either 1 or 0. It follows that e‘ either equals e o r zero. Thus e is primitive.

102

2. The Representation Theory of Finite Groups

(3.25) THEOREM. Let the group algebra KG of a finite group G be the direct sum J @ J’ of two-sided ideals, J and J’. Then J and J ’ annihilate each other; that is, x E J, x’ E J’ imply that xx’ = x’x = 0. Furthermore, J and J’ have unique idempotent generators e and e’ respectively, each of which commutes with all the elements of KG.

Proof. The element 1 of KG has the decomposition e + e’, where e belongs to J and e’ belongs to J’. It follows from Theorem (3.1) that xe,

0 = xe’,

x E J,

xf = x’e’,

0 = x’e,

x‘ E J’.

x =ex,

0 = e‘x,

x

x‘ = e‘x‘,

0 = ex’,

x’ E J‘.

x

=

and In the same way, E

J,

and From these relations,

ee’ = e‘e = 0 and

xx’ = xex’ = 0

= x’e’x = x‘x,

whenever x E J and x’ E J’. We note, in particular, that e is a two-sided identity for J and that e’ is a two-sided identity for J‘. Finally, any y E KG can be written x x‘, where x E J and x’ E J’. We have

+

ey

= e(x

+ x’) = ex = xe = (x + x’)e = ye,

so that e is an idempotent in the center of KG. Similarly, e’ is an idempotent in the center. If e and e* are idempotent generators of J, then ee* is e and e*e is e*. Consequently.

e

= ee” = e*e = e*.

Thus e is unique (3.26) THEOREM. Let J be a two-sided ideal and L be a minimal left ideal of the group algebra KG. Then either L n J = L

or

LnJ=(O).

Proof. Since the minimal left ideal L contains the left ideal L n J, it follows that L n J must either be (0) or L itself.

103

3. The Structure of the Group Algebra KG

Thus, if KG is the direct sum J @ J’ of two-sided ideals J and J’, then each minimal left ideal L of KG is either contained in J or in J’. (3.27) THEOREM. If the two-sided ideal J of the group algebra KG contains the minimal left ideal L, then J contains every minimal left ideal L’ equivalent to L. Proof: We have shown that the minimal left ideals L and L’ of KG are equivalent only if there exists a right-multiplication b, of L onto L . Therefore L is contained in the two-sided ideal J iff

L’ = Lb,

b E KG,

is also contained in J. Let the group algebra KG of a finite group G be the direct sum (3.28)

KG=L,@...@L,

of minimal left ideals. Denote by J, the direct sum of all summands of (3.28) which are equivalent to L,, , where Li, is L, ; denote by J2 the direct sum of all summands of (3.28) equivalent to L,, , where Lizis the first summand of (3.28) not occurring in J, ; and, in general, denote by J, the direct sum of all summands which are equivalent to Lis where Li3 is the first summand of (3.28) not occurring in the sum

J, @ - * - @ J , - , . This construction leads to a decomposition of KG as the direct sum KG

= J, @ . - .@ J,

of left ideals J i , 1 I iI m . We wish t o show that each left ideal J i of this decomposition is also a right ideal. Let J, contain the nonzero element x belonging to some minimal left ideal L,, with idempotent generator e,, and let y denote any element of KG. Then (3.28’)

(xe,)y

= xy = x1

+ ... + x k ,

where each x, of (3.28’) belongs t o the summand L, of (3.28). If x, differs from zero, then x(e, ye,) = (xy)e,

= (xl

+

* * *

+ xk)er= x, .

Consequently, e, ye, is different from zero which implies that the minimal left ideal L, is equivalent to the minimal left ideal L, belonging to J, . However, Theorem (3.27) asserts that L, is contained in Ji under these circumstances. Therefore, xy belongs to Ji. Since each element of J, is a sum of elements of the type just considered, it follows that J i is a right and, consequently, a

104

2. The Representation Theory of Finite Groups

two-sided ideal. Any nonzero two-sided ideal J contained in J i must contain at least one minimal left ideal L of J , . By Theorem (3.27), J must contain every minimal left ideal L' equivalent to L and, consequently, must contain J , . Therefore, J, is a minimal two-sided ideal of KG. We are lead to the following theorem. (3.29) THEOREM. The group algebra KG of a finite group G can be decomposed into the direct sum KG = J , @ - * * @ J , of minimal, two-sided ideals in essentially one way. Proof: The argument that such a decomposition exists has been given. Suppose that

KG=J,@.**@J,

(3.30) and

KG

(3.31)

= J,' @ . . . @ J,'

are two direct sum decompositions of the group algebra KG. We wish to show not only that the number of summands must be the same in each of the sums (3.30) and (3.31), but also that the same summands must appear, although perhaps in different order. First, observe that if J and J' are minimal, two-sided ideals, then J n J' is a two-sided ideal contained in each. Therefore, J n J' must either be (0) or else must coincide with J and J'. The multiplicative identity of KG has the expansion l=e,+...+e,,

(3.32)

where e, is an idempotent generator of J i , 1 5 i I k. It is left as a problem to show that each e, is the only idempotent generator of J, and that each e, belongs to the center of KG, 1 i i I k . There is also a decomposition

1 = el' +

. + et',

with each summand e,' the unique, central idempotent generator of J i , 1 5 s 5 t. Since each such e,' can be expressed in the form, e,'

= e,'

1 = es'el + . . . + es'ek,

it follows that, for some e k ,e i e k is a nonzero element of J,' n J,. Conse-

quently, J,' and J, coincide. Therefore, every summand of (3.31) is a summand of (3.30). A similar discussion shows that every summand of (3.30) must be a summand of (3.31), which completes the argu,nent.

We pause to introduce several new concepts. Let A be a finite-dimensional algebra over the field K of complex numbers. An ideal J of the algebra A is

3. The Structure of the Group Algebra KG

105

said to be nilpotent if there exists a positive integer n such that any product of more than n factors from J is always zero. The sum of all the nilpotent left ideals of the algebra A is a left ideal N called the radical of A. It can be shown that N is a two-sided nilpotent ideal of A and that any nilpotent ideal of A is contained in the radical N. (3.33) THEOREM. The radical N of the group algebra KG of a finite group G over the field K of complex numbers consists of zero alone. Proof. The group algebra KG is the direct sum, KG = N O ” , of the radical N and a complementary left ideal N’. The multiplicative identity 1 of the group algebra KG can be expressed as the sum, l=e+e’,

e E N , e’EN’,

where e2

= e,

(el)’

= e’,

and

ee‘ = e’e = 0.

Consequently, the equality en = e

holds for every positive integer n. It follows, since N is nilpotent, that e is zero and that 1 belongs to N’. Every element x of KG is of the form xl, an element of N’. Consequently, the radical N of KG is the zero ideal, as was to be shown. We turn t o another important definition. (3.34) DEFINITION. Let A be an algebra with an identity over the field K of complex numbers. The algebra A is said to be semisimple if and only if (i) the radical N of A is the zero ideal, and (ii) the left ideals of A satisfy the descending chain condition. A semisimple algebra A which contains no nontrivial, two-sided ideals is said t o be simple. In Definition (2.14), each left ideal L of an algebra A over the field K is defined to be a K-subspace of A which is closed under left-multiplication by any element of A. When the algebra A contains a multiplicative identity 1, a second, equally satisfactory definition, is merely that each left ideal L is only a subgroup, closed under left multiplication. In this instance, the set K’ of all elements of A of the form ctl, ct E K, is a subalgebra isomorphic to K. This subalgebra K ’ can be identified with K so that K becomes a subalgebra of A. Under these circumstances, the left ideals become left Ksubspaces which means that the left ideals of the second definition are left ideals according to the first. Since a group algebra KG has dimension [G : 11,

106

2. The Representation Theory of Finite Groups

it follows that both the ascending and descending chain conditions hold for

subspaces and hence for left ideals (using either definition of left ideal). According to Theorem (3.33), the group algebra KG of a finite group G over the field K of complex numbers has zero radical. Thus, such a group algebra KG is a semisimple. In passing, it is useful to remark that this result does not necessarily hold for group algebras over other fields. In the decomposition (3.30), the minimal two-sided ideals; J,,1 I i 5 k ; are simple algebras which are called the simpIe components of KG. The proof of the simplicity of these ideals is left as a problem. 4. THE SIMPLE COMPONENTS OF THE GROUP ALGEBRA KG

The decomposition of the group algebra KG into the direct sum of its simple components has been obtained in Section 3. In Section 4, it is demonstrated that each simple component i s isomorphic to an algebra K, of all n x n complex matrices for some positive integer n. This analysis leads to the result that the group algebra KG of a finite group G over the field K of complex numbers is isomorphic to an algebra A of complex matrices, each of which appears in the same quasi-diagonal form. The number of diagonal blocks in the pattern is equal to the number of minimal, two-sided ideals appearing in the decomposition (3.30). Denote any one of the summands of Eq. (3.30) by the symbol J. Let the sums

(4.1)

J=L,@”*@L,

and

e=e,+.-.+e,

be the decompositions of J and its generating idempotent e according to some set of equivalent minimal left ideals of J. Remember that (4.2)

L, = Je, = (KG)ei,

1 <_ i I k.

Since the minimal left ideals of J are mutually equivalent, there exists a subset of J, B = {bij},

1 _< i, j I k,

such that each b,, , a nonzero element of the form eixej, determines a right multiplication of L, on L,. Furthermore, bii can be taken to be the given e , , 1 5 i 5 k. The set

V

= {eixej : x E

KG}

is a nonzero subspace of L,. The subspace V i s mapped into the onedimensional subspace W

= {e,xei: x E

KG}

107

4. The Simple Components of the Group Algebra KC

of Li by the right multiplication b,, b,Z = zbji,

Z E

KG.

Since b, is an isomorphism of Lj, it follows that V is a one-dimensional subspace of Lj ; that is,

V

(4.3)

= {V = ubij :

E K}.

Let x denote any element in Lj . Then one has the equalities x

(4.4)

= xej = exej = e,xej

+ . + ekxej. *.

This equation can be written in the form

(4.5)

X = 5 1j

b, j

+ + *

’’

lkjbkj,

where the set {tlj , . . . , t k jis } contained in K. Therefore, the set Bj of elements, 1I i I k,

{bij},

constitutes a set of generators of Lj. Suppose that 0

(4.6)

=t,jblj

Equation (4.6) implies that 0

+

’ * *

+

tkjbkj.

= erO = &jb,j,

according t o which, 0 =trj.

(4.7)

Consequently, the set Bj is a linearly independent set of generators of L,, that is, Bj is a basis of Lj. Every element y belonging to J can be written (4.8)

y

= eye = ey(e,

+

. + ek)

=elye, + . . . + e , y e , =?l,bii

+-**+

+“’+?kibkl

elyek+...+e,yek

+ “ ’ + ~ l k b l ~ + “ ‘ + ~ k ~ b k ~ .

This shows that the set B

= B, v

*..

v Bk

is a generating set of J. It is easy to prove that B is linearly independent over K so that it is a K-basis of J. In particular, one notes that the K-dimension of J is k 2 where k is the number of minimal Ieji ideals occurring in any direct sum, decomposition of J into minimal Ieft ideals. It proves possible t o select a set, M

= {eij},

1I i, j

s k,

108

2. The Representation Theory of Finite Groups

such that M is a basis of the minimal ideal J with the multiplicative properties (4.9)

eijers = ajreis,

for all permissible values of i, j , Y, and s. As a matter of fact, there exists a set {vij}of multipliers from K such that e 1.l. . = v..b.. (4.10) IJ I J . We illustrate the procedure, where k is 2, as the first step in an induction argument. In this case,

B

= (bll,

b12 > b212 b,,),

while the following multiplicative identities hold : bllbll = bll, b22b22 = b22 > bllb12 = b12, b22 b21 = b21, b,,b2, = b22 bll = b1,b21 = b2, b12 = bI2bll = b,,b,, = 0.

In addition, we know that b,, b,, = crb,,, where CI E K. Denote ( l / ~ ) b ,by ~ b;, and observe that b,, b;, = bll. Moreover, the equality (4.1 1)

bilbl2 = Xb22

implies that Xb12 = X(b12 b22) = bl,(Xb22) = blA%Ibl2) = (b12b;l)bl2 = bllb12 = b,2 *

Consequently, in Eq. (4.1 l),

x = 1, so that (4.12)

b;, b12 = b22.

Now define the set {e,j), 1 5 i, j I 2, by ell =bll,

e2, = b 2 , ,

e21 =b;,,

e12 = b 1 2

to obtain the set M . Proceed by induction, assuming the construction for the case of k idempotents, 1 I kI n - 1. Consider the decomposition of J into the direct sum

J

= L,

+

+ L,

of the minimal left ideals {Li) with associated idempotents (e,), I I i 5 n. There exists a basis B

= {bij},

1I i,j 5 n,

4. The Simple Components of the Group Algebra KG

109

of J. By the induction hypothesis, the set B' = ( b i j } ,

1

i , j < n - 1,

can be scaled to obtain a set 1 I i , j 5 n - 1, M' = {eij}, having the required multiplicative properties. We wish to scale the elements of the nth row and nth column in order to obtain a larger set

M = (eij>,

(4.13)

1I i, j I n,

so that

e I.J. ers = 6 j.r e i.s for all admissible values of i, j , r, and s. First take en, to be the idempotent generator e, of L, determined in (4.1). Then scale b,, to obtain elements el, = b,n

such that

and

enleln= e n ,

en1

z=

vnlbnl,

elnenl= e l l .

and

Next define elements

e,~ . = y .J n bJ.n ,

1 <j < n,

such that

ejleln= ejn, 1 <j < n.

1 < i < n,

such that

el,eni = e l , ,

Finally, define e, I . = v . b .

1 < i < n.

i < n, by means of these definitions, such Obtain elements eniand e , , 1 I that

(4.14)

1 <j I n,

ejleln = ej,,

elneni= e l , ,

1 I i 5 n.

It now follows that for, 1 5 j < n, 1 I k < n, (4.15)

ejkekn

= ejk(ekleln) = (ejkekl)eln = ejleln = e j n .

Furthermore, for 1 5 k I n, there exists a constant enk ekn

Consequently, Xk

= Xk

=X k

=

*

ekn)

so that xk is 1 for each k. This implies that en, = enkekn,

xk such that

= elk

ekn

1I k I n.

=

7

110

2. The Representation Theory of Finite Groups

In summary, for 1 ~j i n and 1 I k I n,

ejkekn= e j , ,

(4.16)

ejkern= 0,

k#r .

This result extends the construction from the case of n - 1 idempotents to that of n, completing the induction. It follows that there exists a basis M with the multiplicative properties defined by (4.16) for each minimal twosided ideal J in the decomposition of the group algebra KG. Such a basis is referred to as a set of matrix units for J. We introduce a basis { E i j } , 1 i i , j 5 n, of the n x n matrices over the field K of complex numbers, such that the matrix E,, has exactly one nonzero entry, which is 1 at the intersection of the rth row and the 8th column. Let the element in the ith row and j t h column of the matrix A be denoted by a i j . Then A has the unique expansion A

(4.17)

= i,

i

uiiEij.

For the case where n is 2, the explicit expression for the elements of this basis are

A linear transformation T from a finite-dimensional K-space U to a finitedimensional K-space V is completely determined by specifying the images {T(u,)}of a basis {uJ, 1 I i i n, of U. This theorem can be applied, in particular, to the basis {eij}, 1 I i, j I n, of a two-sided ideal J and the basis { E i j } , 1 i i, j i n, of the space Kn. We denote by T the unique linear transformation from J to Kn defined by

T(eij)= E i j ,

(4. IS)

1 i i, j I n.

It follows from the definition of T that T(ejkek,) =

T(ejn)= Ejn = Ejk Ekn = T(t?jk)T(ek,).

Also, if k # r, then

T(ej,ern)= T(0)= 0 = EjkErn= T(eik)T(er,). Therefore, T is a linear transformation of J onto Kn preserving the multiplication of the basis elements. Let

111

4. The Simple Components of ?he Group Algebra KG

Then

C i , j m1 , n tijqmnEijEmn

Hence, Tis a linear transformation of J onto Kn which preserves the operation of multiplication of vectors, that is, T is a homomorphism of J onto K,. However, T is an isomorphism of the vector space J onto the vector space Kn. Consequently, T is an isomorphism of the algebra J onto the algebra Kn of all complex n x n matrices. The extension of this result to the group algebra KG of a finite group G over the field F of complex numbers is now merely a matter of introducing a suitable notation. We use the symbol

A

= A ( l ) 0 . .*

0 A(r)

for the set of all quasi-diagonal matrices of the form

iI r. where each A(i) denotes the algebra of all ni x ni complex matrices, 1 I The set A is a subalgebra of the algebra K, of all cx x ci complex matrices where u = n,

+ + n,.

Let the u x u matrix E j k iconsist entirely of zeros except in the A(i)th block where it has the single nonzero entry 1 at the intersection of the j t h row and kth column. The set of all matrices {Ejki},1 Ij,k I ni, 1I i 5 r, constitutes a basis of the matrix algebra A. Suppose that the group algebra KG of a finite group G over the field of complex numbers has the decomposition KG

= J1 @

0 J'

2. The Rrpresrntution Theory of Finite Groups

112

as the direct sum of simple two-sided ideals, and, furthermore, that each J' has the decomposition

J'=L,'@**.@L,,,'

as the direct sum of minimal left ideals {Lj'}, 1 < j 5 n,, 1 5 i 5 r. According to the preceding discussion, there exists a family (Ti), 1 5 i 5 r, such that each Tiis an isomorphism of the two-sided ideal J' onto the algebra Kni of all n , x n i complex matrices. Every x of KG has the unique decomposition

x

= x1

+

9

.

.

+

XI,

x iE J',

15i

r.

We define a linear transformation T of KG onto A by means of (4.19)

T(x)= T,(x')+ . . . + T,.(x').

This linear transformation T has the properties that (4.18')

T(ejki)= Ejki

and (4.20)

T(ej,'e,,') = EjkiEs,'.

Consequently, T is a K-isomorphism of KG onto A which preserves the multiplicative operation for a basis of KG. It follows that Tis an isomorphism of the algebra KG onto the matrix algebra A . We phrase this final result in the form of the following theorem. (4.21) THEOREM. The group algebra KG of a finite group G over the field K of complex numbers is isomorphic to an algebra A of complex matrices, each of which is a matrix in the same quasi-diagonal form. The number of blocks in the quasi-diagonal form of the matrices of A is equal to the number of simple components of the semisimple algebra KG. The basis {ejk'].1 <,j, k 5 n , , 1 i 5 r, is called a symmetry basis of KG or a s~w7nwtuyadapted basis, since it reveals the symmetry of the group C as reflected in its distinct classes of equivalent, irreducible representations. On the other hand, the basis { g * } , g E G, is called the natural basis of KG. A general solution of the problem of relating effectively the symmetry basis to the natural basis of an arbitrary finite group G has not been obtained. Methods are known for certain special groups, for example, the symmetric group. However, since the results for the symmetric group are too complicated to present in a short space, we turn to other considerations. (4.22) THEOREM. A linear representation T of the group algebra KG on the complex representation space V is faithful if and only if the decomposition of V into irreducible subspaces of T contains a subspace KG-isomorphic to each minimal left ideal of KG.

113

4, The Simple Components of the Group AJgebra KG

Proof. It follows by Theorem (2.33) that if the direct sum decomposition

v = v, 0

*

* . 0 v,

of the representation space V into irreducible subspaces fails to contain a subspace that is KG-isomorphic to some minimal left ideal L of KG, then T is unfaithful. Conversely, suppose that any given minimal left ideal L of KG is KG-isomorphic to some ireducible subspace, say W, of V. Then there exists a KG-isomorphism h of L onto W such that the vector

h(ax) = ah(x) = T(a)h(x),

a E KG, x E L

is zero if and only if the vector ax is zero. If the vector ax is zero for every choice of the minimal left ideal L, then a is zero. Therefore, T is a faithful representation of KG. (4.23) THEOREM. A linear representation T of a finite group G on a complex vector space V of dimension n is irreducible if and only if lm T contains a linearly independent set of n2 linear transformations. Proof. Note that T is an irreducible representation of C if and only if T* is an irreducible representation of the group algebra KG. However, T* is irreducible if and only if V is KC-isomorphic to some minimal left ideal L of KG, that is, if and only if Im T* is isomorphic to K, where n is the dimension of either L or V over K. Since Im T is a generating set of Im T*, it follows that V is irreducible if and only if Im T contains n2 linearly independent linear transformations where n is the dimension of V. (4.24) THEOREM. If T is an irreducible linear representation of a finite group C with an n-dimensional, complex representation space V, then T occurs n times in the regular representation R. Proof. According to Theorem (2.31), every irreducible representation T of the group algebra KG is equivalent to some representation !RL where L is a minimal left ideal of KG. Such a minimal left ideal L of dimension n is contained in a simple, two-sided ideal J which is the direct sum of n minimal left ideals, each of which is KG-isomorphic to L and hence to V. Furthermore, every minimal left ideal L’ equivalent to L is contained in J. This establishes the theorem. The center o f a group G is the subset of G consisting of all elements which commute with every element of G. The center is denoted by C ( C ) ,that is, (4.25)

C(G)

= { z :z E

G, zx

= xz

for x E Cj.

The center of an algebra A is the set, (4.26)

C(A)

= {z: z E

A, zx

= XZ,

x E A}.

114

2. The Representation Theory of Finite Groups

An element z of the center of KG has the form z=

For any g

E

(4.27)

G, one has

c z(t)t 9-1 c z(t)gtg-' c z(g-'tg)t.

z =gzg-1 = g =

c z(t)t.

tcG

(teG

=

teG

tcG

This shows that when z E C(KG), z(g-'tg)

=~

t, g

(t),

E

G.

In different words, if z belongs to the center of KG, then z has the same value for any two conjugate elements of G. The converse is also seen to be true. If an element z has the same value for every conjugate pair of G, then z belongs to the center. The reason is that the equation, g , t E G,

z(gtg-'> = z(t>,

implies that gzg-' = z ,

gEG,

or

gz=zg,

gEG.

Since every element in KG is a linear combination of elements of G, it follows that xz = Z X , XEKG. Thus z belongs to the center of KG. Denote the set of distinct classes of conjugate elements of G by { K l ,. . . , K,.} where the ith class, (4.28)

Kz

= (gl i 5

..

* >

ghi.

il,

contains hi elements. The set of elements zj=g,j+*..+g,,,,j,

I <j
constitutes a set of generators for the center, Since this set is linearly independent, it forms a basis of r elements of C(KG). Next we prove that the dimension of the center coincides with the number of simple, two-sided ideals in the decomposition KG

(4.29)

= J' @

. . . @ J',

of the group algebra KG over the field K of complex numbers. Let x

=

C tik'eik'+ ... + C t i l e i :

i, k

i. k

5. Introduction to Group Characters

115

be an element of the center. For every permissible choice of u, u, and w, the equation

euw"x = xe,," is valid. Consequently, (4.30)

C tw/eo; k

C
i, k i,

i

Now, since the set of the e,: is linearly independent, it follows that all of the previous coefficients must vanish except the pair 5,," and to,"which are equal. It follows that the part xu of x which lies in J" must be of the form

+ + 5,1"enu,n, = 511"eu,

xu =
where e" is the idempotent generator of J". Thus, x in the center implies that (4.31)

x = tle'

+ ... + <,e',

where each e", 1 5 u r, is the idempotent generator of the corresponding simple ideal J". On the other hand, every x of the form of Eq. (4.31) is an element of the center. Hence the set {el, . . . , e'} is a basis of the center. It follows that the number of distinct classes of conjugate elements in G coincides with the number of simple ideals in the decomposition given by Eq. (3.30), which in turn coincides with the number of distinct classes of equivalent irreducible representations of the finite group G. We turn now to another aspect of representation theory which depends upon a new concept, that of the character of a representation. By means of the theory of characters we shall be able to associate with every representation of the finite group G over the field K of complex numbers a unique set of numbers which characterizes the representation. 5. INTRODUCTION TO GROUP CHARACTERS

The concept of the character aflorded by a linear representation T of a finite group G with the representation space V over the field K of complex numbers is defined in Section 5. Orthogonality relations are developed for the characters of such irreducible representations of G . Two representations T and T of G are proved to be equivalent if and only if they afford the same character. A method of deciding the irreducibility of a representation T from the character it affords is described. Illustrations of the basic theorems are obtained by examining the character tables of several groups of low orders. The center C(KG) of the group algebra KG of a finite group G is investigated

2. The Representafion Theory of Finite Groups

116

and the structure constants of C(KG)are introduced. The use of the structure constants in determining the irreducible representations of G is illustrated. Let T be a linear representation of the finite group G on the n-dimensional representation space V over the field K of complex numbers. The character afforded by the representation T is the mapping x from G to the complex numbers such that (5.1)

x(g)

= tr[T(g)l,

9 E G.

Remember that the trace of a transformation T(g) belonging to Hom,(V, V) is the trace of the matrix of T(g) with respect to any basis. It is a familiar fact that this trace is independent of the choice of the basis of V. Also, if A and B are n x n matrices, then A B and BA have the same trace. (5.2) THEOREM. If Ti and T , are equivalent linear representations of the finite group G with representation spaces V, and V,, respectively, then TI and T2 have the same character. Proof. By assumption, there exists an isomorphism h in Hom,(V,, V,) such that g E G. hT1(g)K1 = T,(g), However, it is known from vector space theory that trt~,(g)I= tr[h-lT2(s)hI

= trtT2(g)hh-'I = trtT2(g)I,

which implies XYS)

= x2(9),

9 E G.

The following proof is a variation of that of Theorem (5.2). Let T be any representation of G and let g' and g be conjugate elements of G, that is, let there exist an x E G such that g' = xgx-1.

If

is the character of T, then z(g')

= tr[T(g')] = tr[T(xgx-'>] = tr[T(x)T(g)T(x-')I = tr[T(g)l = x(g).

It follows that the character x of a representation T is a class function on G, that is, a function f whose value at x coincides with its value at y if x is conjugate t o y .

(5.3) THEOREM. Let T be a decomposable linear representation of the finite group G with complex representation space U. Let U be the direct sum V @ W of the reducing subspaces V and W on which T defines the representations T , and T, , respectively. The character x of T is the sum of the characters and x2 of TI and T,, respectively.

117

5. Introduction to Group Characters

Proof. Let the sets {v,, . . . ,vm} and {wl,. . . , wn} be bases of the subspaces V and W, respectively. Their union {vl, . . . , v,, ; w,, . . . , w,,} is a basis of W. Also T(g)v, = Tl(g)vi = uiivi

+ vi',

and T(g)wj = T2(g)wj = pjjwj

+ wj',

1ii I m, 1 < j 2 n,

where the expansions of vi' and wj' in terms of the given bases have no diagonal terms. Consequently,

xk)= @ I + . . . + a m 3 + ( P I I + . . . + b,") = x1(9) + Z2(9)> 1

as was to be shown. Let { K , , ..., K,} denote the distinct classes of conjugate elements of the finite group G. If x and y are elements of K i ,then

x(x> = x(v>

for any character x. This common value of x for elements of the class Ki is denoted by the symbol xi. Since the number of distinct classes of conjugate elements of G is equal to the number of distinct classes of equivalent complex irreducible representations, the characters of these distinct classes of equivalent irreducible representations can be labeled

x1

2

... >xr.

The standard symbols for the values of the character xi of the ith class of irreducible representations are x1

i

i

,... ,XI.,

where xii denotes the value of the character xi for the members of the class K j of conjugate elements, I I j I r. A complete presentation of the characters of the distinct classes of equivalent irreducible representations of the finite group G is made by means of an r x r table, where r is the number of distinct classes of conjugate elements of G. It is worth remarking that this result of a square character table is not necessarily d i d in the case of certain representations over special fields. In the following two theorems, we derive orthogonality relations among the rows and columns of the character tables of irreducible representations of finite groups over the-field of complex nunzbers.

(5.4) THEOREM. Let T and S be nonequivalent, irreducible linear representations of the finite group G with complex representations spaces U and V. Let {u,}, 1 I i I nz, and {vi}, 1 I iI n , be bases of U and V, respectively. For g E G, let {bji(g)} and {uji(g)} be the matrices of T(g)and S(g) with respect

I18

2. The Representation Theory of Finite Groups

to the given bases. The following equality holds for all permissible values of i. j , k , and r .

c ajk(g)br'(g

- 1)

= 0.

qsG

Proof. By assumption, for g E G, S(g) belongs to GL(V) and T(g) belongs to GL(U). Let C be any element of Hom,(U, V) a n d denote by P the element of Hom,(U, V) defined by

P

(5.6)

=

1S(g)cT(g-').

qEG

For each h E G,

=

c S(q)CT(g

qtG

-117)

=

qeG

S(g)cT(g- ')T(I7)

= PT(11).

From Schur's lemma, it follows that P must be the null homomorphism of U into V. We introduce a family C(i,,j), 1 5 i 2 rn, 1 2.i 2 n , of linear transformations of Hom,(U, V). Each linear transformation C(i,j) is defined by (5.8)

C(i, j)u, = v j

and

C(i,j)uk= 0,

i # k.

Select a n arbitrary member C ( i , j ) of this set to define a particular P by Eq. (5.6). Then it follows that

Since the set { v ~ }1. 5 k I n , is linearly independent, it follows that

(5.9)

c a,"g)b,'(g-')

=0

yf=G

for all permissible choices of i, j , k , and r, as was t o be shown.

(5.10) THEORIYM. Let S be an irreducible linear representation of the finite group G on the n-dimensional complex representation space U with basis {ui). 15 i < 1 1 . If ( a j ' ( g ) }denotes the matrix of S(g), the following equation holds (5.11)

119

5. Introduction to Group Characters

Proof. Let C ( i , j )denote the element of Hom,(U, U) defined by

C(i,j)u;= u j

Since

p

C(i,j)ur

and =

i # r.

= 0,

c %)c(m%7-1)

9eG

commutes with each element S(go),g o E G, by the second form of Schur's lemma,

where 1, denotes the identity operator on U. Consequently, (5.12) ciju, =

9eG

S(g)C(i,j)S(g-')u, =

2 S(g)C(i,j) 21 a,'(g-')u,

9eG

t=

It follows from the linear independence of the set {ui}, 1 5 i 5 n, that

c ajf(g)as'(g-

9 EG

I)

= askij.

Take t equal to s and sum on the index s so that

c i:a,yg

g€G

s=1

-')aj"(g) = ncjj .

Hence

(5.13)

so that

cij = Sji{[G: 11,'~). Finally,

(5.14)

1 aj'(g)a,'(g-')

gtG

= d,'dj'[G : l]/n,

which completes the argument. Denote by xu and xu the characters of the irreducible representations S and T of Theorem (5.4). Consider the special case of Eq. (5.5) obtained by taking r equal to i, j equal to k so that (5.15)

1 a , k ( g ) b , ' ( f ' )= 0.

9s G

120

2. The Representation Theory of Finite Groups

By summing on k and i i n Eq. (5.15), one obtains

c x"(s>x"(s '> -

gtC

= 0,

whenever x" and xu are characters of nonequivalent irreducible representations of the finite group G. A similar summing, using the results of Theorem (5.10), yields the result that

(5.16) where x" is the character of an irreducible linear representation S of the finite group G. These results may be summarized in a single equation

(5.17) where xu and x" are characters of the irreducible representations T" and T" of the finite group G. A linear representation T of a finite group G on an inner-product space V is said to be a unitary representatiot? if and only if T(g) is a unitary transformation for every g E G. When T is a unitary representation, it follows that T(g-')

=

[m)l-'= [Ug)l*.

Since the matrices of T(g) and [T(g)]* with respect to a unitary basis of V are conjugate transposes, one has (5.18)

x(s-I) = tr[T(g-')l

= tr[(T(g))*l = X(g).

There is a theorem, not established in this book, that every linear representation T of a finitc group G with a finite-dimensional representation space V over the field of complex numbers is equivalent to a unitary representation. Consequently, (5.18') for all such representations. This result implies that Eq. (5.17) can be written in the form (5.19)

c x"(g)j"(g)

9EG

= 6,"[G : 11.

Since each conjugacy class Ki contains h i elements for which the values of the character coincide, Eq. (5.19) implies that (5.20)

121

5 . Introduction to Group Characters

The character table of a finite group G with r distinct classes of conjugate elements xi1

*-.

x2l

x12

~2~

xlr

xzr

*..

* * .

xrl

xr2

xi

can be regarded as an r x r matrix C with elements (xji>.Let D be the matrix whose elements {dj'}are given by d.' = h i X j .

By Eq. (5.20), D is equal to [G : l ]C-l , that is, C D = [G : l ] I r , where I, is the r x r identity matrix. This means, of course, that

D C = [G : 111,. a result which can be stated in the form r

(5.21)

or (5.21')

r

. .

11 h,x,'j,J

j=

= 6,"[G

: 11.

The complete reducibility of a representation T of a finite group G over the field K of complex numbers together with Eq. (5.17) permit the identification of the equivalence class of such a representation T merely from the character it affords. To be specific, by means of Theorem (1.45), T can be written uniquely as (5.22)

T = a, TI 0 * . . @ a, T',

where the set { T I , . . . , T') contains exactly one representative from each of the r distinct classes of equivalent irreducible representations of G while the integers {u1,. . ., a,) indicate the number of times each representation occurs in the decomposition (5.22). The representation T is equivalent to any other representation T' in which these irreducible summands appear the same number of times. Moreover, the numbers (a,} can be determined from the character x of T i n the following manner. From the natural extension of Theorem (5.3), the character of T is given by (5.23)

,

x = alxl + . . . + arxr = C aj xj. j = 1

122

2. The Representation Theory of Finite Groups

Form the expression, (5.24)

c X(S)%"(S) c c ajxj(s)x"(s) =

SEG

gtC

= a,[G : 11.

j=1

Thus we find that (5.25) Consequently, the number of times that the irreducible representation T" occurs in the representation T is completely determined by the character of T. This result together with Theorem (5.2) leads to the following theorem. (5.26) THEOREM. Two linear representations T and T' of the finite group G over the field of complex numbers are equivalent if and only if the characters x and x' of the respective representations coincide. We continue with the observation that Eqs. (5.23) and (5.24) imply (5.27) Thus, in conjunction with Theorem (5.26), one obtains the following theorem. A linear representation T of the finite group G over the (5.28) THEOREM. field K of complex numbers is irreducible whenever the right-hand side of Eq. (5.27) is equal to [C : 11 and reducible otherwise. Equations (5.24) and (5.27) can be written

C hi xi j i u = a,[G : 11,

i= 1

(5.29)

i, l ? i ) ( i j i

i= 1

=

[G: l](a,2 + . . . + a;).

Now let T be a linear representation of the finite group G on the n-dimensional representation space V over the field K of complex numbers. We summarize a number of facts about T and its character x. The image T(l)of the group identity 1 of G is the identity 1" of Hom,(V, V) so that

~ ( 1= ) x, = tr[l,]

(5.30)

= n.

Consequently, the first column of the character table X lists the dimensions r distinct classes of equivalent irreducible representations of G. By convention, the first class K , of conjugate elements of a finite group G consicts of the identity alone so that h , is 1. Thus, by Eq. (5.21'),

i n , . . . . , n,) of the

(5.31)

n12

+ . . . + nr2 = (x,')' + . . . + (x,~)' = [ G : 11,

a result previously noted. Let 91 denote the regular representation of a finite group G. The equation %(s)t = st, S, t E G,

123

5. Introduction to Group Characters

implies that the matrix of %(s) with respect to the natural basis has no diagonal entries, except in the case of the identity element of G. Therefore, if x denotes the character of the regular representation, then

(5.32)

X(S) =

0, s # 1,

~ ( 1= ) [ G : 11.

Applying the first of Eqs. (5.29) to the regular representation, one obtains

(5.33)

a,[G : 13 =

1 h i x i j i U= h , X , j I u= n,[G : 13,

i= 1

so that the number a, of copies of the irreducible representation T" in the regular representation % is equal to the dimension of the representation T", a result obtained in Section 4.

(5.34) EXAMPLES. One must have available either some examples of irreducible characters or elementary methods of determining them in order to illustrate the application of these results to the calculation of character tables. To this end, note that every finite group G has a class of irreducible representations, each of which makes every element of G correspond to the identity transformation 1 of some one-dimensional vector space V. This class of irreducible representations is called the 1 -representation and the character it affords the 1-character orprincipal character of G. The 1-character, of course, makes every element of G correspond to the number 1. Each irreducible representation T of a factor group G / H of G modulo one of its normal subgroups H determines .an irreducible representation T of G. More generally, let h be a homomorphism of G onto G'. Then every irreducible representation T' of G' with representation space V determines an irreducible representation T of G since the composition T' h is a homomorphism T of G into GL(V). An interesting special case arises when H is a subgroup of index two of the finite group G. Then G / H is a cyclic group of order two generated by some left coset g H of H . Consequently, G / H has two distinct classes of equivalent irreducible representations '%I and '3' where 93' makes gH correspond to 1 and 93' makes it correspond to - 1. These give rise to two irreducible representations of G, 0

T'

= %'

o

v

and

T 2 = 912 v, 0

where v denotes the natural homomorphism of G onto G/H. T' is merely the I-representation back again. On the other hand, T 2 is an irreducible representation said to belong to the normal subgroup H . It is also called an alternating representation since its character takes only the two values, 1 for elements of H and - 1 for elements of the other left coset of H . The most famous example is that of the alternating subgroup A, of the symmetric group S , . Not uncommonly, a group G may have several different subgroups of index two, each of which gives rise to an alternating representation. This is the case of C , @ C , which is discussed below. The method is also

124

2. The Representation Theory of Finite Groups

useful when the factor group G / H is cyclic or, more generally, abelian. Naturally, whenever the irreducible representations of the factor group G / H are known, the structure of G / H is a matter of indifference. We begin with the application of these ideas to the group S , whose Cayley table is shown in Table (5.35). The group S , has three distinct classes of (5.35)

CAYLEY TABLEOF S3 123456 1123456 2231645 3312564 4456123 5564312 6645231

conjugate elements: K , = {1} with 11, = 1 , K, = (2, 3) with h, = 2, and K, = {4, 5 , 6) with / I ,= 3. Consequently, the number of distinct classes of equivalent irreducible representations is three so that the character table of S , is a 3 x 3 matrix. Since the subset (1, 2, 3) is a subgroup H of index two of the group S , , the I-character and the alternating character provide the first t w o rows 1 1 1 1 1 -1 of the character table, leaving a third row x, y , z to be determined. Equation (5.31) implies that 1’ + 1’ + X’ = 6, so that x has the value 2. Equation (5.21) then gives l(1)

+ l(1) + 2y = 0

and

l(1)

+ 1(-1) + 22 = 0.

It follows that the character table of S , has the form of Table (5.36). (5.36)

CHARACTER TABLEOF S3

The character table of C,@ C, is needed to apply these methods to the quaternion group Q, one of the two distinct classes of isomorphic, nonabelian groups of order eight. The Cayley table of C, @ C,, sometimes

125

5. Introduction to Group Characters

called the Viergruppe, is shown in Table (5.37). Each element other than 1

(5.37)

CAYLEY

TABLE OF

cz 0cz

1234 11234 22143 33412 44321

has order two, so that C, @ C2 contains three distinct normal subgroups of index two, namely,

H2 = (1, 2},

H , = { I , 3},

and

H4 = { I , 4).

In addition to the 1-representation, there are three different alternating representations defined by the three normal subgroups. Denote the characters of these representations by x 2 , x3, and x4 respectively. Since C, @ C, has four classes of conjugate elements, it follows that these four characters constitute a full set for the distinct classes of equivalent irreducible representations. The character table of the group is shown in Table (5.38). With this

(5.38)

CHARACTER

TABLEOF cz 9c z

KI Kz K3 & 1 1 1 x 2 1 1 - 1 -1 x 3 1 -1 1-1 x4 1 - 1 -1 1 X I 1

table available, we turn to the analysis of the quarternion group Q whose Cayley table is shown in Table (5.39).

(5.39)

CAYLEY TABLE OF

THE

QUATERNION GROUPQ

12345678 112345678 223416185 334127856 441238567 558763214 665874321 776581432 887652143

126

2. The Representation Theory of Finite Groups

According to its Cayley table, the group Q contains a subset ( 1 , 3} constituting a normal subgroup H whose left cosets in G are

H

= {I,

3}, 2H = {2,4], 5 H

= (5,

7}, and

6H

= (6, S}.

The Cayley Table of Q / H is shown in Table (5.40).

(5.40)

CAYLEY TABLEOF QIH 1H 1H 1H 2H 2H 5H 5H 6 H 6H

2H 2H 1H 6H 5H

5H 5H 6H 1H 2H

6H 6H 5H 2H IH

A comparison of (5.40) with (5.37) shows that the mapping f from the factor group Q / H to the Viergruppe defined by f { l , 3) = 1 , f(2, 4) = 2, f { 5 , 7) = 3, and f(6, 8) = 4 is an isomorphism. If S',%', !R3, and 914 are nonequivalent irreducible representations of the Viergruppe and v is the natural homomorphism of Q onto Q / H , then the compositions

T'

=

s'

of

3

v , T 2 = '%*

of o

v, T 3 = S 3 o fa v ,

and

T4= S4o f a v,

are nonequivalent irreducible representations of the quaternion group Q. These define corresponding characters x', x', x3, and x4. The character table of the quaternion group Q is shown in Table (5.41), where the entries for (5.41)

CHARACTER TABLE GROUPQ QUATERNION 1 2 3 1 1 x ' l x2 1 1 1 x3 1-1 1x4 1 - 1 1x 5 2 0--2

4 5 6 7 8 1 1 1 1 1 1 - 1 -1 -1 -1 1 1-1 1-1 1 -1 1-1 1 0 0 0 0 0

characters xl,. . . , x4 were made from the Table (5.38) while those of xs were determined afterwards by (5.3 1) and the orthogonality relations. The character table gives the value of x i for each element of the group Q rather than merely for the classes, a more lengthy presentation convenient for some purposes. This completes the discussion of the most basic methods of finding the character tables of a small group. The next procedure to be discussed is applicable in theory to the computation of the character table of any finite group. Its practical application is

127

5. Introduction to Group Characters

limited by the complicated matrix calculations involved. To begin the discussion, we recall that a subset B of an algebra A is a subalgebra of A if B satisfies the axioms of an algebra under the operations and with the field of A. In other words, B is a subspace of the vector space A such that xy E B whenever x, y E B. Although an ideal J of an algebra A is necessarily a subalgebra, a subalgebra B of A is not necessarily an ideal. (5.42) EXAMPLE. The set B of all matrices of the form

I1 a", I

is a subalgebra of the algebra A of all complex 2 x 2 matrices. Nevertheless, B is not an ideal of A, since neither

nor

belong to B whenever a,, is different from zero. We observed in the discussion preceding Eq. (4.29) that the set of elements

(5.43)

zj=glJ+*-.+gh,,J, 1< j < r ,

is a basis of the center C(KG), a subspace of KG. Although the center C(KG) is not an ideal of KG, it is easy to see that the product xy E C(KG) whenever x, y E C(KG) so that C(KG) is a subalgebra of KG. In particular, the product z,z, of two basis elements is a linear combination (5.44)

z,z, = C,"1Z,

+ . . . + c,,,z,

of the basis elements. Since each factor of the product is a linear combination of g i , 1 5 i < [G : 13, with nonnegative integer coefficients, it follows that each such g i appears on the right of (5.44) with a nonnegative integer coefficient and, consequently, so must each z k . The set (5.45)

{c,,,},

1 < u, u, u' 5 r,

of coefficients consists of nonnegative integers which are called the structure constants of the algebra C(KG). To relate these results to earlier ones, recaIl that the group algebra KG decomposes into the direct sum (5.46)

KG=J'@..*@J'

128

2. The Representation Theory of Finite Groups

of its simple components in essentially one way. Each class of irreducible representations of KG contains a member !RL which is a representation induced by the regular representation 93 on some minimal left ideal L contained in the simple component J" of KG. The ideal L is equivalent to an ideal L," which occurs as a summand in the decomposition of J" according to the symmetry basis {e","}, 1 5 u, w I n u , 1 I u 5 r, of KG. The representation 91L,,,,,belongs to the same class of irreducible representations as SL. We are able to obtain the matrix representation afforded by 'iRLWUwith little difficulty. Note that if (5.47)

=

C tj,"ej,", j

1 Iv 5 nu.

Therefore, the matrix representation M afforded by %,+, x of KG correspond to (5.49)

M(x) = { t j / } ,

makes the element

1 Ij , u I nu.

The elements of M ( x ) are the components of x with respect to the symmetry basis of J". Thus every class of irreducible representations contains a member !XLV,<which affords a matrix representation M making each element x of KG correspond to its components with respect to a symmetry adapted basis of J". This class of representations is said to belong to the simple component J". The elements

(5.50)

ei = e l l i

+ . . . + eninii,

1I iI r,

also form a basis of the center of KG. Consequently, there exist matrices {aj'} and { x j i ) such that (5.51)

ei =

(5.52)

z, =

1 ccj'zj, I

.

j = 1

r

C w,,jej,

j= 1

I ii s r ,

1 i u I r.

The component of the element z, in the ideal J k is (5.53)

z,ek = w,kek,

129

5. Introduction to Group Characters

as can be seen from (5.52). The image of z, in the matrix representation M determined by Jk is the matrix w,kInk

5

where In,denotes the nk x nk identity matrix. It follows that the character xk has the value (5.54) where

x"zJ

k

= x1 w ,

=

k

3

xlkis n k , the degree of the matrix representation

(5.55)

XIkw,"

= xk(zu>= Xk(glu

+ . + g h , , u) * *

M . Finally,

= huXk(g),

where g belongs to the class K, . Thus one obtains (5.56)

x,"

= Xk(S) = xlkw,k/hu.

We observed in passing that

z,ek = w ,kek , which means that the left-translation zuL,determined by z,, has ek as an eigenvector with the corresponding eigenvalue ~ ~ ( z , ) / n ,The . eigenvalues and eigenvectors of zULcan be computed from its matrix {c",,,,,}, 1 5 u, w 5 r, of structure constants with respect t o the basis {zl, . . . , z,}. When a given eigenvalue w," has a one-dimensional eigenspace U,", an eigenvector Y corresponding to w," establishes by (5.53) and (5.56) the kth row of the character table up to a scale factor. This factor is determined by the orthogonality relations for the rows of the character table. Such an immediate solution is frequently thwarted by the fact that some of the eigenspaces of any particular zUL have dimension greater than one. Let the left transIation zULcorresponding to

(5.57)

z,

= Ute'

+ + wie' * *

have an eigenspace U of dimension s greater than one. Then U has a basis {eil,.. ., eis} of central idempotents, each of which appears in the expression (5.57) with the same coefficient mui, which is the eigenvalue corresponding to U. It is clear that the left translation zL defined by most linear combinations of the form

(5.57')

z=alel +-**+are'

does not have repeated eigenvalues. This implies, of course, that most left translations zL defined by linear combinations of the form (5.57")

z = blz,

+

'

*.

+ b'z,

130

2. The Representation Theory of Finite Groups

do not have repeated eigenvalues. The eigenvalues of the left translation zL defined by (5.57") are given by w 1 = b 'o , '

+ ... + b'o;

or= blw*'

+ . . . + b'w,'.

(5.58)

The matrix { w j i }of the coefficients on the right-hand side of (5.58) is . nonsingular since it represents a change of basis. It follows that a suitable choice of { b ' , . . . , b') will lead to any prescribed set of values of {o',. . ., or} of the eigenvalues of z L . As a practical matter, one can take linear combinations of the matrices {c,,,), 1 5 v, WJ i r, 1 5 u 5 r, in order to obtain a matrix with distinct eigenvalues whose eigenvectors determine, up to a scale factor, the central idempotents { e l , . . ., er} and consequently the character table of the group. Another approach is to introduce a set of indeterminates { t ' , . . . , t'} in order to determine a family {zL(t', . . . , t')) of linear transformations which satisfy the eigenvalue equations

(5.59)

z , ( t ' , . . . , tr)ek=

(

C tlZi i

.

1

ek =

C t'o? ek = w ( t ' , . . ., t')ek.

( i

To develop the component form of (5.59), one expands ek in terms of the basis { z n I ]1, I nz 5 r , obtaining

By comparison of coefficients of z,,, , one finds (5.61) or

(5.62) The deterrninantal function, (5.633

( 6 , ' ~-

1 ticijmI =f(w;t ' , . . . , tr), I

5. Introduction to Group Characters

131

is a homogeneous polynomial in the r + 1 variables. w, {ti}, 1 I i 2 r. For suitable specializations {b', .. . , b'} of the indeterminates {t', .. ., t'}, the set of equations (5.62) has nontrivial solutions, namely, the coefficients of the set of vectors {ek}, 1 5 k S r, with respect to the basis {z"}, 1 s u s r , of the center C(KG). Furthermore, for these suitable specializations, the corresponding roots (mi}, 1 5 i S r, off(w) are distinct and given by (5.58). Thus one finds, in such instances, that (5.64)

f(w)= (O - a'). (O - o r )= [W - (blw,' + . . . + b'w,')] x [o- (b'o,2 . . . + b'o,2)]. * . [w - (b'w,' + . . . + b'o;)]. '

4

+

From continuity, it follows that this factorization is valid for an infinite number of distinct choices of each of the b', 1 2 i 2 r. Consequently, the factorization given by (5.64) is valid when the set {b'} of constants is replaced by the set {ti) of indeterminates. Finally, (5.65) f(w,t',

. . . ,t') = I d , , , j ~- 1 t i c j i m ( i

+ ' . + t'o,')]

= [O- (t'o,'

1

+ . . . + t'Q,2)] - . -

x [o- ( t ' o , 2

+ ... + t'o;)]

x [o- (t'w,'

is the complete factorization. To fix the ideas, we apply these considerations to the symmetric group S3 whose Cayley table is (5.35) and whose distinct classes of conjugate elements are K , = (11, K2 = (2, 31, and K , = (4, 5,6). The basis {zl,z 2 , z,) is defined by z1 = I, z2 = 2

(5.66)

+ 3,

and

23

=4

+ 5 + 6.

Note that (5.67) z2 2,

+ 3)(4 + 5 + 6) = 2(4) + 2(5) + 2(6) + 3(4) + 3(5) + 3(6) = (2

=6+4+5+5+6+4 = 22, = OZ, OZZ 22,

+

+

+

= ~ 2 3 1 ~ 1~ 2 3 ~2 2

+

c233z3.

Thus one finds that

c Z 3 ,= 0, Furthermore,

~ 2 3 = 2

0,

and

~ 2 3 = 3

2.

132

2. The Representation Theory of Finite Groups

since C(KG) is commutative. An easy computation gives the complete collection of structure constants, conveniently arranged,

(5.68)

These structure constants give rise to a determinantal function (5.69)

-t 3

-t2

f(w; t i , t2, t3) =

-2t3

w - t' - 2 t 2

-3t3

I

By adding the second and third rows to the first, one finds that (5.70) f ( w ; t', t2, t3)

+ 3t3)

w

- (tl

+ 2 t 2 + 3t3)

0 - (t'

0 - t' - 1 2

-3t3

+ 2 t 2 + 3t3)

-2t3

w - t' - 2 t 2

from which it follows that

(5.71) f ( w ; t ' ,

t 2 ,t 3 )

= [w - (t'

+ 2 t 2 + 3t3)]

1

-2t2 1 - 3 ~

= [w

-

(t'

+ 2 t 2 + 3t3)]

= [w

-

(t'

+ 2t2 + 3t3)][w - (t'

1 0 - t' -

-3t3

1

- t2)][w - (t'

+ t y 2 ) + r3(3), m2 = t ' ( 1 ) + t2(2) + w3 = t ' ( ~ + ) t 2 ( - 1) + t3(0). w' = t'(1)

Thus the matrix (wj'}, 1 (5.73)

t3(-3),

i, j

3, assumes the form

1

-2t3

0 - t l - 2t2

1

1

2t2 - 2t 3 w - t' - 2 t 2 3 t 3

The eigenvalues are given by the following expressions : (5.72)

t2

+

+ 2 t 2 - 3t3)].

I33

5. Introduction to Group Characters

In order t o compute the character matrix of S , , it is necessary to compute the values of xI1, x12, and x13, which equal the dimensions of the distinct classes of equivalent irreducible representations. From Eqs. (5.56), it follows that

c

(5.74)

hu

U

Xh,"= c I X I k I 2%%Lvu) U

or [G : 11 =

(5.75)

I xlk I

1w,k(Sij:/h,).

Finally,

I Xtk I

(5.76)

= tG : 11/[

By means of (5.76), we find that (5.77)

(xll)' ')IX(

U

c ~,k(W,k/hu)l.

+ 2(2/2) + 3(3/3)] = 1, = 6/[1(1/l) + 2(2/2) - 3(-3/3)] = 1,

= 6/[1(1/1)

(xI3)' = 6/[1(1/1) - I(-

1/2)] = 4.

Since their values are positive integers, one finds

xI1 = 1 ,

xI2 = 1, and xl 3 = 2. xI1, x12, and xI3, one can compute

Given these values of the character table by (5.56). For example, one obtains (5.78)

xll = 1(1/1) = 1 ,

xZ1= 1(2/2) = 1,

and

the entries of

x31 =

1(3/3) = 1.

Continuing this procedure, one finds the character table of S , to be as shown in Table (5.79). (5.79)

CHARACTER TABLE OF s 3

Table (5.79) agrees with Table (5.36) which was calculated by other methods. It is clear that practical use of this method depends upon the development of computer programs to carry out the details, otherwise the calcdation is too burdensome for any except small groups. An interesting by-product of this method is an expression for the central idempotents {el, . . ., er} in terms of the basis {zl, . . . , 2,) and the character

134

2. The Representation Theory of Finite Groups

table {zi'}, 1 I i, ,j I r . The matrix {oji}, 1 5 i, j 5 r , can be obtained imi, j I r, mediately from the character table. However, the matrix {aji}, 1 I such that

ei =

(5.51)

1 cxjizj

j =1

is the inverse of the matrix {wji}, more precisely, the transpose of (cxji> is the inverse of {coji}. The calculation can be done explicitly because of the orthogonality relations. Equation (5.52) can be written (5.80) by means of (5.56). This implies that (5.81)

1 j i z , = 1 :j

u= 1

u=l

=

Finally,

j=l

[G : 11

h,xi(e'/z,j)

=

c c /iux~j,"(ej/,ylj)

j=1 u=l

2 Gj(ej/xlj) = [G : 11 1 GJ(ej/X,j) = [G : l ] ( e k / ~ l k ) .

j= I

j =1

(5.82)

so that (5.83)

a;

= Xlkj,"/[G : 11.

This completes Chapter 2. Additional methods and results on representation theory can be found in Chapters 3 and 4 which are concerned with applications and examples. PROBLEMS

1. Let , j ' : G G' be a homomorphism of the cyclic group G = (x) generated by .Y. Show that f is completely determined by its valuef(x). --f

2.

Let j ' : G G' and 11: G' + G" be group homomorphisms. Show that is a group homomorphism. --f

/ I : , j ' : G + G"

3. Let G be the external direct product H @ K and let f :H + H' and h : K K' be group homomorphisms. Show that the m a p 2 : C + H' @ K' defined by i(x, y) = ( , f ( x ) h(y)) . is a group homomorphism. --f

Recall that the d e r i w d g r m p G' of a group G is the subgroup generated by all romnirtator,c .'i-'J--',YJ. formed from pairs {x,y} c G.

(a) Prove that the derived group G' is a normal subgroup of G. (b) Prove that the factor group G/G' is abelian.

4.

5. Let f : G - + A be a homomorphism of the group G into an abelian group A . Show that the derived group G' is contained in the kernel off.

135

Problem

6. Every one-dimensional complex representation T of a group G is basically a homomorphism of G into the multiplicative group K* of nonzero complex numbers. By Problem 5, the kernel of T contains the derived group G'. (a) Prove that T defines a representation of the factor group G/G'. (b) Prove that every irreducible representation of G/G' determines a one-dimensional representation of G. (c) Argue that all one-dimensional complex representations of G arise from those of G/G' in this manner.

7. The subgroup G' = {1,4, 5} is the derived group of the group G of order twelve with the following Cayley table. Using the ideas of Problem 6, find the one-dimensional complex representations of G. 1 2 3 4 5 6 7 8 9101112 2 1 4 3 6 5 8 710 91211 3 4 5 6 2 11211 7 8 910 4 3 6 5 1 2 1 1 1 2 8 710 9 5 6 2 1 4 310 91211 7 8 6 5 1 2 3 4 9101112 8 7 7 8 9101112 2 1 4 3 6 5 8 710 91211 1 2 3 4 5 6 9101112 8 7 5 6 2 1 4 3 10 9 1 2 1 1 7 8 6 5 1 2 3 4 1112 8 7 1 0 9 3 4 5 6 2 1 1211 7 8 9 1 0 4 3 6 5 1 2

8. Let S, denote the group of permutations on a set A = (a, b, c> [see (2.9)]. Let B = (va, v b , vc} denote a basis of the vector space V. For each g E S,, define T(g):V -+ V by T(g)v, = Show that T defines a representation of S , . 9. Let G denote a subgroup of GL(V).Suppose that W is a proper subspace of V such that g(W) c W for g E G. Show that the correspondence T : G + Hom(W, W), given by T(g) = g I W is a representation of G with representation space W. 10. Let / I : G G" be a homomorphism of the finite group G into the group G". Let x E G have order n. Show that h(x) has order dividing n. -+

11. Let T : G GL(V) be a complex representation of the finite group G. Let x E G. Show that the matrix of T(x) can be reduced to diagonal form, that is, T(x) is a semisimple linear transformation. -+

Look up the concept of a group being presented by generators and relations, for example, Rotman (1965) or Coxeter and Moser (1965).

2. The Representation Theory of Finite Groups

136

12. Let a group G be given by the generators {x,,. . . , x,) and the relations ym(x,,. , . , x,) = 1 (m = 1, . . . , t ) . Show that a map h : C + G from the group G into the group can be defined in such a way that h is a homomorphism iff the images { y , , . . . ,y,}, y i = h(xi), satisfy the relations y m ( y l , .. . ,y,) = i (m = 1, . . . , t ) . Note this result holds in particular for the case of representations. 13. The Cayley table of the symmetric group S , is given in (2.10). S , is generated by R, and R, which satisfy the relations RZ3= R , = 1 ; R’, = R , = 1 ; and R2 R, R, R, = R , = 1. Verify that these elements satisfy the given relations. Let

Show that T(R,) and T(R,) satisfy the same relations as R, and R,. Conclude that there exists a homomorphism T : S , --t GL,(K) determined by this correspondence. Work out the representation T for all elements of S , . Rernurk {*). The representation of a finite group G on its group algebra KG is a special case of a more general construction. Let H be a subgroup of G with index [G : H ] = 17. Let V denote the complex n-dimensional vector space with basis B = {vl = l H , v2 = x2H, . . . , v, = x , H } consisting of the distinct left cosets of H in G. Define a representation T : G -+GL(V) such that T(y) : V -+V is defined on the basis B by T(g)vi= T(g)x,H = gxi H for g E G.

14. Verify that the procedure defined above determines an n-dimensional representation T of G. 15. Find the kernel of the representation T of Problem 14. 16. Let G denote the group of order twelve whose Cayley table is given in Problem 7: (a) Verify that the conjugacy classes of this group are K , = {I}, K2 = {2), K , = (3, 6}, K4 = {4, S}, K5 = 17, 10, 111, and K6 = (8, 9, 12}, (b) Verify that N = (1, 2, 3,4, 5, 6) is a normal subgroup of C. (c) Verify that H = (1, 4, 5 ) and A4 = (1,2) are normal subgroups of G such that G / H is cyclic of order four and G / M is isomorphic to S , . 17. Make use of the information in Problem 16 to determine five nonequivalent irreducible representations of G. 18.

8H

Let H

= (8.

= ( I , 4, 5 ) = v , . 2 H = (2, 3, 6) = v , , 7 H = f7, 10, 11) = v , , and 9, 12) = v 4 . Use the information of Remark (*) to determine a

137

Problems

four-dimensional representation of G on the space V with basis {vl, v 2 , vg , v4).

An idempotent I in the ring K, of all complex n x n matrices is an n x n matrix I such that Z2 = I. 19. Find the set of all idempotents in the ring K2 of all complex 2 x 2 matrices. Use this result to describe the set of all minimal left ideals in K 2 .

20. Use the ideas of Problem 19 to describe the set of all proper left ideals in K,,. 21. Let A denote the algebra of all complex 3 x 3 matrices of the form

1:

a13

ao,, 211.

Prove the set of matrices of the form

constitute the radical R of A.

(1

a13

0 0

a23

0

Show that in the decomposition (3.32), e, is the only idempotent generator of the simple component Ji . 23. Show that if A is an algebra with identity over the complex numbers K then a left (right, two-sided) ideal J is merely a subgroup of the additive group of A which is closed under left (right, two-sided) multiplication. 22.

24. Show that the A.C.C. and D.C.C. holds for ideals of a complex finitedimensional algebra A with identity.

25. Show that any complex finite-dimensional algebra A with identity contains a radical N which is a two-sided ideal containing every nilpotent ideal of A.

Prove that the two-sided ideals Ji contained in the decomposition (3.30) are simple.

26.

27. Complete the details of the scaling of ei, and eni in the determination of the matrix units of the simple components Ji . 28. Show that the trace of a linear transformation T is independent of the choice of the matrix of T.

29. Show that the tr(AB) A and B.

= tr(BA)

for any two complex n x n matrices

2. The Representation Theory of Finite Groups

138

30. The group G of symmetries of the square is the set of all rotations of three-space about the origin which carry the square into itself. These rotations

can be described in ternis of the permutations they effect on the vertices of the square. These are given by

- (;;:)’

’ ) : ; :3! (

1234 1234 - (3412)’ - (2341)’ 1234 (1432)’

1234 (4321)’

1234 (4123)’ 1234 (2143)’

They may also be described by matrices, namely,

The Cayley table of this group is 12345678 21436587 34217865 43128756 56871243 65782134 78563412 87654321

(a) Find the derived group G’ of G and compute the one-dimensional representations of G from those of G/G‘. (b) Determine the complete set of matrices corresponding to the rotations. These matrices present a two-dimensional irreducible representation of C in a natural way.

139

Problems

(c) Write out a full set of irreducible representations of G together with the corresponding character table.

31. Problem 17 can be used to find a complete character table of the group G of Problem 7. (a) Compute the matrices of the permutation presentation of G on the left cosets H , 3H, and 5H where H = {l, 2, 7, S}. (b) Compute the character of this permutation presentation. (c) Reduce this character into its irreducible components. 32. The following Cayley table is that of a group G which is one of the fourteen groups of order sixteen. Its derived group G' i s (1, 2). The Frattini subgroup 4 of a group G is the intersection of all maximal subgroups of G. 2 3 4 5 6 7 8 910111213141516 1 4 3 6 5 8 710 9121114131615 4 1 2 7 8 5 61112 91015161314 3 2 1 8 7 6 5121110 916151413 5 6 7 8 2 1 4 314131615 9101112 6 5 8 7 1 2 3 41314151610 91211 7 8 5 6 4 3 2 1161514131112 910 8 7 6 5 3 4 1 215161314121110 9 910111213141516 1 2 3 4 5 6 7 8 10 9 1 2 1 1 1 4 1 3 1 6 1 5 2 1 4 3 6 5 8 7 1112 9 1 0 1 5 1 6 1 3 1 4 3 4 1 2 7 8 5 6 121110 916151413 4 3 2 1 8 7 6 5 1314151610 91211 6 5 8 7 1 2 3 4 14131615 9101112 5 6 7 8 2 1 4 3 15161314121110 9 8 7 6 5 3 4 1 2 161514131112 910 7 8 5 6 4 3 2 I 1 2 3 4

(a) Find all the one-dimensional representations of G. (b) Use the one-dimensional representations to locate all the maximal subgroups of G. (c) Find the Frattini subgroup 4 of G.

33. Let G be a group of order ten with the following Cayley table. ~

~~~~~~

1 2 3 4 5 6 7 8 910 2 3 4 5 110 6 7 8 9 3 4 5 1 2 910 6 7 8 4 5 1 2 3 8 910 6 7 5 1 2 3 4 7 8 910 6 6 7 8 910 1 2 3 4 5 7 8 910 6 5 1 2 3 4 8 910 6 7 4 5 1 2 3 910 6 7 8 3 4 5 1 2 10 6 7 8 9 2 3 4 5 1

140

2. The Representation Theory of Finite Groups

Show that the derived group H i s (1, 2, 3 , 4 , 5 ) . (b) Use H to find the one-dimensional representations of G. (c) Verify that G has four classes of complex irreducible representations with dimensions 1, 1 , 2, 2. (d) Let T be an irreducible two-dimensional representation of G with representation space V. Suppose that u E V generates a subspace U invariant under TI H where T(2)u = MU. Show that T(5)u = ci4u and that if T(6)u = w, then (u, w} is a basis of V. (e) Show that T(2)u = MU and T(2)w = a4w while T(6)u = w and T(6)w = v. Thus 2 and 6 correspond to the matrices (a)

respectively . (f) Work out the matrices of all group elements. (g) Determine the admissible values of M in order that T be irreducible. Compute the character table of G from Problem 33 and check all orthogonality relations. 34.

35. Let the Cayley table of the group G be given by the following. 1 2 3 4 5 6 7 8 910111213141516 2 3 4 5 6 7 8 110111213141516 9 3 4 5 6 7 8 12111213141516 910 4 5 6 7 8 1 2 31213141516 91011 5 6 7 8 1 2 3 413141516 9101112 6 7 8 1 2 3 4 5141516 9101l1213 7 8 1 2 3 4 5 61516 91011121314 8 1 2 3 4 5 6 716 9101112131415 916151413121110 1 8 7 6 5 4 3 2 10 9 1 6 I 5 1 4 1 3 1 2 1 1 2 1 8 7 6 5 4 3 1110 9 1 6 1 5 1 4 1 3 1 2 3 2 1 8 7 6 5 4 121110 9 1 6 1 5 1 4 1 3 4 3 2 1 8 7 6 5 13121110 9 1 6 1 5 1 4 5 4 3 2 1 8 7 6 1413121110 91615 6 5 4 3 2 1 8 7 151413121110 916 7 6 5 4 3 2 1 8 16151413121110 9 8 7 6 5 4 3 2 1

?(,

The classes of this group are K , = { l } , K2 = ( 2 , S } , K3 = ( 3 , 7}, K4 = (4, 6}, -C5> K --C 19, 1 I . 13, IS}, and K , = (10, 12, 14, 16f. Omitting the trivial I.

141

Problems

case of the identity K , , the structure constants are Cizj

0100000 2010000 0101000 0010200 0001000 o o m 2 0000020

Ci3 j

0010000 0101000 2000200 0101000 0 0 1 m 0000020 0000002

cx4j

0001000 0010200 0101000 2010000 0100000 0000002 0000020

Ci5 j

0000100 oO01000 00 10000 010oooo 1000000 0000010 000000 1

C L ~ J

o000010 000i)002 0000020 0000002

C
0000001 0000020 0000002 0000020 o m 1 o 0000001 4040400 0404000 0404000 4040400

(a) Use the given structure constants to find the determinantal function. (b) Find the character table of G from this result.

The Computation $ Representations and Characters $Finite Gt-oztps

This chapter contains a description of several additional methods o f computing the irreducible representations and characters of a finite group G over the field of complex numbers. These methods depend strongly on the concept of tensor product which makes its appearance in three separate forms: the tensor product of vector spaces, the tensor product of linear transformations. and the tensor product of modules over an algebra. Certain important geometric groups connected with three-space are introduced and discussed. Applications of the new methods t o the representation theory of some of these geometric groups are given in some detail. However, readers interested i n research applications in physics will doubtless find it necessary to consult more detailed treatises for a complete discussion of their problems. The iirst section defines the tensor product of two abstract vector spaces and of two linear transformations on such spaces. The tensor product of two representations o f ii tixed group G is explained in terms of these two concepts. An application of the ideas is given to the representation theory of a group C o f order sixteen. The importance of tensor products in the applications of group theory to quantum mechanics is discussed. The outer tensor product R # 7' o f representations I< and 7 of the groups J and K is defined to be a certain representation of the direct product J @ K. The distinction between the tensor product and the outer tensor product of representations is noted. I t is argued that every class o f equivalent irreducible representations o f the

I . Basic Concepts of Tensor Products of Group Representations

143

external direct product J @ K of two groups J and K contains a member which is the outer tensor product R # T of irreducible representations R and T of J and K , respectively. The construction is applied to the case of C , 0C,. The second section is concerned with methods of determining the irreducible characters or representations of a group G from the irreducible characters or representations of a subgroup H of G. The method depends upon the concept of the tensor product of two A-modules, where A is a K-algebra. We review briefly the basic concepts of A-modules before introducing the definition of an induced representation which is the fundamental idea of the section. We define the concepts of subduced representation and conjugate representation before establishing Clifford's theorem on the decomposition of an irreducible module. The section concludes with the statement without proofs of certain important theorems on the irreducibility of various induced representations. Applications of these results are made in Section 4. Section 3 is concerned with the group G of Euclidean motions of threedimensional Euclidean space and some of its important subgroups. The Euclidean group, the affine group, the translations subgroup T of G, and the rotation subgroup D about a fixed point p are discussed. We define the concept of a three-dimensional lattice and introduce the classification of simple lattices into their Bravais classes. The notions of the space group, the translation group, and the point group of a lattice are defined. The group Td of symmetries of a regular tetrahedron and the group Oh of symmetries of the cube are introduced as examples of point groups. The space group Td2 of the zinc blende lattice is described as well as the space group 0,,7of the diamond lattice. In Section 4 we apply the methods of Section 2 to determining the irreducible representations of some of the groups which are introduced in Section 3. Naturally most of the methods are connected with the ideas of induced representations although various complementary methods are introduced to complete some of the calculations. 1. BASIC CONCEPTS OF TENSOR PRODUCTS OF GROUP REPRESENTATIONS

This section contains an introduction to the concept of tensor product employed in both the theory and applications of group representations. The reader is recommended to Curtis and Reiner ( I 962) for a rigorous presentation of the details of this as well as the following section. Let U and W be vector spaces over the complex field K. and let D denote the Cartesian product U x W. Denote by V" the vector space of all functions

3. Computation Representations and Characters

144

with domain D and range the complex numbers K. Let V' be the subspace of V" consisting of those functions which vanish on the complement of a finite subset of D. Let [u, v] denote the element of V' whose values are zero except at (u, v) where it is 1 . Then the set ([u, v]}, (u, v) E U x V, is a basis B of V'. Denote by M the subset of V' consisting of all functions of the following four types : c[u, wl - [cu, wl.

(1.1)

[u, w1

+ wz1 - [u, w11

-

[u, w*l,

c[u, wl - [u, cwl,

[u1

+ u2, wl - [u,, wl - [uz, wl,

where c E K , u, u , , u2 E U, and w, wl,w2 E W. Let F denote the subspace of V' spanned by the elements of M . The tnisor product U 0VJ of U and W is the factor space V'/F. Let v denote the natural mapping of V' onto U 0V. The image of [u, w] under v is denoted by u @ w and is called " u tensor w." Since the elements of M are in the kernel of the linear transformation v of V' onto U 0W, it follows from (1 . l ) that c(u 0w) = u 0cw, c(u 0w) = cu 0w, UO(W1 + w , ) = u @ w , + u O w , , (ul u2) 0w = u1 0w u* 0w.

(1.2)

+

+

Let {ul, . . ., u,,,) and {wI,. . ., wfl} be bases of U and W, respectively. By induction on (1.2), every element u 0w can be written (tlUl

+ . . . + 5,num)OO71W1 + . ' . + vflwn) = 51q1(u1 O w , ) + ... + 51Vn(U1 Ow,) + ... + 5, ql(un, 0w1) + . . + 5 , q,(u, 0wn), '

where u

= <,Ul

+ ... + t,U,

and

w = qlw,

+ ... + qnwfl.

Thus every element u m w can be written as a linear combination of the elements of the set G ' = ( u 1 0 w , , . . .,U,@W,}. Since B is a basis of V', it follows that v ( B ) , the set B' of all elements of the form u @ w, is a set of generators of U @ W. This implies in turn that G' is a generating set of U @ W. It can be shown that G' is a basis of U 0W called the e s t ~ ~ i dbasis t d determined by {ul, . . . , urn}and {w,, . . . , wfl}. Consequently, the tensor product of an m-dimensional space by an n-dimensional space is an nm-dimensional space. This construction is sometimes introduced in an abbreviated form in the literature of physics. There the tensor product of two vector spaces U and W is a "vector space" with an extended basis consisting of the set {ulwl, . . . , u, w,,)of ordered " products," the first factor taken from

1. Basic Concepts of Tensor Products of Group Representations

I45

a basis of U and the second from a basis of W. The analogs of conditions (1.2) are assumed to hold. A generalization of the notion of a bilinear form is required in order to introduce a second method of forming the tensor product of two vector spaces. Let U and W be m-dimensional and n-dimensional vector spaces, respectively, over the complex numbers K. We generalize Definition (5.52), Chapter 1 as follows.

A bilinear form or mapping on U and W is a mapping (1.3) DEFINITION. f f r o m U x W to a vector space P such that 6 ) f ( c u , w) = f ( u , cw) = cf(u, 4, (ii> f ( u 1 + u2 > w) = f ( u , , w) + f @ z w), f ( u , w1 + w2) = f ( u , W d +f@,wz>, where c E K , u, u l , u2 E U, and w, w l , w2 E W. 3

(1.4) DEFINITION. A tensor product of U and W is a pair (P,f) consisting of an mn-dimensional vector space P over the complex numbers and a canonical, bilinear mapping f of U x W into P such that (i) Every bilinear mapping g from U x W into a vector space V can be written as the composition g* of of the canonical mappingf’and a uniqueIy determined, linear transformation g* of P into V. (ii) The set of all imagesf(u, w) = u @ w constitutes a generating set of P. The previous construction of the factor space V‘/F is a specific tensor product in which the canonical mappingf( = makes the element (u, w) of U x W correspond to v[(u, w)] = u @ w. There are many pairs (P, f ) which satisfy the criterion of a tensor product for two vector spaces U and W. However, any two such, say (P,f) and ( P ’ , f ’ ) ,are connected by a canonical isomorphism k of Hom,(P, P’) such that )1.1

(1 5 )

f‘

= 11

of.

In this sense, the tensorproduct is a class of pairs of vector spaces and bilinear mappings tied together by relations such as (1.5). It is customary to identify this class with its members and to speak o f the tensor product of U and W. An element of the form u @ w is called a simple or basic tensor. Most tensors are linear combinations o f simple tensors. Given three vector spaces, U, V, and W, one can construct two tensor product spaces. (1 4

TI = ( U @ V ) @ W

and

(1.7)

T2 = U @ ( V @ W),

146

3. Computation Representations and Characters

by iteration of the above construction. These two tensor products are conceptually distinct; however, there exists a canonical isomorphism f : TI + T, defined by J”(u

0v) @ w] = u @ (v 0w)

on simple tensors and extended to the whole space by linearity. Similarly, given four vector spaces, U, V, W, and X, one can construct by iteration five tensor products such as ((U 0V) 0W) 0X and (U 0V) 0(W 0X) and a number of canonical isomorphisms between them. The process can be continued so as to define numerous tensor products of a set {Vl,. . . , V,} of n vector spaces as well as a family of canonical isomorphisms relating them. I t is customary to identify all of these n-fold tensor products and to write them merely as

V,@...@V,,,

25n.

The elements of these spaces are referred to as ntl7 order tensors. Those of the form v, @ . . . @ v,, as simple or basic tensors. Since these ideas are adequate for our purposes, we forgo a discussion of contravariant, covariant, and mixed tensors and tensor products. One of the most useful properties of the tensor product is the uniqueness of the linear map g* : U @ W -+ V determined by a bilinear mapping g from U x W into a vector space V. As an example of the use of this property, let k E Hom,(U, U’) and I? E Hom,(W, W’). The pair ( k , I?)determines a unique linear transformation k @ / I , called “ k tensor 17,” such that k 0/ I E Hom,(U 0W, U’ 0W’). Observe that the mapping T from U x W into U’ @ W’ defined by

T(u, w)

= k(u)

@ /?(W)

is bilinear. Consequently, there exists a unique T* such that

T = T* o

A

where T* is a linear transformation of U 0W into U’ 0W‘ and f is the canonical map of U x W into U @ W. From the definition of T*, it follows that (1.8)

T*(u 0w) = k(u) 0/?(W).

We define k @ k to be the linear transformation T*. Let S and T be representations of the finite group G with representation spaces U and W, respectively. For each g E G, S(g) E GL(U) c Hom,(U, U)

and

T(g)E GL(W) c Hom,(W, W).

147

1. Basic Concepts of Tensor Products of Group Representations

The linear transformation S(g) 0T(g) is an element of Hom,(U 0W, U @ W).

(1.9) THEOREM. The mapping R from G into Hom,(U @ W, U 0W) defined by (1.10)

R(g) = S(g) 0T(g)

is a representation of G with representation space U @ W called the tensor product of S and T. Proof. Let u 0w be an element of U 0W and g , g’ be elements of G.

(1.1 1)

R(gg’)(u 0w) = [S(gg’) 0T(gg’)l(u 0w) = S(gg’)u 0T(gg’)w = S(g)[S(g’)ul

0 T(g)[T(g’)wl

= [ a ) 0T(g)l[S(g’)u 0 T(g’)wl =

0T(g)I{[S(g’) 0T(g’>l(u0w)> = R(S”(S’)(U 0 4 1 = “g)R(g’)l(u 0w).

Since elements of the form u 0w generate U 0W, R(gg’) = R(g)R(g’)-

Finally, R(l)(u 0w) = [S(1) 0’ T(l)](u 0w) = S(l)u 0 T(l)w = u @ w.

Thus, R(1) = 1Lf,w,

and R is a representation of G.

A broad brush explanation of the central importance of the concept of tensor product in the applications of group theory to quantum mechanics is as follows: From a mathematical point of view, a basic problem in elementary quantum mechanics is the determination of the eigenvalue spectra of certain self-adjoint operators on a Hilbert space V. The space V is a subspace of the set of all complex-valued functions on the configuration space 6 of the problem. Each Euclidean motion g of 6 induces a linear transformation T(g) on V according to the procedures of Chapter 2. The group G of all such motions commuting with a given operator D is called the symmetry group of D. The most frequently studied operator in elementary quantum mechanics is the energy operator or the Hamiltonian H . Each element g of the symmetry group G of H induces a linear transformation T(g) on the Hilbert space V, such that (1.12)

HT(S) = T(g)H,

9 E G.

148

3. Computation Representations and Characters

The correspondence T enjoys the properties that and

T(1) = 1,. Consequently, Tis a representation of the symmetry group G with representation space V, usually an infinite-dimensional space. Whenever G is either finite or compact, all of its complex representations are completely reducible and all of its irreducible representations are finite dimensional. As a result, the space V is the direct sum of finite-dimensional subspaces {Ei}, 1 i i, each of which transforms according to some irreducible representation of G. Ordinarily, each subspace E, corresponds to an eigenvalue of the Hamiltonian H and thereby to a possible energy level of the system. Thus the energy levels can be labeled with the distinct classes of irreducible representations of G. The most satisfactory example of these remarks is to be found in the results for the quantum mechanics of the hydrogen atom whose Hamiltonian H has for its symmetry group G the group of rotations of three-dimensional space about a fixed point. Each energy level corresponds to one or more eigenfunctions of H , called wavefunctions in the terminology of physics, that are labeled by the irreducible representations of G. These representations { T o ,..., T,, ...} of the rotation group can be fully described by the nonnegative integers. There exists a class of wave functions corresponding to To called s-electrons, a class corresponding to TI called p-electrons, and so on. The eigenspace or representation space of To is one-dimensional, that of Tl is three-dimensional, and that of 7” is (2k + 1)-dimensional. Consequently, for a given energy level corresponding to To there is a single s-electron, for an energy level corresponding to T, there are three p-electrons, and, in general, there are 2k + 1 electrons corresponding to the representation Tk. The physicist forms approximate wave functions of complicated atoms from products of wave functions of simpler atoms. Approximate electron wave functions of a helium atom can be made from the products of two families, {pl, p 2 . p 3 } and {q,, q 2 ,q3}, of three p-electrons of the hydrogen atom, each family transforming according to the irreducible representation T, . Then the elements of the set of nine products {plql , . . . , p3 q 3 ) transform according to the tensor product representation T, @ T, of the rotation group G. Since the representation Tl @ Tl is reducible, this set of products does not span an eigenspace of the Hamiltonian, but rather spans a larger space which decomposes into eigenspaces. Such decomposability of the product representation is not peculiar to this example, but is the situation ordinarily encountered, that is, the tensor product S @ T of two irreducible representations, S and T, of a group G is generally (but not always) a reducible representation of G.

1. Basic Concepts of Tensor Products of Group Represenrations

149

Although the reduction of the tensor product of two representations is an important problem in both theory and application, we defer its consideration. Our present aim is to illustrate a method of determining new characters from known ones by means of the concept of tensor product of linear transformations. This method depends upon computing the matrix of a tensor product k @ I7 from the matrices of its factors k and / I . Let { u l , . . . , urn}and {wl,. . . ,wn}be bases of the vector spaces U and W, respectively. The tensor product space U @ W has a basis {ul @ wl, . . . , u, 0w,) whose elements are distinguished by two subscripts rather than one. It is customary to order such a basis {vij), labeled with two indices, according to the rule that vij precedes v,, if i < m or if i = m a n d j < n. This ordering is sometimes called the dictionary order for obvious reasons. It assumes the form for the above basis: 11 21 ml

12 . . * In 22 ... 2n m2

... mn.

This scheme can be extended to tensor products consisting of more than two factors in a straightforward manner. Let k be an element of Hom,(U, U) with matrix { a j i ] ,1 I i, j I m, with respect to the basis { u l , . . . , urn}.Let / I be an element of Hom,(W, W) with matrix {b:}, 1 I r, s i n, with respect to the basis {w,, . . . , w,]. An element cjsirof the matrix k @ h with respect to the extended basis { u j @ w,~),enumerated in dictionary order, has two superscripts denoting its row and two subscripts denoting its column. To determine its value, we compute [k @ h](uj@ w,) from the definition to be

(1.13) [ k @ h](uj @ w , ~ = ) k(u,) 0h(wJ = [C ajiui] @ [C b,'~,] = so that

(1.14)

C C ajib,'(ui @ wr),

i=l

r=l

c j ,ir - ajib,'.

The matrix C whose elements {cjsir)are given by the above relation is called the Kronecker product of the matrices A and B corresponding to {aj') and { b l } , respectively. Thus we see that the matrix of k 0/ I , with respect to the extended basis, is the Kronecker product of the matrix of k and that of / I . By the rules of dictionary ordering, one has that the elements ( c l s l r } , 1 5 r , s 5 n, occupy the upper-left n x n corner of the matrix C. These elements are of the form, clslr = a,'b,*,

I I r, s I n.

3. Computation Representations and Characters

150

c=

(1.15)

(1.19)

allB a,'B aI2B a22B . . almB

... amlB . . . am2B . . ... . . . ammB

CAYLEY TABLE OF GROUPG 1 2 3 4 5 6 7

2 3 4 5 6 7 8 910111213141516 3 4 1 6 7 8 5101112 914151613 4 1 2 7 8 5 61112 91015161314 1 2 3 8 5 6 712 9101116131415 8 7 6 3 2 1 4 1 3 1 6 1 5 1 4 1 1 1 0 912 5 8 7 4 3 2 114131615121110 9 6 5 8 1 4 3 215141316 9121110 8 7 6 5 2 1 4 31615141310 91211 910111213141516 1 2 3 4 5 6 7 8 101112 9 1 4 1 5 1 6 1 3 2 3 4 1 6 7 8 5 I l l 2 91015161314 3 4 1 2 7 8 5 6 12 9 1 0 l l 1 6 1 3 1 4 1 5 4 I 2 3 8 5 6 7 131615141110 912 5 8 7 6 3 2 1 4 14131615121110 9 6 5 8 7 4 3 2 1 15141316 9121110 7 6 5 8 1 4 3 2 1615141310 91211 8 7 6 5 2 1 4 3

151

1. Basic Concepts of Tensor Products of Group Representations

K4 = ( 5 , 7}, K5 = (6, 81, KG = J9). K7 = (10, 12}, KB = (1 I), K9 = {l3, 15}, and K,, = (14, 16}. The character table is as shown in Table (1.20). (1.20)

CHARACTER TABLEGROUPG

Class 1

2

3

4

5

6

7

8

9 10

1 1 1 1 1 1 1 1 1 1 1-1 1 1-1 1-1 1 1 --I 1 1 1 - 1 -1 1 1 1 - 1 -1 1-1 1-1 1 1-1 1-1 1 1 1 1 1 1 - 1 -1 -1 -1 -1 1-1 1 1 - 1 -1 1 - 1 -1 1 1 1 1 - 1 -1 -1 -1 -1 1 1 1-1 1-1 1-1 1-1 1-1 2 0-2 0 0 2 0-2 0 0 2 0-2 0 0-2 0 2 0 0

x1

xz x 3 x4

xs X6

x, x* x9

XJO

As a general rule, the tensor product of two irreducible representations of a group is not an irreducible representation. However, the tensor product of the one-dimensiona1 representations is always one-dimensional and consequently irreducible. In the case of G, the set {x,, . . . , x s } of characters corresponding to the one-dimensional representations is closed under multiplication (corresponding to tensor product) and forms an abelian group G" which is isomorphic to C2 0C , 0 C2 . The factor group GIH of G modulo the normal subgroup H = { I , 3} is isomorphic to G". Turning to the two-dimensional representations in Table (I .21), the character afforded by D9 coincides with the character afforded by each

(1.21)

TWO-DIMENSIONAL IRREDUCIBLE REPRESENTATIONS OF G

Element

1 2

DlI9 DzJ9

O O

1

3 4

i-1-i

DZz9

O O O 1 -i--1

DJ1" DzI1O

1 0

Dl2Io

DzZlo

i-1

0 0 1 -;-I

8

9 10 11 12 13 14 15 16

0 0 0 I-i-1 0 - 1 -i 1 i 0 0 0

0 i i 0

1 i-1-i 0 0 0 0 O O O O I-i-1 i O O O 0-I--i 1 i 1-;--I i 0 0 0 0

6

O O

D1z9

0

7

5

-i 0 0 0 0 1-i-1 0 0 - 1 -i 1 i 0 0 0 0

0 - 1 -i i 0 0 i 0 0 0-1 i

1 i 0 0 0 0 0 0-1 i I -i 0 0 1 i - 1 --i 1 -i 0 0 0 0

member of the set of tensor product representations {D'0 D9, . . . , D4 @ D9}, so that each of these representations is equivalent to D9.Similarly, each

152

3. Computation Representations and Characters

member of the set { D 5 0 D9, . . . , D s 0D 9 } is equivalent to D". Analogous results hold for the tensor product representations of the one-dimensional representations with D". One the other hand, the tensor product D9 @ D" is a four-dimensional representation so that it must be reducible. By Lemma (1.17), D9 0D'" affords the character as shown in Table (1.22). (1 2 3 )

CHAKACTER OF D9 0D'O

We know from Chapter 2 that

D9 @ D"

= a,

D' + ' . . + a,, D",

where the set (al, . . . , a,,,; of nonnegative integers is determined by Eqs. (5.22), (_5.23),and (5.25) of that chapter. An evaluation gives the result that

a,= a2 = a3 = u4 = ug = a , ,

and

=0

u5 = u6 = a , = a8 = 1.

I t follows that

D 9 @ D'O

=

D 5 0 D6 @ D7 @ D8.

The element 9 of G has the matrices (1.23) with respect to the representations D9 and D". The matrix corresponding to 9 is given by

0 0 0 0 - 1 0 0 0 0 - 1 0 0 0 0 - 1

11-1

(1.24)

[ D 90 D"](9)

=

in the tensor product representation. Equation (1.24) can be verified either from the definition of the Kronecker product of the matrices of ( I .23) or from the definition of the tensor product of two representations. To employ the second method, let {ul, u2) and { w l . wz) be bases of the representation spaces of the representations D 9 and D'",respectively. Table (1.21) defines the matrices of D9(9) and D"(9) with

153

1. Basic Concepts of Tensor Products of Group Representations

respect to these bases of U and W. The set {u, @ w,, u, @ w 2 , u2 @ wl, u2 @ w2} is an extended basis of U @ W. It follows that

D9 @ D1O(9)[u1@ wl] = D9(9)u, @ D'O(9)wI = -U1 0W1, D9 @ DI0(9)[U, @ W2] = D9(9)U, @ D'O(9)W2 = -U1 @ W 2 . D9 @ D1O(9)[u2@ w,] = D9(9)u2@ D'O(9)wI

= -u2 @ w,,

and

D9 @ DI0(9)[u2@ w2] = D9(9)u2@ D10(9)W2= -u2 @ u 2 . This completes our discussion of Example ( I . 18). We shall now make a few observations about the reduction of tensor products into their irreducible components. Our remarks are mostly concerned with physical applications although they deal with what is almost entirely a mathematical problem. The reduction of tensor products is a standard problem in the applications of group theory to atomic and molecular physics as well as in the theory of the scattering of atomic and nuclear particles. We have mentioned that the physicists form approximate wave functions for complicated atomic or molecular systems from wave functions for simpler systems. Related constructs appear in atomic and nuclear scattering theory where an intermediate system arising from an amalgamation of the target and scattering particles disintegrates into more elementary particles. Generally the approximating wave function used in the solution of these types of problems is a linear combination of products of wave functions of simpler systems. The wave functions of the simpler systems belong to irreducible representation spaces of some representation of the symmetry group G of the system. The products of such functions are elements of the representation space of the tensor products of the corresponding irreducible representations. Almost always the physicist is interested in irreducible representations or representation spaces since these are the ones usually associated with the eigenvalues of the energy operator. Consequently, he is regularly faced with the reduction of tensor products into their irreducible components. The method of character analysis discussed in Example (1.18) can be applied. The character of the tensor product R @ T of two representations R and T of the symmetry group G can be obtained from the characters of the representations R and T. Consequently, it is reasonable straightforward to determine the decomposition (1.25)

R@T

= n, Tl @ . . . @ nk Tk

of R 0T into its irreducible components in the cases of finite or compact groups G when the field in question is the complex numbers. The computation of the coefficients { T I # } , 1 5 i < k , of (1.25) by (5.22), (5.23), and (5.25) of

3. Computation Representations and Characters

154

Chapter 2 is not difficult in the finite case. Analogous formulas (using integration) can be employed in the compact case. Unfortunately the decomposition (1.25) is not always sufficient for the needs of the physicist. He sometimes requires a more detailed resolution of the problem. If U and W are the representation spaces of the representations R and T, respectively, then there is an analogous decomposition

u o w = u, 0 . * * o u t

(1.26)

of the representation space of R @ T into its irreducible components. Frequently, it is a detailed description of (1.26) which is desired. Naturally these problems have been solved and the answers tabulated for those applications of common occurrence. Our discussion is purely for the purpose of acquainting the reader with the basic ideas. Those interested in more details are referred to Boerner (1963). Gel'fand and Sapiro (1952), Hamermesh (1 962), Lomont (1959), and Lyubarskii (1960). Let {ul, . . ., urn]and {wl, . . . , wn>be bases of the representation spaces U and W of the representations R and T, respectively. The problem is a generalization of that of finding a symmetry adapted basis of the group algebra KG given the natural bossis of KG. Given the extended basis (ui@ wj}, 1 5 i _< m, 1 rj I n, of the tensor product U 0 W of the representation spaces, we are required to find a second basis, also called a symmetry adapted basis, which determines the irreducible subspaces of the decomposition (1.26). The problem is most easily solved if each irreducible representation occurs at most once in (1.25), when each of the representations R and Tis irreducible. Groups having this property are called simply reducible groups. Since our aim is only to illustrate the problem, we consider the simply reducible case for which the tensor product of irreducible representations assumes the form

ROT

(1.27)

=

TI

@

*

a

'

0 Tk,

where the set {T,), 1 < i _< k , consists of mutually inequivalent, irreducible representations of G. The corresponding representation space U 0 W decomposes into the direct sum

uOW=U,O...@Uk

(1.28)

of inequivalent, irreducible subspaces. We must find a set of coefficients i 5 rn, I IJ n, 1 5 E 5 n,, 1 I 2 1 k , in K such that the set of linear combinations

{ c E A i j ]1, 5

(1.29)

u,, =

1 cEnij(ui

wi),

1 _<

E

I n,,

i s a basis for the irreducible subspace U,, 1 2 , I5 k.

The coefficients { c e n i j }are called the Clebsch-Gordon coeficients of G. We have not introduced a sufficiently large set of parameters to demonstrate the

155

1. Basic Concepts of Tensor Products of Group Representations

true complexity of these coefficients. In the general case, there are six sets of indices involved in their definition. Furthermore, one must introduce canonical bases in the representation spaces before one can define the ClebschGordon coefficients. We will not enter into any further discussion of these quantities but refer the interested reader to the more specialized treatises mentioned previously in this section. We turn now to the introduction of a parallel concept, not to be confused with that of the tensor product of two representations of a single group G. Let G be the external direct product J @ K of two groups J and K. Let R and T be representations of J and K with representation spaces U and W, respectively. The outer tensor product Q of the representations R and T (1.30) DEFINITION. is a representation of the external direct product G with representation space the tensor product U O W. The value of the representation Q at an element ( j , k ) of G is given by the formula Q<j,k ) = W

)0W ) .

The proof that the outer tensor product Q is a representation of G is left to the reader. The outer tensor product Q is sometimes denoted by the symbol

R#T.

Arguments similar to those leading to Eq. (1.16) show that the character x of the outer tensor product R # T is the product de of the characters d and e of its factors R and T , respectively. Suppose that d and e are irreducible characters of J and K. Then one has

= [ J : 1 ] [ K :11 = [ G :

I].

Thus one finds that the outer tensor product of two irreducible representations of J and K is an irreducible representation of G = J @ K. Let d and d' be inequivalent irreducible characters of J and e and e' be inequivalent irreducible characters of K. Then a computation similar to the above shows that the characters de, de', d'e, and d'ef are inequivalent, irreducible characters of J @ K. One recalls that the number of classes of conjugate elements of the external direct product J O K is equal to the product of the number of such classes of J and the number of such classes of K. These two observations show that if r is the number of classes of J and s is the number of classes of K, then rs is the number of classes of equivalent irreducible representations of J @ K . Consequently, one can obtain a member

156

3. Computation Representations and Characters

from each class of equivalent irreducible representations of J @ K by forming all distinct outer tensor products R @ Twhere R runs exactly once through the classes of equivalent irreducible representations of J and T exactly once through the classes of equivalent irreducible representations of K. These observations suggest a useful method of calculating the irreducible representations of groups which can be written as the external direct products of groups of lower order. The same idea can be applied with almost no modification to the case of internal direct products. We give an example of the procedure. (1.31) EXAMPLE. Consider the group C, @ C4 whose Cayley table is given in (1.66) and whose irreducible representations are given in (1.67), Chapter 2. The character tables of C, and C , are as shown in Table (1.32). (1.32)

CHARACTER

7-1’

T*’

TABLE OF 1 2 1 1 1 -1

c,

CHARACTER TABLEOF C4

1

2 3 4 1 1 1 i -1 -i Ta” 1 - 1 1-1 T4” 1 - i -1 i

TI” 1 T,” 1

The character table of the group C, @ C, shown in Table (1.33), is the Kronecker product of the character tables of C , and C , . (1.33)

CHARACTER TABLE OF C2 0C4 1 2 3 4 5 6 7 8 TI1’1 1 1 1 1 1 1 1 Ti*’1 i - 1 -i 1 i - 1 --i T,3’ 1 - 1 1-1 1-1 1-1 TI4’ 1 - - i - 1 i 1 -i-1 i TZ1’ 1 1 1 1 - 1 -1 -1 - 1 TZ2’1 i - 1 - i - 1 --i 1 i TZ3’1 - 1 1 - 1 -1 1-1 1 Tz4’ 1 - i - 1 i-1 i 1 -i

2. REPRESENTATIONS AND CHARACTERS OF A GROUP INDUCED FROM THOSE OF A SUBGROUP

This section presents a short introduction to the theory of induced representations for finite groups. From both the theoretical and practical point of view, the concept of an induced representation is a powerful tool which can be extended to the theory of topological groups. Several significant applications

157

2. Representations and Characters Induced from Those of a Subgroup

of the theory are given in Section 4. There are a number of ways of approaching the subject. Here it is treated by the use of tensor products of modules over algebras. Such a treatment requires an extension of the concept of tensor product introduced earlier, but not complete generality since the tensoring is confined to modules which are vector spaces over the complex numbers. We use both methods of introducing a tensor product of Section 1 and begin by recalling the basic definition of a module. Let A be an algebra with identity 1 over the field K of complex numbers. A left A-module is a K-space U which admits a left-multiplication by elements of A. The K-space U is almost afwaysjinite-dimensionalover K i n our considerations. The following rules are required to hold for the left-multiplication:

(2.1)

l u = u, a(bu) = (ab)u, (a + b)u = au + bu, a(u + u') = au + au' a(m) = ic(au) = (Ica)u,

1 EA, U E U , a,bEA; UEU, a,bEA; UEU, aEA; u,u'EU, a E A; u E U; K

E

K.

The most familiar, nontrivial example of a left A-module is that of an ndimensional K-space u , where A is taken to be the set Hom,(U, u ) of Kendomorphisms, written on the left. A right A-module U is a K-space with a right-multiplication defined which satisfies rules analgous to those of (2.1) with the elements of A appearing on the right-hand side of the elements of U rather than on the left-hand side. We give some definitions for left A-modules, but leave a statement of the corresponding definitions for right A-modules to the reader. The subset of all linear transformations of Hom,(U, U) which commute with left-multiplication by elements of A is denoted by Hom,(U, U). An elementfE Hom,(U, U) is called a kft A-homomorphism of U and satisfies the equation f (au) = af(u), a E A, u E U. A left A-submodule V of U is a K-space of U such that a v E V for V E V , a E A . When f is an element of Hom,(U, U), 1mf and Kerf are left A-submodules of U. Given a left A-submodule V, the factor space U/V permits a multiplication, defined by a[u + V] = au + V for u + V E UjV and a E A. The factor space V/V is a left A-module, called the left factor module, with this definition of multiplication. Most of the modules in the sequel are left A-modules; however, the reader is generally expected to recognize from the context the left or right nature of the modules and mappings involved. (2.1') REMARK. In general, a left A-module is defined to be an abelian group U, rather than a vector space, for which a multiplication by elements of A enjoyingmost properties of (2.1) is defined. Using this more general definition, U can still be made into a vector space over the field K whenever A contains a

3. Computation Representations and Characters

158

multiplicative identity 1. The more restricted definition is adequate for our purposes and eliminates a certain amount of technical argument. Let U and W be right and left A-modules, respectively. Let D denote the Cartesian product U x W and V" the vector space of all functions with domain D and range the complex numbers K. Let V' denote the vector space of all functions in V" which vanish on the complement of a finite set. Let M denote the subset of V' consisting of all functions of the form (2.2)

c[u, wl - [cu, wl,

c[u, wl - [u, cwl, [u, w1 + w2l - [u, W l l - [u, w21, [u1

+

[ua, wl - [u, awl, u2, wl - [u,, w] - [u* , w],

using the notation of Section 1, where c E K , u, u l , u2 E U, w, wl, w2 ISW, and a E A. Then the tensor product U 0W over A is defined in the same manner as in Section 1. A second definition, corresponding to the second definition of Section I , can be given by introducing the concept of a balanced map. Let U and W be right and left A-modules, respectively. (2.3) DEFINITION. A balanced rnapffrom U x W into a K-space V is a map from U x W into V such that f(cu, w) =f(u,

(2.4)

f(u

f(u, w + w')

for c E K , u, u'

E

+ u', w) =f(u,

= f(u,

U, w, w'

CW)

E

= Cf'(U,

w) +f(u', w),

w) +S(u, w'),

W, and a

E

w),

f(ua, w) =f(u, aw)

A.

(2.5) DTFINITION. A tensor product of the right A-module U and the left A-module W over the algebra A is a pair (P,f) consisting of a K-space P and a canonical. balanced mapffrom U x W into P such that

(i) Every balanced map g from U x W into any K-space V can be written as the composition g* 0 f of the canonical map f' and a uniquely determined, linear transformation g* of P into V. (ii) The set of all images of the formf(u, w), denoted by u 0w, constitutes a genesating set of P.

Two tensor products defined in this manner are essentially the same. I t is customary to identify the set of all tensor products of U and V and to denote the resulting object by the symbol U @ A W which is read, " U tensor W over A." We note that tensor multiplication and algebra multiplication are related by (2.6)

ua@w=u@aw.

We require a special case in which the concept of a bimodule is needed. Let A and B denote algebras with identities over the field K of complex numbers.

2. Representations and Characters Induced from Those of a Subgroup

159

A K-space U is said to be an (A, B)-bimoduZe if U is both a left A-module and a right B-module for which (2.7)

a E A, u E U, b E B.

a(ub) = (au)b,

Ordinarily the tensor product U OBW of a right B-module U with a left B-module W is a vector space that is not a B-module. However, if U is an (A, B)-bimodule and W is a left B-module, the tensor product U Oe W can be made into a left A-module. Multiplication on a set of generators of U Be W is well defined by the formula (2.8)

a(u 0 w) = au 0 w,

a E A, u E U, w E W,

and can be extended to all of U OBW by linearity. The group algebra KH can be embedded as a subalgebra of KG when H is a subgroup of the group G. The set KG is a K-space that is a (KC, KH)-bimodule with the required multiplications defined by the algebra product. Let U be the representation space of the representation T of the subgroup H . The K-space U can be made into a KH-module as follows. For any element, x = k,h,

of KH, the equation XU

= k, T(h,)u

+ ... + k,h,

+ . .. + k, T(h,)u,

u E U,

defines a multiplication making U into a KH-module. We are now prepared to make the basic definition of an induced representation. (2.9) DEFINITION. Let U be the representation space of a representation t of a subgroup H of the group G. Then U is a left KH-module and KG is a (KG, KH)-bimodule. The tensor product KG O X H U is a KG-module, called the induced module, which we denote by the symbol Uc. Each element g of G defines a linear transformation tG(g)on the vector space Uc by means of the definition, (2.9’)

tC(g)u = gu,

u € uc,

where gu denotes the module product of the module element u by the algebra element g. Thus the induced module Uc affords the representation tG of the group G. This representation can be extended to the group algebra KG by means of linearity. The representation tC is called the induced representation or the representation induced by t . The induced module UGaffords a matrix representation of G with respect to any selected K-basis of Uc. We wish to investigate the relationship between a matrix representation of H afforded by the module U and a matrix representation of G afforded by the induced module UG.One way of so doing is to

160

3. Computation Representations and Characters

select a K-basis { u l , . . . , u,) of U and a complete set { g l , . . . , g,,,} of coset representatives such that G = g,H u . . . u g, H . It can be shown that the set (2.10)

1 < i < m , 1 <j
(gi@uj),

is a K-basis of UG.We recall the rule introduced for ordering such bases, namely, that gi @ u j precedes g,@ u , if ~ i < r or i = r a n d j < s. The matrix of a linear transformation tC(x)with respect to such a basis is of the form {aij''}. We assume that the matrices of t(h), h E H , are known with respect t o the basis {ul, . . . , u,]. Then one has (2.1 1)

1 1 t"(9)iji"(9'

0U')

= tC(g)(g,0 U j ) = S k i 0 U j ) = ggi @ u j = g,h @ uj = gk @ huj = gk @

=

where

f(h)jEU,

1 C A(gA-'ggi>t(/l>jE(gA 0

uE)~

(2.12) In Eq. (2.1 I ) , we have used the fact that there exists a unique h E H such that ggi = g k h or h = gk-lggi. We have also made use of the identity g k / i @ u j = ,qk@ hiwhich holds since the tensoring is over KH containing 11. Thus one obtains the identity (2.13)

tc(g)ij'.c =

A(gi-'ggi)t(h);,

where A is defined by Eq. (2.12). It follows that the matrix of t C ( g )with respect to this basis appears i n block form where the number of blocks in any row or column is equal to the index ni of H in G, while the dimension of each block is equal to the dimension n of the K-space U . These remarks can be rephrased in the following way. Let t be an n-dimensional matrix representation of the subgroup H of index m in the finite group G. Denote by {glH , . . . , g , H ) the collection of all distinct left cosets of H in G. The function T which makes correspond to each s of G the block matrix (2.14)

T ( s )=

1

t'(g1 - ' x d

...

t'(s1

t'(g,n-'*Yq,)

' '

t'(g,n

where each r ' ( g i - ' . y , ) is an

12

x n matrix, 1

-1

-1

xgm) -\-.qrn>

1,

< i,,j I n7, with

/ ' ( g i - l q j ) = 0,


t ' ( g i- 1 s g j ) = f ( g i - ' s g j ) ,

gi-'xgjE H ,

and where

2. Representations and Characters Induced from Those of a Subgroup

161

is an (nm x nm)-dimensional matrix representation of G which is said to be induced by the matrix representation t of H and sometimes denoted by tc. Only under special circumstances discussed below is T( = t C ) an irreducible representation of G even though t is an irreducible representation of H. Given a representation T of G, there exists a representation t of H whose values are given by

t(h) = T(h),

h E H.

The representation t is said to be subduced by the representation T and is denoted by T,. The term subduced can be applied also to a module or representation space M. One should observe that the vector space structure of M is not altered, but that the algebra of multipliers is changed from KG to KH when passing from a KG-module M to the subduced KH-module denoted by M,. As an easy example, consider the case where H consists only of the identity 1 of G. Then KH can be identified with the field K , and the subduced module M, is the set M with only its structure as a vector space over K. Thus the KHmodule M, can be a reducible KH-module, even though M itself is an irreducible KG-module. We sometimes write the irreducible KG-module M as a direct sum M = N, ON,, where the set {NJ, I I i I t , consists of irreducible KH-submodules (of MH),each of which is a K-space, of course. One has hN, = N, , h E H , but gN, is not necessarily contained in N, when g an element of G not in H. There is a famous theorem of Frobenius which we do not prove, but which can be stated as follows.

(2.15) THEOREM. Let H be a subgroup of the finite group G. Let { T ( i ) } , 1 2 i 5 m, and { t ( j ) } , 1 < j 2 n , be a complete collection of representative elements from the classes of irreducible complex representations of G and H, respectively. Then there exists a matrix {I,,,}, 1 i i 5 m, 1 < j I n, with nonnegative integer entries such that T(i)HM

I.,, t ( j )

and

t(j)' z

1 i., T(i). ,

Theorem (2.15) has the following consequences. Given any complex irreducible KH-module N of a subgroup H of G, there exists a complex irreducible KG-module M such that N is KH-isomorphic to a direct summand of M, . Conversely, given any complex irreducible KG-module M, there exists a complex irreducible KH-module N such that M is isomorphic to a direct summand of NG. These observations lead one to suspect that the irreducible complex representations of a finite group G can be obtained by induction from the

162

3. Computation Representations and Characters

irreducible complex representations of its subgroups. Unfortunately, there is no general method of reducing an induced module NG into its irreducible components without the use of the irreducible complex characters of G. Consequently, one tries to determine methods of induction which avoid this reduction problem. To do so, requires the development of a number of new ideas and theorems. We begin with Clifford's theorem which describes the reduction of an irreducible KG-module M of a group G into irreducible KH-submodules of a normal subgroup H of G. The statement and proof of the result require the introduction of the concept of conjugate representation. Let t be any representation of the normal subgroup H and g be any element of G. There exists another representation t , called the conjugate (or G-conjugate) of t under conjugation by g and defined by

(2.16)

tq(/?)=

t(g-'hg),

h E H.

It is convenient to say that t' is a conjugate o f t if it is equivalent to a conjugate of f. The idea of conjugate representations arises in several natural ways. First, let N be an irreducible KH-module which is a subspace of a KGmodule M. Let {n,, . . . , n,] be a K-basis of N and suppose that for h E H

h(ni) =

2 aji(h)nj.

The subset ( p , , .. .,gn,) is a K-basis of g N for g E G. One has (2.17)

/7(gni) = ,~~[(g-'hg)niI = g [ E aji(S-'hg)nj] =

C aji(g-'/?g)gnj*

Thus glv is a KH-module for the conjugate representation t, of G. Second, let N be a KH-module affording the representation t where H is a normal subgroup of G. Suppose that G = g,H u . . . LJ g , H. Then the induced module NC is the direct sum

NG= g1 @ N @ . . . @.qr @ N.

(2.18) Note that (2.19)

/lugi o n ] = gi[(gi-'hgi)o n] = g i o [gi-'/igi]n.

I t follows that gi @ N is a KH-submodule which affords the repr,esentation . Third, let N be a KH-module. There exists a second KH-module N(g) whose underlying space is N, but whose scalar multiplication is defined by

tq,

(2.20)

Irn

= (g-'Q)n,

h E H , n E N,

and extended to all of KH by linearity. It is easy to see that if N affords the representation t , then N(g) affords the conjugate representation t,. Any conjugate t , of an irreducible representation t is irreducible.

2. Representations and Characters Induced from Those of a Subgroup

163

One obtains a complete set ofconjugates o f t by selecting from the collection {t,}, g E G, one member from each class of equivalent representations which occur. If {gl, . . .,gr}is a complete set of coset representatives of H in G, then the collection { t g , } , 1 5 i t , contains a complete set of conjugates of t. There may, of course, be duplications in the form of equivalent representations. We sketch a proof of an important theorem of Clifford’s. (2.21) THEOREM. Let M be an irreducible KG-module where K is the field of complex numbers, and let H be a normal subgroup of G. Then M is the direct sum

M = N,

(2.22)

@ ... O N ,

of irreducible, mutually conjugate KH-submodules. The set {Nil, 1 I i I t, contains a complete collection of conjugate KH-modules. Furthermore, if V is any irreducible KH-submodule of M; then V is isomorphic to some member of this set. Proof. Let N be an irreducible KH-submodule of M. If gN = N for every g E G, then N = M. Otherwise there exists g2in G, not in H, such that g2N @ N is direct. Assume we have a direct sum decomposition

M,

=N

@ g 2 NQ * - * @g,N,

then either every irreducible KH-module gN is contained in M,, which thereby coincides with M, or else there exists an irreducible g,+lN such that M, @ gr+,N is direct. By induction, we arrive at (2.23)

M

= g l N 0 . .. @g,N

for a suitable collection {gi}, 1 i i i t, where N = glN. If N affords the representation t of H ; then we have seen that g i N affords the conjugate representation tg,,that is, all the irreducible summands occurring in Eq. (2.23) are mutually conjugate KH-modules. Let V be any irreducible KH-submodule of M. By complete reducibility, (2.24)

M=V@V’,

where we consider M as a KH-module. I t follows from the Krull-Schmidt theorem, essentially Theorem (1.45) of Chapter 2, that V is isomorphic to at least one of the summands of Eq. (2.23). By the remarks introducing Clifford’s theorem, Eq. (2.18) is a decomposition of NGaccording to Eq. (2.23) whenever N is a KH-module for a normal subgroup H of a group G. There is a sort of partial converse. Let M be a KG-module containing a KH-module N such that (2.25)

M =glN@.**@g,N,

164

3. Computation Representations and Characters

where {gl, . . . ,g,}is a complete set of coset representatives of the subgroup H (not necessarily normal) in G. Then M is KG-isomorphic to the induced module NG. The mapffrom NGinto M suggested by f(gl

0n1 + .

*

+ g, 0n,) = glnl + . . . + g t n r

is a well-defined K isomorphism of the K-space NCinto the K-space M. Since we have f ( g ( g i 0n)) = f ( g g i @ n, =f(gkh = gk(hn) = (gkh)n =

0n,

= f k k

0

= s k i n ) = g f ( g i @ n),

it follows thatfis a KG-isomorphism. The results of Clifford's theorem can be given a more specific formulation. According to Theorem (2.21), we have the decomposition (2.22)

M = N, @ . * . O N , ,

where the set {Ni}, 1 5 i 5 t , of irreducible KH-submodules contains a complete set of conjugates of NL. These submodules can be partitioned into disjoint subsets G j, 1 ij 5 s, each of which is a maximal subset of equivalent submodules from the collection {Ni}, 1 5 i I t . The homogeneous comporients of M, (M regarded as a KH-module) are the KH-submodules Mj, each of which is the direct sum of the submodules belonging to Gj, 1 5 j i s. There exists a set { g k } , 1 I k _< s, of elements of G such that (2.26)

M, = g k M I ,

1 _< k IS.

Equation (2.22) can be rewritten (2.27)

M = M, @ . . . O M , ,

where each Mj , 1 < j < s, is a homogeneous component of M, . We present an example of such a decomposition in Section 4. An orbit of H in G is a maximal subset 0 of inequivalent G-conjugate KH-submodules. One obtains an orbit of H by selecting one KH-submodule Nj* from each class Gj. Let the KH-submodule M* be defined by (2.28)

M* = N,* @ . . . ON,*.

Then M is the direct sum of a finite number n(D) of KH-modules isomorphic to M*. The number n(D) is called the order of the orbit 53 in M. Let H * be the subgroup of G consisting of all elements g E G such that yM, = M , . Then H* is called the inertia subgroup of M I . It is easy to see that yk H*gk-' is the inertia subgroup of M k ,whereg, is defined by Eq. (2.26), 1 5 k 5 s. Furthermore, { g l , . . . , gs)is a complete set of coset representatives

2. Representations and Characters Induced from Those of a Subgroup

165

of H* in G. Consequently, M is KG-isomorphic to the induced module M I G where MIG= KG OKH* MI.

(2.29)

We are lead to a rather useful induction tool by means of some of these considerations. (2.30) THEOREM. Let N be a complex irreducible KH-module where H is a normal subgroup of the group G with { g , , . ..,g l ) a complete set of coset representative of H in G. Then the induced module NG is a complex irreducible KG-module if and only if the set {N(gi)}, 1 5 i 2 t , consists of mutually inequivalent KH-modules. Proof. Let t denote the matrix representation of KH afforded by N and tG the matrix representation of KG afforded by the induced module N". It follows from tc(g)ij'e = A(g,- 'gg,)t(h>;

(2.13)

that the nonzero blocks of a matrix t C ( g ) occur along the main diagonal if and only if g is an element of H . This observation shows that the character ZG afforded by NG has nonzero values only for elements of H . For /7 E H , ZGhas the value ZG(h) = Z,(/?)+ . . . + Z,(/l),

(2.31)

where Z i is the character of H afforded by the KH-module g i 0N, 1 2 i I t . Hence we find ZG(g)ZG(g- 1 ) = ZG(h)ZG(Iz-1)

c

1

hsH

geG

hsH

=

j

i

11 1 Z,(h)Z,(hP). i

j

htH

I f the KH-modules g i @ N, 1 5 i _< t. are all inequivalent, then the simple characters Z iare mutually orthogonal. It follows that (2.32)

i

c 1 Zi(/7)Zj(h-') c 1 aij[H 11 j

heH

=

i

j

:

=

[G : 11.

Otherwise, the sum of Eq. (2.32) exceeds [G : 11, and the induced module NG is reducible. This theorem is an effective tool for computing certain irreducible representations of various groups which appear in physical problems. Naturally there are many instances when it is not applicable. One arises if it is necessary to deal with the situation where the subgroup H is not normal. Another, if a

166

3. CompufafionRepresentations and Churucters

given class of irreducible representations is represented more than once among the set {N(gi)) of conjugate modules. A thorough study of the important results in this area is beyond the scope of this book. We discuss some of the principal theorems and definitions, but must refer the reader to more detailed treatises, say to Curtis and Reiner (1962), for proofs. Let M and N be two completely reducible KG-modules for some finite group G. Then M and N are the direct sums (2.33)

M

= MI @ . . .

OM,

N

= N, @ . . .

ON,

and (2.34)

KG-modules. The KG-modules M and N are said to be disjoint or 1ndcpci7Jri7t if there is no pair, M i and N j , of equivalent submodules occurring i u the decompositions (2.33) and (2.34). Let L be a KH-module afTording the representation t of a subgroup H of the group G. The induced module LG= KG B K HL affords the induced representation T = tG of KG. For any g E G, the set g 0L is a subspace, denoted by L(g), of Lc which is an H-submoduie when g belongs to the normalizer o f H i n G, but not in general. However, L(g) is a K ( g H g - ' ) submodule and a KH(g)-submodule where H(g) is the subgroup of H defined by N ( g ) = H n g H g - ' . Since H(g) is a subgroup of H , the representation t subduces a representation s = t H ( s )on L. There exists a representation t ( g ) of H ( g ) on L(g) which is a conjugate of s. To see this, let {v,, . . . , vJ be a K-basis of L with respect to which it affords the matrix representation S of H(g). This means, of course, that of' irreducible

xvi =

(2.35)

c s(x)jivj,

x E H(g).

The set {g 0v l . . . . , g 0v,S is a K-basis of L(g). Since every x of H(g) has the form 9/79 I , one has ~

(2.36)

.u[g 0vi] = [ g h g - ' g ] 0vi = g 0hvi = g 0 [g-'xg]vi = g 0 S(g - ' x g ) j vj = S(g - 'xg) g 0vj .

1

c

Thus we see that the matrix representation S(g) of H(g) afforded by L(g) with respect to {g 0v,, . . . , g 0vr} has the form o f a conjugate of the matrix rcpreszntation S afforded by L with respect to {v,, . . . , v,.). We again refer the reader to Curtis and Reiner (1962) for the following generalimiiorrs.

(2.37) rFlI:OKEM. Let H be a subgroup of the group G with M a complex irreducibi:: KH-module. Suppose for all g in G but not in H, the KH(g)modules and M(gj are disjoint. Then MGis an irreducible KG-module.

3. The Group of Euclidean Motions of Three-DimensionalEuclidean Space

167

We formulate this result in terms of the character x of a complex irreducible representation t of the subgroup H of the group G. Denote by x ~ ( and ~ ) x(g) the characters of the representations t H ( g and ) t(g) introduced above. Then we have the following theorem. (2.37’) THEOREM. Let x be the character of a complex irreducible representation t of a subgroup H of the group G. Then tc is an irreducible representation of G if for all g in G not in H , (2.38)

(2.39) REMARK. One need only check the conditions of Theorems (2.37) and (2.37’) for a complete family of coset representatives of H in G. For small groups, this can be done many times by observation. (2.40) COROLLARY. Let H be a normal subgroup of the group G and t be a complex one-dimensional representation of H . Then the induced representation T = tC is irreducible if t and t, are distinct irreducible representations of H for every g 4 H. (2.41) THEOREM. Given H and K subgroups of the group G, let U be an irreducible KH-module and V be an irreducible KK-module. Suppose that UGand VG are irreducible KG-modules. Then UGand VG are not KG-isomorphic if for all g E G, the J-modules g @ U and V are disjoint where J = g H g - ‘ n K. 3. THE GROUP OF EUCLIDEAN MOTIONS OF THREE-DIMENSIONAL EUCLIDEAN SPACE AND SOME OF ITS SUBGROUPS

This section contains a description of the group of Euclidean motions of three-dimensional, real Euclidean space, hereafter referred to as Euclidean space, and certain of its important subgroups. The next section of the book discusses the representation theory of some of these groups. The subject is an old one which requires for thorough treatment a great deal more space than we are prepared to give to it here. Modern works on this subject are largely due to physicists or mathematicians writing for physicists of which we mention those of Slater (1965), Koster (1957) and Lomont (1959). A fairly recent book written from a more mathematical point of view is that of Burckhardt (1947). The present exposition assumes that the reader has a background of experience with Euclidean space and is generally familiar with its synthetic and analytic descriptions. The group of rigid motions G of the Euclidean space X is the set of all bijections or permutations on X which not only leave invariant the distance

168

3. Computation Representutions and Characters

between any two given points of X but also the orientation of any triple of mutually orthogonal lines in X. Each rigid motion is the composition or product of two rigid motions, the factors taken from two special classes of such motions. The first of these is the class of translations, each element g of which moves every point x of X the same distance and same direction as every other point. The second of them is the class of all rotations, each element g of which is a rotation of the space X about a line through some point x of X. We introduce notation in order to discuss these more fully. The set T(x) of all directed segments S in X with origins at the point x of X can be made into a real, inner produce space V which is called the tangent space at x. Let S, S,, S, belong to T(x)and 1,E R.The sum S, + S, is defined by the parallelogram law, the product i.S of S by a real number A by stretching, and the norm (/S((of S by its Euclidean length. We can also associate with the point x a coordinate system determined by any three mutually orthogonal lines through x. The set of all translations of X constitutes the translation subgroup T of the group G of all rigid motions of X. An element g of Tis described by means of a field of congruent directed segments {i,}, x E X, each i, representing the motion of x under g. A more useful description is obtained by fixing an orthogonal coordinate system in X with origin at some point 0 and associating with each x of X either its triple (tl, t 2 ,t 3 of ) coordinates or its radius vector i from 0 to x. The action of the translation g on x is described either by or, in vector notation by, g(i) = i:

+ i,

c3)

where i: denotes the radius vector with components (tl, t 2 , and i the . is convenient to denote a translation g vector with components ( T , , T ~ T ,~ ) It by the vector i defining the motion, a notational convention which we adopt. Consequently, Eq. (3.2) assumes the form (3.2')

20)

= i:

+ 2.

A discussion of the rotations in X requires more effort. Let g be any rigid motion of X which leaves the point x fixed. Since the p l a n e 9 through x can be defined as the locus of all points equidistant from two specified points y and z, it follows that the image g(v) must be contained in the plane q' of all points equidistant from the points g(y) and .q(z). Let !Ill be a lattice of squares A of side E in'$; then g(W) is a lattice of squares g(A) of side E in'$'. Any point p' of !$3' lying interior to an image square g ( A ) is uniquely determined by its distances from the vertices of g(A). These same distances uniquely determine a point p i n A which maps onto p' under g. Points on the boundary of g(A) can

3. The Group of Euclidean Motions of Three-Dimensional Euclidean Space

169

be given a similar treatment. Thusg(’iJ3)must coincide with’$’, andg maps any plane ’$ through x onto another plane g(‘iJ3) through x. Since any line L! through x can be considered as the intersection of two planes p1 and ’$, through x,g(2) is the common part of g(’$,) and g(‘$,). Thus g maps planes and lines through x onto planes and lines through x, respectively. Now, let S,, S, , and S, + S, denote three vectors which form two sides and the included diagonal, respectively, of a parallelogram EJ contained in the tangent space V at x. This parallelogram transforms under g into another parallelogram g(Q) lying in the plane g(’iJ3)with sides g(S,),g(S,), and included diagonal g(S, + S,), all beginning at x. It follows that

We leave to the reader the argument that if 2 any real number and S any vector beginning at x, then (3.4)

g(AS) = %g(S).

Equations (3.3) and (3.4) assert that g acts as a linear transformation on the tangent space V at x. The fact that g is a rigid motion implies that g is an orthogonal transformation. The matrix M ( g ) of g with respect to any orthogonal basis of V is an orthogonal matrix by Theorem (5.87), Chapter 1. We continue the analysis of g by observing that the 3 x 3 real matrix M ( g ) has at least one real eigenvalue A. Thus there exists a unit vector ii such that which implies that II = 1 by the orthogonality of g. If {ii,, a,, ii,} is an orthonormal basis of V, then {gii,, gii, , gii,) is a similarly oriented orthonorma1 basis of V from which it follows that the determinant I g 1 is 1. We conclude there exists a unit vector ii such that gii = a.

(3.6)

Let the orthonormal basis {GI, li, , a,} of V have li as its first vector. Then the plane ‘iJ3 spanned by {a,, ii3f is mapped onto itself by g since g is orthogonal. Further analysis persuades one that (3.7)

giiz =

(cos “)a2 + (sin %)a,

and

(34

gii, = -(sin a)ii2

+ (cos cr)ii,

3. Compuiaiion Representaiions and Characters

170

for some angle a. Thus the matrix of g with respect to a suitably chosen orthonormal basis in the tangent space V at x assumes the form (3.9)

This result permits us to describe g as a rotation through an angle CI about the line determined by 6 . The set of all such rotations in X is not a subgroup of the group G of rigid motions, but rather is a complete class C of conjugate subgroups of G. Each subgroup b of the class C consists of those elements of G which leave a specific point x of X invariant. This subgroup b is referred to as the isotopy subgroup of G at x. It suffices for many purposes to consider some particular member of the class C. We select the group D of rigid motions leaving the origin 0 of our coordinate system fixed and denote by R an element of D . The fundamental result on rigid motions of real three-dimensional Euclidean space X is the following theorem. (3.10) THEOREM. Every rigid motion g of X is the composition 3 R of a rotation R about the origin and a translation i. 0

Proof. Let i be the vector from the origin 0 to g(0). Then f also denotes a translation belonging to T whose inverse is - f. The rigid motion -i g is a member R of D since it maps the origin onto itself. The equality -i 0 g = R implies that 0

g=ioR.

(3.1 1)

If one begins the argument with g-' rather than g, then one obtains g=Rof'

for suitably chosen R and i'. The action of a rigid motion i c R on a point x with coordinates can be expressed in the form g1

(3.12)

=alltl

'12 = a21i'l

'13

+ a12 + a2Z

= a31i"l f

a32

+ u13

+ ( 2 + u23 5 3 + 5 2 5.7 + a 3 3 < 3 + t 3

52

(El, t 2 ,t3)

e3

3

>

where (v,, q 2 , t / 3 ) is the set of coordinates of the image (f 0 R)(x). The matrix {a,,). 1 I I , j < 3, occurring in Eq. (3.12) is the orthogonal matrix M ( R ) of the rotation R in this particular coordinate system. The determinant 1 M ( R ) / is 1 in our present discussion. I t is common to enlarge the group of rigid motions to include those transformations given by Eqs. (3.12) in which

3. The Group of Euclidean Motions of Three-Dimensional Euclidean Space

I71

the matrix {aij}is required to be orthogonal, but permitted to have determinant - 1. An orthogonal matrix M with I MI equal to - 1 is called an improper orthogonal matrix. Motions defined by Eqs. (3.12) with an improper matrix {ai,} are referred to as improper motions, reflecting the fact that such motions can not take place in the real world analog of the Euclidean space X. The larger group consisting of both the rigid and the improper motions is called the group of' Euclidean motions. The Euclidean motions are described in the literature of physics and crystallography by the symbols { R I i} where R designates the matrix {aij} and i denotes the translation part {ti}.These symbols can be given an abstract rather than a coordinate interpretationthe symbol R denoting a proper or improper rotation and the symbol 1 denoting a corresponding translation. In this context the function { RI t} maps the radius vector t into the radius vector S according to (3.13)

S = {RIi}?= Rt

+ i.

We shall not elaborate the notation to distinguish between the analytic and synthetic interpretations of the symbol { R13). The product of two Euclidean motions { RI i} and { S [6) is given by (3.14)

{RIi}{SIii} = {RSlZ

+ Rii}.

The inverse { RI i}-' is given by (3.15)

{ R I t}-'

= {R-'

I -R-li}.

We use the symbol E to denote the identity matrix or linear transformation in Sections 3 and 4 so that the identity of the Euclidean group is { E I O } . The subgroup of translations T, sometimes called the pure translations, consists o f all elements of the form {E I Z}. It follows from

(3.16)

{RIZ}{&p}{R-11 - R - l Z } =={EIRii}

that the subgroup Tof pure translations is a normal subgroup of the group of Euclidean motions. The Euclidean group is a subgroup of a larger group called the group of a8ne motions. This group consists of all transformations of the form { A I ii} where A denotes any nonsingular linear transformation. The concept of the affine group is used briefly in the sequel. The set of the group of Euclidean motions G can be identified with the Cartesian product D x Twhere D denotes the set of the rotation group and T the set of the translation group. Nevertheless, G is not the direct product of D and T since multiplication does not conform to the direct product rule (3.14')

{RIf}{SIli}= {RSli + ii},

I72

3. Computation Representations and Characters

but rather to Eq. (3.14). As a matter of fact, the Euclidean group (as well as the affine group) is an important example of the concept of semidirect product in group theory which we now define. Our definition is first phrased in a slightly nonstandard manner to fit the definitions and notations for the Euclidean group used by crystallographers and physicists. (3.17). DEFINITION. The group G is the external semidirect product of the group D and the group T if and only if (i) (ii) group (iii)

(3.18)

every element g of G is an ordered pair {dl t } with d E D and t E T ; there exists a homomorphism d + d of D into the automorphism % ( T ) of T ; the product of (dl t } and {d’l t ’ ) is given by {dl t}{d’ I t’}

= {dd’

I t(2t‘)).

We see that (3.14) agrees with (3.18) since R acts as an automorphism on the vectors of the translation subgroup. The standard definition is as follows. (3.19) DEFINITION. The group G is the external semidirect product of the group A and the group B if and only if (i) (ii) group (iii)

every element g of G is an ordered pair (a, b) With a E A and b E B ; there exists a homomorphism b + 6 of B into the automorphism % ( A ) of A ; the product of (a. h) and (c. d ) is given by (a, b)(c, d ) = (a(&), bd).

The parallel concept of internal semidirect product is given by the following definition.

(3.20) DEFINITION. The group G is the internal semidirect product of its normal subgroup A and its subgroup B if and only if every element g in G can be written uniquely in the form ab with a E A and b E B. One notes that the product of the elements ab and cd can be written (ab)(cd) = a(bch-’)bd.

where a(bcb-’) E ‘4 and bd E B. Finally we observe that the Euclidean group G is the internal semidirect product of the normal subgroup T of translations and the rotation group D leaving the origin fixed. As a simpler example, we consider the group S, whose Cayley table is shown in Table (3.20’). The subgroup A consicting the of set { I , 2, 31 is a normal subgroup of S , . The set { 1 , 4 ) is a subgroup B of S , . It can be checked quickly from the Cayley table that each element of S, can be written uniquely in the form ab with a E A and h E 13. Therefore S , is the internal semidirect product of A and B.

3. The Group of Euclidean Motions of Three-Dimensional Euclidean Space

(3.20')

173

CAYLEY TABLEOF S3 123456 1123456 2231645 3312564 4456123 5564312 6645231

We turn from the rotation and translation subgroups to a discussion of two classes of improper Euclidean motions which occur frequently in applications in physics. These two motions are properly thought of as occurring in the tangent space T(6) at the radius vector 6 , namely, inversion in the point ii and reflection in a plane in in the tangent space T(ii) with normal 5. Let the symbol Z denote the improper Euclidean motion which transforms the radius vector 7 into the vector -7. The Euclidean motion (Zl2ii) is the inversion in the point ii which transforms a vector S in the tangent space T(ii) into the vector - S . Let 5 denote a unit radius vector and A the rotation at the origin of 180" about the direction of 6. The Euclidean motion {AZI ii + Aii} denotes the rejection in the plane lit through ii whose normal is 6. These improper Euclidean motions have the geometric action which their names suggest. There are numerous other Euclidean motions with specific names in the literature of physics and crystallography. Among these are the screw axes, the rotation reflections, and the glide reflections. A screw axis is a rigid motion consisting of a rotation about a given axis followed by a translation parallel to the axis. A rotation reflection is an improper Euclidean motion consisting of a rotation about a fixed axis followed by a reflection in a plane perpendicular to the axis. A glide rejection is an improper Euclidean motion consisting of a reflection in a plane T ~ Ifollowed by a translation which is parallel to the plane. This concludes our general discussion of the Euclidean group G. We turn to the consideration of certain discrete subgroups of G (or of the affine group) of interest in the applications. These fall into three categories: abelian subgroups B of the group T of pure translations, subgroups '$ of the group D of rotations about the origin, and subgroups 6 of the affine group. These groups arise as sets of Euclidean or afine motions which leave invariant a lattice 2 of points in X. One must begin by describing the nature of such lattices of points. The definition of a lattice 2 in three-space X as a regular array of points is intuitively correct but appears, at first glance, not subject to precise analysis.

174

3. Computation Representations and Characters

Another definition is that a lattice E is the collection of all points in threespace determined by radius vectors of the form (3.21) where {VI' vz, v3 } is a set of noncoplanar vectors and the coefficients n I :::;; i s; 3, are integers. Although this definition is more useful, it does not fully describe those arrays which are commonly called lattices. Consider the two-dimensional examples sketched in Fig. (3.22). i ;

(3.22) l'

\'

"

,

-,

-,

", ",

/ /

"

//

-,

/

Y ('

~l

"

""

,

,,

-,

-,

·h' _ _ x

/

/

/

"V • /

c'

The two lattices of Figure (3.22) are very much alike. They are each built out of squares congruent to their "so-called" primitive cells, the squares abed and a'b'c'd'; which are themselves congruent. Let VI and Vz be the directed segments da and de, and VI' and vz' be the directed segments d'a' and d'e', respectively. Then the translational symmetry of 1]( is described by all translations i = niv i + n z vz, and that of'B by all translations of the form i' = mlv l' + mz v/ with 11 1, 11Z' m l , and m z integers. Consequently, the two lattices have the same translational symmetry. Nevertheless, there are important differences between the two. Each lattice point, say d for example, of 1]( is shared between four cells congruent to abed. However, each such cell has four vertices, so that we say that the lattice 1]( has one vertex per primitive cell. On the other hand, each primitive cell of lattice 'B contains an extra vertex, so that lattice 'B has two vertices per primitive cell. The extra vertex in a' b' c' d' can be located by a vector "A., not of the form mlv l' + m z v/, m l , m z E Z, called a nonprimitioe translation. A lattice such as 1]( containing only one vertex per primitive cell is called a simple lattice while a lattice such as 'B containing more than one vertex per primitive cell is called a compound lattice. Another important distinction between these two lattices is that they have different rotational symmetry. For example, a rotation of 90° about the z-axis is a symmetry of lattice Ill, but not of lattice 'B.

3. The Group of Euclidean Motions of Three-DimensionalEuclidean Space

175

A simple thee-dimensional lattice L? is a regular array of points in threespace X determined by a family of radius vectors of the form of Eq. (3.21), where the noncoplanar set {vl, v, , v3}is sometimes referred to as the primitive translation vectors of the lattice. This terminology is not too satisfactory for at least two reasons: (i) the vectors are not unique; they are not the only triple determining the collection defined by (3.21); and (ii) the term nonprimitive is used to mean a vector not of the form of Eq. (3.21) rather than not one of the set {vl, v, , v3}. A compound three-dimensional lattice L? is a regular array of points determined either by means of a simple lattice I!' together with a set of nonprimitive translation vectors, or by means of two or more simple lattices, or by a combination of these two devices. Since there are an infinite number of simple three-dimensional lattices, it is useful to seek a method of classification to simplify their study. For such a 'purpose, it is convenient to consider the rotational symmetry of a lattice. The holohedry of a simple three-dimensional lattice L? is the group G of all rotations about the origin (including the improper ones) which map the lattice !i onto ?itself. Two simple lattices I! and ??with holohedries G and G", respectively, are said to be in the same Bravais class if and only if there exists a nonsingular linear transformation A of the space X such that

(i) L? is mapped bijectively on 5 by A ; (ii) conjugation by A is an isomorphism of G onto G", that is, G" = A G A - ' . This equivalence relation partitions the set of all simple three-dimensional lattices into fourteen equivalence or Bravais classes. Naturally, one selects a particularly suitable representation from each class and calls it the Bravais lattice of its class. We sometimes use the word crystal rather than lattice, since an ideal simple crystal has the form of such a lattice. We consider two examples below, the face-centered cubic and the body-centered cubic lattices, but return at the moment to a further discussion of the translation group, point group, and space group of a lattice. All of these are subgroups of the group of affine motions. The translation group B of a lattice I! is the set of all pure translations which map the lattice L? onto itself. Let {vl, v 2 ,v3}denote the set of primitive translations of 2 . The group B is the set of all Euclidean motions of the form { E I Z ) with i given by (3.23)

i = nlvl + n, v2 + n3 v j ,

where n i , 1 2 i 2 3, is an integer. We sometimes refer to an element { E 1 i} of B as a primitive translation. This usage is justified since many elements of the form (3.23) can be taken as one member of a set of three which generate the lattice 2.

I 76

3. Computation Representations and Characters

The crystallographic point g r o u p 9 of a lattice L! is the finite subgroup of the rotation group D at the origin which maps 52 onto itself. There are eighteen classes of isomorphic point groups. However, these are divided into equivalence classes by agreeing that the groups 'p and are in the same class if and only if there exists an orthogonal transformation R on the Euclidean space X such that conjugation by R is an isomorphism of 'p onto f@, that is,

$5

(3.24)

-

M

'p = R ~ R - ' .

There are thirty-two classes of point groups using this classification. To make the distinction clear, let '$ denote the subgroup consisting of { E , A} and $ that consisting of { E , I } where A denotes a 180" rotation at the origin and Z denotes inversion in the origin. Each of these groups is isomorphic to the cyclic group of order two and hence algebraically isomorphic to the other. Since A and I have determinants 1 and - 1, respectively, there exists no R such that (3.24) is satisfied, that is, ?, and $ are not equivalent under this stronger relation. A point group I; is said to be of the first kind if each of its elements has determinant 1. It is said to be of the second kind if some of its elements have determinant - 1 . In the last case, it is easy to see that one-half of the elements have determinant 1 and the other half - I . The crystallographic space group 6 of a lattice 2 is the subgroup of the affine group which maps 2 onto itself. There are 219 classes of algebraically isomorphic space groups. Again the crystallographer places a stronger equivalence relation on these groups. Two space groups 6and 6 are said to be equivalent if and only if they are conjugate subgroups of the group of affine motions on the space X. This means there exists an affine motion { A ji} such that conjugation by { A it} is an isomorphism of 6 onto G.There are 230 classes of space groups under this equivalence relation. Again one selects a particularly suitable member from each class and refers to it as the space group. It proves possible to select a representative from each class that belongs to the Euclidean group G. Thus one may, as we shall, discuss the space groups as subgroups of the Euclidean group. Any space group 6 consists of the set {{Rli}}of all Euclidean motions which map some lattice 2, not necessarily simple, of the Euclidean space X onto itself. The set of all rotations { R : { R I I } E G } forms a g r o u p v called the point group o j the space group G. This group is to he distinguished f r o m the set of rotations which leave the lattice S invariant. The group may not be isomorphic to a subgroup of 6 in the sense that there may exist an element E I; which does not appear as a first element of a motion in G of the form ( a 10). In contrast, the translation subgroup 23 of the space group 6, consisting of all pure translations in 6, is exactly the subgroup of the translation group T of the Euclidean group G which leaves the lattice L! invariant Furthermore, there may exist an element { R I i] in 6 without { E I i} being in 23

3. The Group of Euclidean Motions of Three-Dimensional Euclidean Space

177

These oddities follow from the fact that a space group 6 need not be a semidirect product of its translation subgroup % (a normal subgroup) by its point group ’$. There are 73 space groups, called the symmorphic space groups, each of which is the semidirect product of its translation subgroup ‘23 by its point group ’$. There are 157 space groups, called iionsynzmorphic space groups, each of which is an extension, but not a semidirect product, of its translation subgroup 23 by its point group ‘$. This finishes our brief introduction into the basic concepts of the point and space groups. The details of the subject are endless. The standard reference is International Tables for X-Ray Crystallography,” published for the International Union of Crystallography by the Kynock Press, Birmingham, England, in 1952. This work employs the Hermann-Manguin description of the various groups. An earlier notation, also in common use, is that introduced by Schoenflies. Each of the notations is functional, that of the HermannManguin system highly so, but it is beyond the scope of this book to examine them in detail. We conclude this section with four examples of crystallographic groups. These consist of two space groups, Td2and oh’, together with their respective point groups, Td and o h . We give the Schoenflies designation for the group followed by that of the Hermann-Manguin system in parentheses. “

(i) The group Td(43m)is a point group of the second kind which consists of the twenty-four rotations, twelve proper and twelve improper, which map a regular tetrahedron with its center at the origin onto itself. It is sometimes called the group of symmetries of the tetrahedron and is algebraically isomorphic to the group of permutations on four objects. (ii) The group Oh(m3m) is a point group of the second kind which consists of forty-eight rotations, twenty-four proper and twenty-four improper, which map a cube with its center at the origin onto itself. It is called the group of symmetries of the cube. (iii) The space group of the zinc blende crystal Td2(F43m)consists of all Euclidean motions which transform an ideal, infinite zinc blende crystal onto itself. (iv) The space group of the diamond crystal Oh’(Fd3m) consists of all Euclidean motions which transform an ideal, infinite diamond crystal onto itself. These four groups are conveniently discussed together since all of them are intimately related to the symmetry properties of the cube. As an example of the symmetries of geometric objects and in preparation for our studies of these geometric groups, let us examine more closely the nature of the rotations which map the cube of Fig. (3.25) onto itself. We refer to such rotations as

I78

3. Computation Representations and Characters

(3.2

rotationul symmetries of the cube and to the axis of such symmetry axis of the cube.

a rotation as a

First note that a nontrivial rotation about any axis not passing through the center of the cube will not bring the cube back onto itself. Thus all nontrivial rotational symmetries of the cube must be rotations about axes passing through its center. In the same way, a symmetry axis which cuts the interior of a face of the cube must pass through the center of that face, and one which cuts the interior of an edge of the cube must pass through the center of the edge. Thus we may determine the nontrivial rotational symmetries of the cube by restricting our attention to (i) (ii) (iii) of the

the three axes that pass through the centers of two opposing faces; the six axes which pass through the centers of two opposing edges; those four axes passing through the center and two opposing vertices cube.

It is now easy to count the nontrivial, proper rotations which belong to O h , the group of symmetries of the cube. Each of the axes in class (i) gives rise to three rotations of magnitudes 90°, 180", and 27(3", respectively, and all three of them to nine symmetries. Each of the axes in class (ii) gives rise to

3. The Group of Euclidean Motions of Three-Dimensional Euclidean Space

I79

a single rotation of magnitude 180°, and all six of them to six symmetries. Finally, each of the axes in class (iii) gives rise to two rotations of magnitudes 120" and 240", respectively, and all four of them to eight symmetries. These twenty-three rotations together with the identity rotation constitute the twenty-four proper rotations in the symmetry group 0, of the cube. This subgroup of proper rotations is a point group of the first kind, sometimes referred to as the octahedral group and denoted by O(432). One can obtain the group 0, from the group 0 by various means. Let E , I , and m denote the Euclidean motions consisting of the identity, inversion in the origin, and reflection in some coordinate plane, respectively. Then the sets { E , I } and { E , m} are cyclic subgroups of order two of the Euclidean group G. We denote the first of these by Ci and the second by C, . The group 0, is the direct product of 0 and Ciand the semidirect product of 0 and C , . Each presentation is equally valid, but the first is more useful for representation theory since it easily reduces the representation theory of Oh to that of 0 by the method described below Definition (1.30). Let A denote the tetrahedron inscribed in the cube of Fig. (3.25) with vertices the set {a, b, c, d}. The proper rotational symmetries of A is a subgroup T(23), called the tetrahedral group, of twelve elements which are all contained in the octahedral group 0. We describe these elements in more detail below. At the moment, we note that the rotation R of magnitude 90" about the x-axis (or any other of the coordinate axes) is an element of 0 which is not an element of T. One can easily prove that the remaining twelve elements of 0 are obtained by multiplying each element of T in turn by the rotation R. In particular, T is a subgroup of 0 of index two. The improper Euclidean motion in consisting of a reflection in the plane determined by the set of vertices {c, d, h, e ] is a symmetry of A. The group Td(a3m) consists of T together with the twelve additional elements obtained by multiplying each element of T i n turn by the improper Euclidean motion in. Thus, T is also a subgroup of Td of index two. The elements of Tdcan be identified as members of the permutation group S, on the set {a, 6 , c, d } of vertices of A. With such an identification, the elements of T make up the subgroup A4 of even permutations in S, . We list the symmetries of Td in terms of permutations in Table (3.26) and those of 0 as matrices in Table (3.27). The transformation of the cube induced by carrying out the indicated permutation of the vertices or the alibi coordinate transformation defined by the matrix is the corresponding symmetry, of course. The function f :Td-+ 0 that assigns to the permutation numbered n the matrix numbered 12 is an algebraic isomorphism of Td onto 0. The map f does not preserve the geometric action of Td, but it is a faithful irreducible representation nevertheless. One can define a second isomorphism h of Td by means of the group of matrices 0. For 1 I n 5 12 define h(n) =f(n), but

180

3. Computation Representations and Characters

for 13 < n 5 24 define h(n) = -f(n). The function h is an irreducible representation which preserves the geometric action of Td. Thus we have discovered a pair of faithful irreducible representations of Td which are inequivalent. (3.26)

THEGROUPTn AS PERMUTATIONS 1 (a) (ubc) 9 (ucb) 13 (ub) 17 (uc) 21 (bc) 5

(3.27)

(uc)(bd) (a&) (bcd) (udbc) (bd)

(ucbd)

3 7 11 15 19 23

THEGROUP0

AS

2 6 10 14 18 22

(ud)(bc) (ucd)

(udb) (ucbd) (ubcd)

(ad)

4 8 12 16 20 24

(ub)(cd) (bdc)

(udc) (cd) (udcb)

(ubdc)

ROTATIONS

The Cayley Table of Td and 0 is given in Table (3.28) using the enumerations of Tables (3.26) and (3.27).

181

3. The Group of Euclidean Motions of Three-Dimensional Euclidean Space

(3.28)

CAYLEY

5 6 7 8

6 5 8 7

7 8 5 6

1 2 3 4

2 3 4 1 4 3 4 1 2 3 2 1

5 6 7 8

8 7 6 5

6 5 8 7

9 10 11 12

11 12 9 10

12 I1 10 9

10 1 9 2 12 3 11 4

13 14 15 16

15 16 13 14

14 13 16 15

16 15 14 13

21 22 23 24

23 24 21 22

22 21 24 23

17 18 19 20

18 17 20 19

20 19 18 17

19 20 17 18

13 14 15 16

14 13 16 15

16 15 14 13

21 22 23 24

24 23 22 21

23 24 21 22

22 21 24 23

17 18 19 20

20 19 18 17

19 20 17 18

7 9 12 10 8 10 11 9 5 11 10 12 6 12 9 11 3 4 4 3 1 2 2 1

8 7 6 5 11 12 9 10 2 1 4 3 24 23 22 21 15 16 13 14 18 17 20 19

TABLE FOR Td

AND

11 12 9 10

15 16 13 14

16 15 14 13

17 18 19 20

18 17 20 19

19 20 17 18

20 19 18 17

21 22 23 24

22 21 24 23

23 24 21 22

24 23 22 21

19 20 17 18

21 22 23 24

24 23 22 21

22 21 24 23

23 24 21 22

13 14 15 16

16 15 14 13

14 13 16 15

15 16 13 14

22 21 24 23

13 14 15 16

15 16 13 14

16 15 14 13

14 13 16 15

17 18 19 20

19 20 17 18

20 19 18 17

18 17 20 19

4 9 11 10 12 5 3 10 12 9 11 6 2 11 9 12 10 7 1 12 10 11 9 8 7 1 2 4 3 9 8 2 1 3 4 10 5 3 4 2 1 11 6 4 3 1 2 12

7 8 5 6 10 9 12 11

6 5 8 7 12 11 10 9

8 7 6 5 11 12 9 10

4 3 3 4 2 1 1 2

2 1 4 3

9 10 11 12

10 9 12 11

12 11 10 9

13 14 15 16

14 13 16 15

1 2 3 4

4 3 2 1

2 3 17 20 18 1 4 18 19 17 4 1 19 18 20 3 2 20 17 19

6 7 8

5

7 8 5 6

8 7 6 5

6 21 5 22 8 23 7 24

23 24 21 22

24 23 22 21

17 18 19 20

19 20 17 18

18 17 20 19

20 1 19 2 18 3 17 4

3 4 1 2

2 1 4 3

21 22 23 24

22 21 24 23

24 23 22 21

23 24 21 22

5 6 7 8

6 5 8 7

8 7 6

13 14 15 16

16 15 14 13

15 16 13 14

14 13 16 15

9 10 11 12

5

0

12 11 10 11 12 9 10 9 12 9 10 11

5 6 7 8

8 7 6 5

7 8 5 6

6 5 8 7

1 2 3 4

We record the order of each element of T, and the list of conjugacy classes for convenience as in Tables (3.29) and (3.30), respectively.

(3.29)

ORDER OF

Element: Order: Element: Order:

(3.30)

ELEMENTS OF Td

1 2 3 4 5 6 7 8 1 2 2 2 3 3 3 3 13 14 15 16 17 18 19 20 2 4 4 2 2 2 4 4

91011 3 3 3 21 22 23 2 4 2

CONJUGACY CLASSES OF Td Cl

=

U),

c*= {2, 3,41,

C1 = {5, 6, 7, 8, 9, 10, 11, 12}, Cq= {13, 16, 17, 18, 21,23}, C5 = {14, 15, 19, 20, 22, 24).

12 3 24 4

182

3. Computation Representations and Characters

The group O,, is the direct product of 0 and the subgroup Ci consisting of the set { E , I ) defined above. Since the element I has the matrix

11-i -;_81; 0

it follows that the remaining improper rotations in 0, may be obtained by

merely changing the signs of all the elements of matrices which appear in the list of 0 tabulated as rotations. We adopt the following notation: Let n be an integer with 24 < n 5 48, then n denotes the matrix obtained from the matrix of n - 24 in Table (3.27) by changing the signs of each of its entries. The set of forty-eight matrices determined in this way constitutes the complete set of the forty-eight elements of the symmetries of the cube. We use the symbol R,, 1 I nI 48, to denote elements of Oh in the discussion below. For the most part, all of the information needed about the group 0, can be obtained from Tables (3.27) and (3.28) with a judicious insertion of minus signs. For longhand calculations, the realization of T, as S4 is perhaps the most convenient. We leave the details to the reader and turn to a survey of the space groups Td2and Oh7. We start by discussing the related Bravais lattice 8, called the face-centered cubic lattice, which determines the translational symmetry of both the ideal diamond and ideal zinc blende crystals. This lattice 8 is defined by means of a basic iinit cell (not the primitive cell) which can be taken to be a cube of side a with one vertex at the origin and the remaining ones on the first octant of a rectangular coordinate system. An incomplete drawing of such a cell in an inverted position is given in Fig. (3.31). The drawing is given incompletely (3.31)

t

183

3. The Group of Euclidean Morjons of Thee-Bimensionaf EucfideanSpace

and in such a position that the standard set {vl, v 2 , v,} of primitive basis vectors can be seen more clearly. A drawing showing all vertices in the basic cell is given in Fig. (3.32). The component forms of the vectors vl, v 2 , and (3.32)

v3 are (a/2)(0, I , l), (a/2)(1, 0, l), and (a/2)(1, 1 , O), respectively. The parallelepiped spanned by the set {vl, v 2 , v,) is the standard primitive cell of the face-centered cubic lattice. As such, it contains only one vertex in the same sense as our two-dimensional example given above. The basic unit cell sketched in Fig. (3.32) is not a primitive cell. Rather it is merely a convenient unit (containing four vertices) whose translates generate the face-centered cubic lattice. Any other set {vl’, v2’, v,’} determined from the set {vl, v2, v3} by means of an integral unimodular matrix transformation is another set of primitive basis vectors for the lattice 8.The point group Ohis the holohedry of the face-centered cubic lattice. The space group Td2and 0,’have a common translation subgroup 23 consisting of all Euclidean motions of the form (6

I

$19

(3.33)

t=n,v, + n , v , +n,v,,

where the coefficients n,, 1 I i 3, belong to 2. The lattice of either the diamond or zinc blende crystal is a compound lattice composed of two face-centered cubic lattices and with their crystal axes parallel and with the origin of coordinates of the lattice g2lying one-quarter of the way along the principal diagonal of the cube, at the point marked 4, in Fig. (3.32). The result is sketched in Fig. (3.34) where only four vertices from the second lattice those marked 1 , 2, 3, and 4, are shown. All vertices of such an array are occupied by atoms in either the diamond or zinc blende crystal. These vertices are referred to as sites in the physics literature. In the diamond crystal, each site is occupied by a single carbon

sl

s2,

s2

184

3. Computation Representations and Characters

atom. In the zinc blende crystal, each site of the lattice gl is occupied by a single zinc atom and each site of the lattice iJ2is occupied by a single sulphur atom. The translational symmetry of both the diamond and the zinc blende crystal is completely determined by the subgroup 3 defined above and is the same as that of the simple face-centered cubic lattice. The reader should convince himself of this and that the selection of an origin has nothing to do with the translational symmetry of these crystals. There are many other compounds which crystallize in the zinc blende structure. One of these is cubic boron nitride which we also discuss. Thus we begin to talk of boron nitride rather than zinc blende. The difference between the diamond and the boron nitride crystals lies in their rotational rather than their translational symmetry. To see how, one concentrates upon the upperfront octant of the cube sketched in Fig. (3.34). We make the task easier by reproducing this portion alone in Fig. (3.35) where the corresponding spheres are numbered as in Fig. (3.34). The five spheres of Fig. (3.35) all represent carbon atoms in the diamond crystal. However, the central sphere 2 represents a boron atom and its neighboring four spheres represent nitrogen atoms in the boron nitride crystal. The elements of the set {I-, h,,3L2,a,) of vectors joining 2 to its nearest neighbors (in the terminology of the physicist) are not vectors belonging to the translation group %, that is, la,} %. The origin of coordinates can be taken to belong to either 31or 5 2 .It is convenient for our

+

3. The Group of Euclidean Motions of Three-DimensionalEuclidean Space

(3.35)

185

6

present discussion to consider the origin at site 2. The coordinates of I , A,, -1, --I), (a/4)(1, - I , I), (a/4)(-1, 1, l), and (a/4)(1, I , - I), respectively. Every site which differs from that of 2 by a vector of the form of Eq. (3.33) is situated exactly as 2 . One notes that all of the Euclidean motions of T, map the tetrahedral configuration of Fig. (3.35) onto itself. Their effect on cells not so located at the origin is somewhat more complicated, but results in the crystal being mapped upon itself. Thus every element R of the group T, is in the point groups of both Oh7and T,'. The space group T,' consists of all Euclidean motions { R 1 f} where R belongs to T, and i is given by Eq. (3.33). The inversion I in the origin is not a symmetry of the tetrahedron. Its action on the cube of Fig. (3.35) is to fill the empty corner sites and empty the full ones. Yet, the diamond crystal (although not the boron nitride) can be brought back into coincidence with itself after inversion by the translation {EIL}. Thus {Ilh} is a symmetry of the diamond crystal which is not a symmetry of the boron nitride crystal. This result implies that

L2,a n d I , are then (a/4)(-1,

(3.36)

{RlX

+ i} = {ZIX}{R'I -i}

is a symmetry of the diamond crystal for every choice of the primitive translation i and of the rotation R' of the group T, . The space group Oh7consists of all the elements of T,' plus those of the form of (3.36) where R = IR' denotes an element of Oh not belonging to T,. To see that the products of elements such as those of (3.36) multiply properly, one first checks that Li- I, 1 5 i 5 3, is a primitive translation. Then notes R I + I is 0 or a primitive translation coinciding with -Ii + I for some i whenever R is an element of Oh not in T d .To see that Td2is a normal subgroup of Oh7,it is sufficient to observe that conjugation of an element { R ( i )of Td2 by {IIL), namely, (3.37)

{IJI}{RJi}{Ilh)= {RI -i - R I

+ A}

186

3. Computation Representations and Characters

is an element of Td2,since RI - 3, is an element of the form of Eq. (3.33). However, 0,,7is not the semidirect product of Td2and a suitable subgroup, but rather an extension, so that the representations of T; cannot be used in a completely straightforward manner to compute those of Oh7. We consider the representations of these groups in the next section. 4. THE IRREDUCIBLE REPRESENTATIONS OF CERTAIN POINT AND SPACE GROUPS

In this section we illustrate the method of induced representations by applications to certain finite and special infinite groups. Detailed calculations are made of the irreducible representations of the groups T,, 0, and oh. The representation theory of the space groups Td2 and Oh7is discussed in less depth. Tables have been prepared containing the information needed at various stages of the calculations for the reader’s convenience. In addition to the method of induced representations, other useful devices are introduced to handle cases of self-conjugate representations where induction is less satisfactory. We start with the tetrahedral group T which is a subgroup of index two in both T, and 0. Although the Cayley table of T can be read immediately from Table (3.28), we repeat it (see Table (4.1)) for convenience. The group T has four classes of conjugate elements: SZ, = ( I ) , R, = (2, 3, 4}, S 3 = { S . 6. 7, X}, and SZ, = (9, 10, 1 1 , 12). One easily finds by experimentation with Eq. (5.31), Chapter 2, that T has three classes of complex one-dimensional representations and one class of complex three-dimensional irreducible representations. The subgroup H = { I , 2, 3, 41 is a normal subgroup of index three in T . One sees from the Cayley table of Tthat If is isomorphic to C, @ C, The character table (irreducible representations) of C, @ C , is given in Table (S.38), Chapter 2, and repeated in Table (4.2).

(4.1 )

CAYLFY TABLEOF T ~

~~

1 2 3 4 5 6 7 8 9101112 2 1 4 3 6 5 8 710 91211 3 4 1 2 7 8 5 61112 910 4 3 2 1 8 7 6 5121110 9 5 8 6 7 9121011 1 4 2 3 6 7 5 81011 912 2 3 1 4 7 6 8 5111012 9 3 2 4 1 8 5 7 612 91110 4 1 3 2 9111210 I 3 4 2 5 7 8 6 101211 9 2 4 3 1 6 8 7 5 I I 91012 3 1 2 4 7 5 6 8 1210 911 4 2 1 3 8 6 5 7

4. The Irreducible Representations of Certain Point and Space Groups

(4.2)

CHARACTER

TABLEOF

187

cz @ c z

K I Kz K3 K4 1 1 1 x2 1 1 - 1 -1 x3 1 -1 1-1 x4 1 -1 -1 1 X I 1

Let t be an n-dimensional matrix representation of a normal subgroup K of a group G. Recall that for each g E G , the conjugate t , is an n-dimensional matrix representation of K defined by

(4.3)

t,(k) = t(g-lkg>,

k

E K.

A representation t of K is said to be self-conjugate if t, is equivalent to t for every element g of G. To check for self-conjugacy of a representation t , it is sufficient to check its conjugates by a complete set of coset representatives of K in G. In applying these ideas to the normal subgroup H of T, we make no distinction between the irreducible characters of H and its corresponding irreducible representations since these are one-dimensional. Thus we need examine the four irreducible representations of H only under conjugation by the elements 5 and 9. It is easily seen that the 1-representation x1 is selfconjugate. One verifies by calculation that x s 2 , defined by (4.4)

x S 2 ( h )= x2(9h5),

h E H,

coincides with x4. Similarly, one finds that x g 2 agrees with x3. Thus the irreducible representations of H split into two orbits with respect to T, namely, {x'> and {x2, x3, x">. It follows either from Theorem (2.37) or one of its corollaries that the representation of T induced from x1 is a reducible representation, but that the representation induced by any one of the set {x2, x3, x"} of conjugate representations is an irreducible representation of T. Thus we are able to find a suitable irreducible, three-dimensional representation by induction. The representation x1 can be made to furnish three distinct one-dimensional representations of T. This is the number required. The group H is generated by the elements 2 and 3 which satisfy the defining relations (4.5)

22 = 1,

32 = 1,

2(3) = 3(2).

The tetrahedral group can be obtained from H (not in the standard presentation) by adding one more generator 5 with the defining relations

(4.6)

5 3 = 1,

2(5) = 5(3).

188

3. Computation Representations and Characters

It is easy to pick out three distinct sets of complex numbers (2', 3', 53, namely, { I , I , l}, { I , 1. E } , and { 1, 1, E ' } , which satisfy these relations where E is a primitive cube root of unity. Each such set gives rise to an irreducible representation of T. These representations are determined by the correspondence 2 + 2', 3 + 3', and 5 + 5', where the primed quantities are taken successively from each of the three sets listed above. We discuss this idea more fully below. The final results are listed in Table (4.9) as the three distinct one-dimensional representations of T. Here n denotes (cos 30 + / sin 30)'' when n is positive, but denotes zero when n = 0. The induced representation xZTis determined by the equations, (4.7)

The arguments gi-'ngj, 1 S i, j 2 3, 1 2 n I 12, are listed in Table (4.8), where g1 = 1 , ,q2 = 5, and g 3 = 9. (4.8)

TABLEOF CONJUGATES 1 2 3 4 5 6 7 S 9101112

1 2 3 4 5 6 7 8 9101112 9111210 1 3 4 2 5 7 8 6 5 S 6 7 9121011 1 4 2 3

5 6 7 8 9101112

9101112 1 2 3 4 1 2 3 4 5 6 7 8 1 3 4 2 5 7 8 6 9111210 9121011 1 4 2 3 5 8 6 7

5 7 8 6 9111210 1 3 4 2 1 4 2 3 5 8 6 7 9121011

An irreducible three-dimensional representation of T can be read off immediately between Tables (4.2) and (4.8) by means of Eq. (4.7). Thus we are able to write down a representative from the class of equivalent irreducible three-dimensional representations of T. The results are given in Table (4.9). We could have obtained dilTerent, but equivalent representations of T by induction on either of x3 or x4. As a matter of fact, it would have been more economical of effort in finding the representations of T, to have used x4. The Cayley table of T, (or 0 ) is given in Table (3.28). One finds that Td has five classes of conjugate elements: K , = {l), K , = (2, 3,4}, K , = (5, 6 , 7, 8, 9, 10, 11, 12}, K4 = (13, 16, 17, 18, 21,23}, and K , = (14, 15, 19, 20,22,24}. Consequently, T, has five classes of equivalent complex irreducible representations: two of which are one-dimensional, one of which is two-dimensional, and two of which are three-dimensional.

4. The Irreducible Representations of Certain Point and Space Groups

(4.9)

189

IRREDUCIBLE REPRESENTATIONS OF 7

T(1) T(2) T(3) T(4)ii T(4)12 T(4)13 T(4)zi T(4)zz T(4)23 T(4)3i T(4)32 T(4)33

1 2 3 4 5 6 7 8 9101112 12 12 12 12 12 12 12 12 12 12 12 12 12121212 4 4 4 4 8 8 8 8 12 12 12 12 8 8 8 8 4 4 4 4 12 12 6 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 12 6 6 0 0 0 0 12 12 6 6 0 0 0 0 0 0 0 012 6 612 0 0 0 0 12 6 6 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 6 6 12 0 0 0 0 0 0 0 0 12 6 12 6 0 0 0 0 12 6 12 6 0 0 0 0 12 6 12 6 0 0 0 0 0 0 0 0

The self-conjugate representation f = T(1) of the subgroup T of Td produces two inequivalent one-dimensional representations of Td. These are obtained by essentially the same method as that used to derive the three inequivalent one-dimensional representations of T from the 1-representation x1 of H = {1,2,3,4}. To review the basic idea, let t be a matrix representation of Td . Then t is determined by its values on the set (2, 3, 5, 13) of generators of T d .The group Td itself is determined, of course, by these generators together with a suitable set % ' of defining relations. Conversely, given a set ( M Z , M , , M , , M,,} of nonsingular matrices satisfying the set '% of defining relations, there exists a matrix representation t of Td defined by (4.10)

[(x) = M , ,

x

E ( 2 , 3,

5, 131,

and extended to the group Td by the defining relations. Generally speaking, it is not practical to seek such a family of matrices in order to determine an irreducible representation of a group. However, if one begins with a knowledge of the set { M 2 , M 3 , M 5 } of matrices, then it is sometimes worthwhile to look for a matrix M I , which together with the others satisfies the defining relations. This is certainly true in the case of a one-dimensional representation and especially so in the case where the known matrices are just the number 1. Given that 1-representation T(l) of T , denoted by t , it follows that t(2) = 43) = @ ) = 1, so that one has to concern oneself only with those defining relations between the generator 13 and the remaining set (2, 3, 5) of generators. These are (4.11)

2(13) = 13(3), 3(13) = 13[3(2)], 5(131 = 13[5(5)], 132 = 1.

190

3. Computation Representations and Characters

If Y denotes some representation of T, which agrees with the I-representation t on T , then the only restriction implied by Eqs. (4.1 1 ) is that [r(13)]' = I . Thus one obtains two inequivalent, one-dimensional representations r and r' of T, : one by taking the value r(13) = 1 and the other by taking the value Y'( 13) = - 1. Each of these representations coincides with the 1-representation on T. The same sort of method sometimes works for higher-dimensional representations when the new generator t o be added is sufficiently wellbehaved. For further details, see Boerner (1963, pp. 95-101). The representations T(2) and T(3) are mutually conjugate so that they give rise to a representative element from the class of two-dimensional irreducible representations of T, . We have found two inequivalent three-dimensional representations of Td in the previous section; however, we use a different approach here to illustrate the use of some of the theorems on induced representations. The subgroup H = { I , 2, 3, 4, 13, 14, 15, 16) is of index 3 in T, which decomposes into the left cosets of H in T, according to

T, = { I , 2, 3, 4, 13, 14, 15, 16) u ( 5 , 6, 7, 8, 17, 18, 19,20} u {9, 10, 11, 12,21,22, 23,24} = H u 5H u 9H. One notes from the Cayley table of T, that the subgroup H is isomorphic to the dihedral group of order eight. Its classes of conjugate elements are K , = ( I ) , K2 = (2, 31, K , = {4}, K4 = (13, 16), and K , = (14, 15). I t has two conjugate subgroups J

= 9 H 5 = (1,

2, 3, 4, 17, 18, 19, 20)

and

K

=

5H9

= {1,2,

3,4, 21,22,23,24]

in T,. The one-dimensional representations of H are listed in Table (4.12) whose entries are the rulues, not their exponents, of the representation. We now show that the irreducible one-dimensional representations d 3 and d 4 of H induce nonequivalent irreducible representations of T , . First, we

(4.12)

ONE-DIMENSIONAL REPRESENTATIONS OF H d' (I2

d3 d4

1 1

I

1-1 I -I

2

3 1

1

I

-1

-I

I

4 13 14 15 16 1 1 1 1 1 1 - I - I -1 - 1 1 1 - 1 -1 1 1-1 1 1-1

4. The Irreducible Representations of Certain Point and Space Groups

191

employ Theorem (2.37) to show that each of the induced representations is irreducible. By this theorem, we must show that, for all g in G not in H , the restriction of t and its conjugate t ( g ) to H ( g ) are disjoint where H ( g ) is the subgroup H n g H g - ’ . However, we have already observed that one need concern oneself only with a complete set, say { 5 , 91, of coset representatives of H in T d . These two elements determine the subgroups H ( 5 ) and H ( 9 ) given by H(5) = H n 5H9 = H n K

= (1,

2, 3,4]

and = H n J = (1,

H(9) = H n 9H5

2, 3, 4).

The pertinent one-dimensional representations of Q in Table (4.13).

(4.13)

= H(5) = H ( 9 ) are

listed

IRREDUCIBLE REPRESENTATIONS OF (2 ~~~~

1 2 3 4 1 - 1 -1 1 1 1-1 -1 1-1 1-1

82 &3 &4

The representations dHH(g,3 and d,,(g)4 coincide for either g = 5 or g = 9 and correspond to the irreducible representation E’ of Q. Consequently, their conjugate representations also coincide. Thus, if we show that d 3 induces an irreducible representation d 3 T dof T d ,it will follow that d4 induces an irreducible representation d4Tdof Td. The conjugate representation d3(5) defined by d 3 ( 5 ) ( x )= d3(9x5), is the irreducible representation defined by

E~

x

E H(5) =

Q,

of Q. The conjugate representation d 3 ( 9 )

d 3 ( 9 ) ( x )= d3(5x9),

x

E

H(9) = Q ,

is the irreducible representation c3 of Q. Thus both d 3 ( 5 ) and d 3 ( 9 ) are disjoint from dH(g13 = E’ (g = 5 or 9 ) which implies, by Theorem (2.37), that d 3 induces an irreducible representation of Td. We use Theorem (2.41) to show that d3Tdand d4Tdare inequivalent irreducible representations of T d . In the present application, the subgroups H

I92

3. Computation Representations and Characters

and K of Theorem (2.41) are each H and the modules M and N correspond to the representation spaces of d 3 and d4, respectively. Since H = K in this application, the subgroup J of Theorem (2.41) corresponds to the subgroup H(g)(=Q) of Theorem (2.37) whenever g does not belong t o H . In such cases, the above argument shows that the KH(g)-modules g 0 M and N are diTjoint. When g E H , say g = I , then H(g) = H and the modules 1 @ M e M and N are disjoint H ( y ) modules corresponding to the inequivalent irreducible representations d 3 and d4. It follows that d 3 and d4 induce inequivalent irreducible representations of T, . The information needed to calculate the induced matrix representations by means of Eq. (2.14) is tabulated i n Tables (4.8) and (4.14). The symbols k , and k , stand for the elements 1 and 13 while the symbols g l , gz , and g3 stand for 1 , 5 , and 9, respectively, in these tabulations. The results of the calculation are given in Table (4.1 5).

(4.14)

CONJUGATION TABLE 1 2 3 4 5 6 7 8 9101112 I 2 3 4 5 6 7 8 9101112 13 14 15 16 17 18 19 20 21 22 23 24 13 15 14 1 6 2 1 2 3 2 2 2 4 1 7 1 9 1 8 2 0 1 3 2 4 9111012 5 7 6 8 13 14 15 16 17 18 I9 20 21 22 23 24 13 14 15 16 17 18 I9 20 21 22 23 24 I 2 3 4 5 6 7 8 9101112 1 3 2 4 9111012 5 7 6 8 13 15 14 16 21 23 22 24 17 19 I8 20 13 14 15 16 17 18 19 20 21 22 23 24 13 1 4 1 5 16 17 18 1 9 2 0 2 1 2 2 2 3 24 21 22 23 24 13 14 15 16 17 18 19 20 17 18 19 20 21 22 23 24 13 14 15 16 2 1 2 3 2 4 2 2 1 3 15 16 14 1 7 1 9 2 0 18 17 19 20 18 21 23 24 22 13 15 16 14 13 15 16 14 17 I 9 20 18 21 23 24 22 17 20 18 19 21 24 22 23 13 16 14 15 13 16 14 15 17 20 18 19 21 24 22 23 21 24 22 23 13 16 14 15 17 20 18 19

In Table (4.15) 17 stands for (cos 15 + i sin 15)" for n positive, otherwise it denotes the integer zero itself. This table also gives the irreducible representations of 0 since T, and 0 are algebraically isomorphic altliough they are not equiz>alentpoint groups. To obtain the irreducible representations of O h ,one recalls that Oh is the direct product of 0 and the group C iwhose irreducible

4. The Irreducible Representations of Certain Point and Space Groups

(4.15)

193

IRREDUCIBLEREPRESENTATIONS OF Td

I 2 3 4 5 6 7 8 9 l o l l 1 2 1 3 1 4 15 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 12 12 12 12 12 12 12 12 12 12 12 12 24242424 8 8 8 816161616 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 024242424 8 8 8 816161616 0 0 0 0 0 0 0 0 0 0 0 02424242416161616 8 8 8 8 2424242416161616 8 8 8 8 0 0 0 0 0 0 0 0 0 0 0 0 24121224 0 0 0 0 0 0 0 024121224 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 024121224 0 0 0 024121224 0 0 0 0 0 0 0 024121224 0 0 0 0 0 0 0 0 0 0 0 024121224 0 0 0 024122412 0 0 0 0 0 0 0 024122412 0 0 0 0 24122412 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 024122412 0 0 0 0 0 0 0 02412241224122412 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 024241212 0 0 0 0 0 0 0 024241212 0 0 0 024241212 0 0 0 024241212 0 0 0 0 0 0 0 0 24241212 0 0 0 0 0 0 0 0 0 0 0 024241212 0 0 0 0 24121224 0 0 0 0 0 0 0 012242412 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 024121224 0 0 0 012242412 0 0 0 0 0 0 0 024121224 0 0 0 0 0 0 0 0 0 0 0 012242412 0 0 0 024122412 0 0 0 0 0 0 0 012241224 0 0 0 0 24122412 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 012241224 0 0 0 0 0 0 0 02412241212241224 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 024241212 0 0 0 0 0 0 0 012122424 0 0 0 024241212 0 0 0 012122424 0 0 0 0 0 0 0 0 24241212 0 0 0 0 0 0 0 0 0 0 0 012122424 0 0 0 0

representations (characters) appear in Example (1.3 1). By previous observations, each irreducible representation % of 0 defines two irreducible representations %+ and 'K of 0, according to

(4.16)

%+(n)

=

%(n) = K ( n )

1 5 n 5 24,

and (4.17)

When % runs through a complete family of irreducible representations of 0, the corresponding set of 93''s and %-'s constitute a complete family of irreducible representations of oh. This family can be obtained immediately from Table (4.15) by appropriate insertion of minus signs. This finishes our discussion of the irreducible representations of 0, T,, and O,, . We next turn our attention to the determination of the irreducible representations of Td2 and 0,'.

194

3. Computation Representations and Characters

The usual point of departure is the study of the irreducible representations of the translation subgroup 23 common t o Td2and Oh7.Recall that 23 is determined by the face-cei7tered cubic lattice rr

tY =

U : f =n l v l + n2vz + n,v,>,

where n j E Z and the vectors v i , 1 5 i I 3, are now given by (a/2)(0,1, I), (~7,/2)(1,0, l), and (a/2)(1, I , O), respectively. The group 23 consists of all Euclidean motions { E 1 i}where i E 5. One notes that 23 is an infinite discrete group isomorphic to the direct product of three copies of the integers Z . It follows from a famous theorem of Pontrjagin's, see Pontrjagin (1946, Chap. V), that the character group of 23 can be identified with the three-dimensional torus. This means that each irreducible unitary representation (character) of 23 can be specified by a triple of complex numbers {exp 2 7 4 i, exp 2x2, i, e x p 2 d 3 i), where 0 I i.i5 I . It is convenient (and customary in the physics literature) t o introduce the torus in an oblique manner which we now discuss. There is associated with the face-centered cubic lattice 5 a second lattice 93. called the body-centered cubic lattice. The lattice 23 is determined by the set of radius vectors (4.18)

23

= {b:

b

= iilsl

+ n 2 s2 + n3 s 3 ] ,

where n , E Z and the vectors s , , 1 I iI 3, are of the form (l/a)(- I , 1, I), (l/o)(l, - 1 , l ) , and (l/a)(l. 1, -1). respectively. The lattices 5 and 23 are said to be reciprocal or dual to each other. One has (4.19)

(v,,s J ) = A,, ,

1I i,j I 3,

so that

(4.20)

(f, b)

=z

EZ,

fE

8,

b

E

23.

The reciprocal lattice 23 is the largest three-dimensional lattice which satisfies Eq. (4.20). Let R denote any rotation about the origin which transforms the lattice 5 onto itself. It follows from Eq. (4.20) that (4.2 1 )

(f, R*b) = (Rf, b)

=z E

2,

for every b E 23. Thus R* and, consequently, R transform 23 onto itself. This observation implies that 5 and '23 have the same rotational symmetry. We are concerned here only with unitary representations of the translation subgroups, that is, with homomorphisms into the group of complex numbers of length one. The term represcrTtation is understood to mean unitary represcntutior7 in the remainder of this section.

4. The Irreducible Representations of Certain Point and Space Groups

I95

Each vector r = l l s l + ?-2 s2 + l 3s3determines an irreducible representation

T, of 23 defined by (4.22)

T,.({E If}) = exp 2n(f, r)i =(exp 2n&A,i)(exp 2 ~ 1 ,I2 4 ~i)(exp 2

~ A, 4i), ~

where f = &vl + 42v2 + 43v3 is any vector of 3. Every irreducible representation is of the form of Eq. (4.22) by Pontrjagin’s theorem. However, if r’ = r + b, b E 23, then T,. = T,, so that every irreducible representation corresponds uniquely to a vector r of the form (4.23)

r = lls1

+ A2 s2 + A3 s 3 ,

where 0 < Ai 5 1, 1 < i 2 3. The set of all such vectors r makes up the primitive parallelepiped spanned by {s,, s 2 , s3) minus three of its faces. Corresponding points on opposite faces of the parallelepiped determine the same irreducible representation. If one identifies the opposing faces, then one again obtains the three-dimensional torus as the space of the irreducible representations. Unfortunately, the parallelepiped spanned by {s,, s 2 , s3} does not exhibit the rotational symmetry of the face-centered and body-centered cubic lattices. Consequently, the physicist looks for a more symmetrical polyhedron with which to identify the irreducible representations of 23. Thefirst Brillouin zone Si consists of all vectors (4.24)

r = A,sl

+ 1,s2 + A3s3

such that no shorter vector r’ determines the same irreducible representation of 23. In other words, r E R if and only if [lrlj i Ilr + bl/, b E 23. This last observation permits a useful geometric description of 53. For b E 23, let ’$ and ’$‘ denote the perpendicular bisectors of b and -b, respectively, as shown in cross section in Fig. (4.26). Let R(b) denote the closed convex solid bounded by the two planes ‘r( and which is denoted by the crosshatched area in the figure. Any vector r’ which projects beyond the region R(b) can not belong to R, since either r’ + b or r’ - b is a shorter vector determining the same irreducible representation of 13. It follows that

v‘,

(4.25)

R

n R(b).

hs8

nbsB nbEB

+

However, if r belongs to R(b), then llrll i llr bll, b E 23, so that r belongs to R. Thus 53 = R(b). The first Brillouin zone fi for the face-centered cubic lattice 5 is determined by a finite number of the shorter vectors of the body-centered cubic lattice 23. A sketch of the polyhedron R is given in Fig. (4.27). The face-centered cubic lattice 3 is dual to the body-centered cubic lattice 23, so that there exists a

196

(4.26)

3. Computation Representations and Characters

CONSTRUCTION FIRSTBRILLOUIN ZONE

first Brillouin zone for the lattice 23, similarly constructed from vectors of the lattice 8, that is sketched in Fig. (4.28). This polyhedron fi is frequently referred to as the Wigner-Zeitz cell of the face-centered cubic lattice 5. Each of these polyhedra, R and fi, exhibit the full rotational symmetry of the facecentered and body-centered cubic lattices, that is, each of them is mapped onto itself by every member of the symmetry group 0, of the cube. Elements of R are referred to as k-vectors. They are also referred to as wave vectors in physics. Every irreducible representation of the translation subgroup (13 is determined by a k-vector belonging to the first Brillouin zone R. A k-vector interior to R determines a unique irreducible representation of 23. A k-vector on a face of R determines the same irreducible representation of 'I) as the corresponding k-vector on the opposite face. Various vectors in R are carried into themselves by nontrivial subgroups of 0,. They are called high symmetry vectors and play a special role in the applications of representation theory to solid state physics. Some of these are drawn in Fig. (4.27) and some are listed in Table (4.29).

197

4. The Irreducible Representations of Certain Point and Space Groups

(4.27)

FIRSTBRILLOWIN ZONEDIAMOND LATTICE

-I.

t

t

The irreducible representations of Tdz and Oh7 are largely determined by the irreducible representations of the group 0,,(or its subgroups) and those of the common translation subgroup 22 of Tdz and Oh7.We limit ourselves to the case of unitary representations in order to apply some of the results of finite group theory to the space groups. This restriction is of small importance for the applications to physics where only unitary representations are ordinarily considered. Any irreducible, unitary representation T of Tdz or Oh7 with representation space M subduces a representation TD of the commutative subgroup 23 with representation space M, . The module M, is the orthogonal direct sum of one-dimensional 22-submodules. We use the term unitary module of Tdzor 0,’ to designate a complex inner product space M, that is, a module on which the elements of Td2or Oh7act as unitary transformations. Every representation space M of a unitary representation of Td2or Oh7is such a unitary module, of course. Given the sets S, = (1, . . . , 12) and S, = (37, . . . ,48}, let C and D denote the ensemble of Euclidean motions { R ,lo}, n E S, u Sz ,and {R, 1 I},n $ S, u S, , respectively. Here {R,}, 1 I n I 48, denotes the elements of Oh previously

I98

3. Computation Representations and Characters

(4.28)

WIGNER-SEITZ CELLDIAMOND LATTICE

t

(4.29)

HIGH SYMMETRY VECTORS Name

r

I3

a a X

r.

z

8ak

Weight

000 200 400 600 800 220 820

1 6 6 6 3 12 12

Name

c

w

K L

U Q

L

8ak

Weight

440 840 660 222 822 642 444

12 6 4 8 8 24 4

introduced and 3, is the vector of Fig. (3.35). The set C is a complete set of coset representatives of 'li in T, while the set C u D = A is a complete set of coset representatives of % in Oh7.We use the set A in the next theorem and

4. The Irreducible Representations of Certain Point and Space Groups

199

frequently omit the arrow -+ from the second factor of an element {R,lf}or { R It} t o simplify the notation.

(4.30) THEOREM. Every irreducible unitary module M of Oh7 is finitedimensional. Proof. Let n be a unit basis vector of the one-dimensional 23-submodule N of M, . Then there exists an element k of the first Brillouin zone 9 of 5 such that { E I i}n = (exp 2n(i, k)i)n for every element { E 12) of 23. Consider the set {n(R, i)} of vectors defined by

(4.31)

n(R, i) = (R]i)n

(Rli)

Every element of Oh7is of the form { R I j + t} : R translation. One notes that

(4.32) {R' I j

E

E

A.

oh,j = 0 or L,t a primitive

+ t}n(R, i) = (R'I j + t}{R I i}n = {R'R I j + t + R'i}n = { R l j + i}n = {RIj}{EIR-li)n = {RIj>[a(R-'i)n] = a(R-'f)n(R,

j)

where R = R'R and u(R-'i) = exp 241, Rk)i. Thus we see that the finitedimensional space Q spanned by the set {n(R, i)} is invariant under every operator from Oh7.Since M is irreducible, it must coincide with the finitedimensional space Q. The reader doubtless notes that we have repeated the argument of Clifford's theorem in a special notation. It is worthwhile discussing the homogeneous components of M, which occur in this particular case. Let the set (kj}, 1<jI s, determine the complete set of inequivalent irreducible representations of 23 occurrring in the 9-module M,. Reenumerate the Euclidean motions {R,Ii} E A such that the vectors n(R,, i), a j I E < u ~ + transform ~ , according to kj, 1 5 j 5 s. The K-space Mj, 1 Ij I s, spanned by the vectors (n(R, I i)}, aj I E < a j + ' , is a homogeneous component of M, . The module M, is the direct sum

(4.33)

M, = M I @ . . . @ M S .

A vector m belongs to Mj if and only if it transforms according to the kvector kj, 1 5 j 5 s, that is, (4.34)

{E[i}m = (exp 2n(f, kj)i)m.

The subgroup S j * , 1 Ij I s, of the space group Oh7defined by

(4.35)

B ~ =*{ { RI i} E oh7: {RI i

}=M ~ ~ ~}

200

3. Computation Representations and Characters

is called the inertia group of the B-module M j by mathematicians. Denote by {RIS} an element of Bj*. For any {elf) E '23 and mj E Mi, one has (4.36) (exp 274, kj)i){R I S}mj

= {E

[ i}[{RI S}mj]

[{e/i)(R]S}]mj = [(RIS){ejR-'i}]rnj kj)i){R (S)mj = (exp 2 4 5 , Rkj)i){R1 S}mj . =

= (exp 2n(R-'5,

We see that exp 27@, kj)i = exp 274, Rkj)i which implies that (5, k j - Rkj) is an integer for all 5 E 5 . It follows that Rkj = k j + b, where b E 23, the reciprocal lattice. Conversely, given an element {R1 S} E Oh7 such that Rkj =kj + b, b E 23, one has for mj E Mi,

[ i}{RI S}mj = {R1 S}[{RI S)-l{el f}{R I S}lmj = { R [S)[{eI R-'f}]mj = (exp 27c(R-'f, kj)i){RIS}mj = (exp 27@, Rkj)i){R) S)mj = (exp 2n(f, kj + b)i){RIS}mj = (exp 271(f, kj)i){R S}mj , so that {RjS)mj E M j . This shows that {RIS) is an element of Bj*. Conse{E

quently,

(4.37)

Bj* = {{R[ S}

E

Oh7:Rkj = k j

+ b, b E %}.

Similar considerations are valid not only for Td2,but all other space groups as well. We turn now to an important theorem which is stated in terms of Oh7,but which is also true in general. (4.38) THEOREM. Every irreducible 0,7-module M is an induced module arising from an irreducible %*-module N where %* is the group of some kvector k of the first Brillouin zone si of the face-centered cubic lattice. Proof The vector n of Theorem (4.30) transforms under the translation subgroup 23 according to the k-vector k denoted now by k,. There may, of course, be no S-submodule of M, transforming according to a different representation of 3 . Then there is only one summand in (4.33). Otherwise, by an appropriate change of notation, n(R, , i) transforms according to the k-vector k, which differs from k, by a vector not belonging to 23. A calculation similar to (4.36) shows that R, k, = k, + b, b E 23. If one applies these considerations to a basis of M I , then one discovers that {R2]i}Ml c M, . Similarly, one finds that {R21i}-'M2 c M,. Since M, and M2 are finitedimensional, it follows that {R,] i} induces an isomorphism of M, onto M, . It is easy t o see in this way, that there exists a set {Rj\i}, 1 < j < s, such that { R j 1 i} induces an isomorphism of M, onto Mj where {R,1 i> can be taken to be {c I O } . Thus we find that (4.39)

Bj* = { R j l i } S l * { ~ j / i ] - l ,

1 < j < s.

4. The Irreducible Representations of Certain Point and Space Groups

201

From the remarks after Clifford’s theorem, it follows that M is equivalent to the induced module MIoh7. Suppose that M, is a reducible %l*-module, that is,

Mi = N , @Pi,

(4.40)

with each N, and P, nontrivial. Let {n),(l),. . . , nA(l)}be a K-basis of N,. For 1 ~j I s, define n,(j) by

1 5 u I A. n,(j) = { R j I i}n,(l), It follows immediately that the subspace N, spanned by the set (n),(j), . . . , n,(j)} is a iBj*-submodule of M. Further consideration reveals that the K-space (4.41)

(4.42)

N

= N, @ . . .

ON,

is a proper 0,7-submodule of M, an impossibility. Consequently, MI is an irreducible %),*-module and the result follows. As we have already observed, this theorem holds for space groups in general. Thus we see that the irreducible representations of the space group G acting on a lattice 2 are determined by certain irreducible representations of the groups of the wave vectors k associated with the first Brillouin zone 53 of 2. This zone is defined in the general case analogously to the special case of the face-centered cubic lattice which we have discussed. Since the group of the wave vector k is yet another space group, it may seem at first glance that Theorem (4.38) is basically insignificant. However, this is not so. First, it proves easy theoretically to find the required kind of irreducible representations of the group 2l3 of the wave vector k. Furthermore, the theoretical method is quite practical in a large majority of cases. Second, one is generally interested in these kinds of irreducible representations of 2l3 rather than the general irreducible representations of G in the applications to solid state physics. We turn our attention to a discussion of these irreducible representations without proofs of many of the details. Note that if G is the space group of any lattice 2 and $93is the group of the wave vector k for any element of the first Brillouin zone 53 of 2, then 5% always contains the translation subgroup % of G as a normal subgroup. A representation T of % with representation space M or, more generally, a %-module M is said to belong to the wave vector k if every m E M transforms according to Eq. (4.34) with k replacing k, .

(4.43) DEFINITION. An n-dimensional complex projective representation of a group G is a map T from G into CL(V), V an n-dimensional complex vector space, together with a mapffrom G x G into the nonzero complex numbers such that T(1) = 1, and (4.44)

T(919,)=f(91,92)T(g,)T(92)7

(91, 9 2 )

= G.

202

3. Computation Representations and Characters

The map f i s called a factor set of the representation T and satisfies an identity we omit. A projective matrix representation is similarly defined. Let G be the space group of any lattice Q and let 23 be the translation subgroup of 6. Let 53 denote the first Brillouin zone and 8 the reciprocal denote the group of the wave vector k in 53. Note that the lattice of 2. Let factor group
r ( { ~i>)m \ = ( E I i>m = (exp 2n(i, k)i)m.

In other words, suppose that M, belongs to the wave vector k. Then, for all w ~ ' m , (4.47)

W W ) = Y(w)

0qw>,

where Y and T are complex irreducible projective representations of '2B with r and Teach of dimension one and Y(v) = 1, v E 23, so that Y deduces a projective representation of the factor group YB/23. If there exists an ordinary representation T of %? such that T(v)= r(v), v E 23, then the representations Y and T can be taken to be complex ordinary representations of ?D. This theorem reduces the calculation of almost all of the irreducible representations of the space groups which occur in physical applications to the calculation of the irreducible ordinary representations of their point groups. This last calculation can be carried out easily by the methods of this chapter. To establish this reduction, let k be a wave vector lying within the interior of the first Brillouin zone 53. Then the group 2B of k is the set

2l-3 = {{Rli}E 6 :Rk

= k}.

The wave vector k determines a one-dimensional representation r of 23 according to the formula (4.48)

Y({E

1 r}) = exp 2n(i, k)i,

{E

I i} E 23.

There exists a one-dimensional representation T of %? defined by (4.49)

T({R 1 i}) = exp 27c(i, k)i,

{R I i} E ?D,

4. The Irreducible Representations of Certain Point and Space Groups

since (4.50)

203

T({RI%}{SIii}) = T({RSl% + Rii}) = exp 2743

+ Rii, k)i

k)i)(exp 271(RIi, k)i) = (exp 27c(%,k)i)(exp 2n(i1, k)i)

= (exp 27-43,

169T ( { S /a>>.

= T({R

Hence T is a representation of 2B such that T(v) = r(v), v E B. Thus there exist ordinary representations Y and T which satisfy Eq. (4.47). As a matter of fact, one can define T by Eq. (4.49) and take Y to be defined by a suitable ordinary representation of the point group 'p. Conversely, given k interior to the first zone and Y any representation of 'p, one can define a representation U of 2B according to (4.51)

U({R[f}) = (exp 2743, k)i)Y(R).

It follows by the same sort of calculation as that carried out in Eq. (4.50) that U is a representation of 2B. Whenever Y is an irreducible representation of 'p, U is an irreducible representation of 'B. Thus, whenever k is within the first zone, all irreducible representations of the group 9.3 of this particular form can be obtained by a knowledge of the irreducible ordinary representations of the point group 'p of 2B. Fortunately, this special kind of irreducible representation is the one usually desired in the applications in solid state physics. Unfortunately, one can not always escape the requirement of projective representations when k is on the boundary of the first zone. We say more about the boundary case below, but pause now to present an example of the interior case. (4.52) EXAMPLE. Consider the group '2J3 of the wave vector A whose component form is (1/2u)(l, 0, 0). This vector is interior to the first Brillouin zone R of the diamond lattice and its group 'u3 consists of all elements {RI t} of 0,' whose rotations come from the set { I , 2, 19,20, 27, 28,41,42}. These are the motions of O,, which leave the x-axis fixed. They constitute the point group of ?R a group 'p isomorphic to the dihedral group of order eight. Its Cayley table is given in Table (4.53) for convenience. The irreducible reprebentations of 'p can be computed immediately from the one-dimensional representations of the subgroup H = { 1, 2, 19,20}. These are easily seen to be as shown in Table (4.53'). Representations d' and d 2 are self-conjugate and each of them supplies two irreducible representations of '$3. The representations d 3 and d 4 are mutually coajugate and together they supply one irreducible two-dimensional representation of 'p. The results are tabulated in Table (4.54).

204

3. Computation Representations and Characters

(4.53)

CAYLEYTABLEOF b 1 2 19 20 27 28 41 42 2 1 20 19 28 27 42 41 19 20 2 1 4 2 4 1 27 28 20 19 1 2 4 1 4 2 2 8 2 7 2728 41 42 1 2 1 9 2 0 28 27 4241 2 1 2 0 19 41 42 28 27 20 19 1 2 42 41 27 28 19 20 2 1

(4.53')

IRREDUCIBLE OF H REPRESENTATIONS d ' d2 d3 d4

(4.54)

1 2 19 20 l 1 1 1 1 1 - 1 -1 1-1 i --i 1 - 1 --i i

IRREDUCIBLE REPRESENTATIONS OF 8 1 2 19 1 1 1 1 1 1 1 1-1 1 1-1 1 -1 i 0 0 0 0 0 0 T(5)22 1 -1 --i

T(1) T(2) T(3) T(4) T(5)1, T(5)lz T(5),,

20 27 28 41 42 1 1 1 1 1 1-1-1-1 -1 -1 1 1 - 1 -1 1 1 -1 -1 -1 -i 0 0 0 0 0 1 -1 -i i 0 1 -1 i --i i 0 0 0 0

Thus we find there are five classes of irreducible representations of B3 associated with its own wave vector k. Each of these classes contains a member defined, for {Rli}in B3, by (4.55)

Yj

({RnI i>)= ( ~ X2P@ , k)i)T(j)(Rn),

1 < j 5 5 , where n E (1, 2, 19, 20, 27, 28, 41, 42) and T(j) denotes the j t h irreducible representation from Table (4.54). Since $93is itself a space group, these are by no means all of its classes of irreducible representations. However, these are those representations of principal interest in the applications to solid state physics and allied topics. Naturally, there may be an infinite number of other classes of irreducible representations of 'Lu. The representation space M of a more general irreducible representation of B3 will decompose according to Eq. (4.33) into its

4. The Irreducible Representations of Certain Point and Space Groups

205

homogeneous components (Mi}, 1 Ij Is, with respect to 23. The associated set & j ) , 1 ~j 5 s, of k-vectors specifying the translational properties of the homogeneous components is called the star of the representation, not only in the special case of the group 1' 13 of the wave vector but also in the general case of any space group. The restricted representations discussed here are of particular interest in physics because each k-vector determines the momentum of a particle in question. If the star of a representation contains more than one vector, then the corresponding representation characterizes a collection of particles with different momenta. There is another useful theorem discovered by Mackey which treats the case of symmorphic space groups. We quote its relevant form here. (4.56) THEOREM. Let G be a symmorphic space group, that is, one which is the semidirect product of its translation group (23 and its point group 'p z 6/23. Let r: B + GL(M) be an irreducible representation of (23 such that r is equivalent to all its conjugates with respect to 6. Then there exists a representation Y:6 -+ GL(M) such that Y(v) = r(v), v E B. This theorem asserts that one can use the preceding construction for this type of space group and this type of irreducible representation. Nevertheless, there arise situations in which k is not interior to Sl and 1' 13 is not symmorphic. Consequently, one must face the fact that projective representations are necessary for a complete treatment of the representations of the space groups. As an example, consider the k-vector X = (l/a)(l, 0, 0) of Fig. (4.27) and Table (4.29). The group '1x3 of X consists of all elements {RIS} of Oh7where R belongs to the subgroup 'p of the sixteen elements (1,2, 19,20,27,28,41,42; 3,4, 17, 18,25,26,43,44} of O h . The first eight of these map X onto itself, and the second eight map X onto its negative. The group 1' 13 of X is not a symmorphic group and X is not an interior point of 9. Thus one is confronted with the task of determining the projective representations of q.This calculation is beyond the scope of this book. We must refer the reader to Lyubarskii (1960, pp. 91-102), Doring (1959), and Harter (1969) for additional information on such matters. We make one more remark for the benefit of the reader interested in energy band calculations (without bothering to dejne all the terms). It is usually sufficient to consider such boundary points as X to have the same symmetry as the interior points of the zone lying along the same symmetry axis. The use of such lower symmetry will partly obscure the nature of the bands at X, of course. Nevertheless, most of the confusion can be eliminated by the use of the compatability relations. This paragraph concludes Chapter 3. In Chapter 4, we turn to a rather expurgated discussion of the representation theory of the symmetric group and to its applications to the representation theory of certain continuous groups.

206

3. Computation Representations and Chraacters

PROBLEMS

1. Let S and T be linear transformations on the vector spaces U and V with bases {ul, u,] and {v,, v, , v,), respectively. Suppose that Su, = u, - u 2 , Su, = 2u, + u, and Tv, = v1 - v,, Tv, = v, + v2 + v,, and Tv, = v, + 2v, vj . Express S @ T with respect to the basis (ul @ v,, . . . , u2 @ vj}.

+

2. Find the corresponding matrix of S @ T with respect to the given basis In two ways: (a) Directly from the linear transformation S 0T defined above, (b) by computing the matrices of S and T individually with respect to the given bases and forming their Kronecker product.

3. Let P and Q be linear transformations on the same vector spaces U and V given above. Let Pu, = 2u, + u , , Pu, = u, - u 2 , and Qv, = vl, Qv2 =

+

+

3 Q, v = ~ ~1 ~ 2 . Determine the linear transformation P @ Q with respect t o the basis (a) of Problem 1. (b) Find the matrix of P @ Q with respect to this basis. (c) Check that tr(P @ Q ) = tr(P) tr(Q).

~2

~

4. Using the definitions of Problems 1 and 3, verify that (S @ T)(P @ Q) = SP @ TQ.

5. Work out the table of irreducible complex representations of the group G = Z , @ 2,whose Cayley table is given below. 1234 2143 3412 4321

Show that the set of complex irreducible representations of G forms a group under the tensor product operation which is isomorphic to G. 6. The Cayley table of the cyclic group G of order seven has the given form. 1234567 2345671 3456712 4567123 5671234 6712345 7123456

(a) Find the table of complex irreducible representations of G. (b) Show that this table is a cyclic group of order seven under the tensor product operation.

Problems

207

7. Let G be any finite abelian group. Argue that the character table is an abelian group isomorphic to G under the tensor product operation. 8. The Cayley table of the symmetric group S , is given in Fig. (2.10) of Chapter 2. A two-dimensional irreducible representation T' of S , is determined in Problem 13, Chapter 2. (a) Work out an irreducible complex two-dimensional representation T by the method of Problem 33, Chapter 2. (b) Check that T and T' are equivalent. (c) Compute the character of T 0T. (d) Decompose T O T into its irreducible components.

9. The Hamiltonian operator is frequently made up of two parts: a kinetic energy term which is proportional to the usual Laplacian operator together with a potential energy term. Show that the Laplacian operator on ordinary three-dimensional space commutes with the action of the rotation group. When the potential energy is spherically symmetric argue that the rotation group of three-space is contained in the symmetry group of the Hamiltonian. 10. Look up several additional reports about the applications of group theory to physics and chemistry and present a short paper or discussion on them. Lomont (1959), Lyubarskii (1960), and Hamermesh (1962) are good sources. For a deeper discussion see Slater (1965) and other books written by him. 11. Let S and T denote finite dimensional irreducible representations of the group G. Prove that S 0T and T @ S are equivalent representations of G.

12. The group G with the Cayley table given on page 208 is C,@S,. Show that G is the direct product of its subgroups H = (1, 5, 9,lO) and K = (1, 2, 3, 13, 14, 15}. 13. The subgroup M = (1, 2, 3,4, 5, 6 , 7, 8, 9, 10, 11, 12) is cyclic with generators 7 and 8 which are mutually conjugate. Thus a representation t of M is self-conjugate if and only if t(7) = t ( 8 ) , where 8 = 7 5 . Otherwise, it is mutually conjugate with a second representation t' such that t'(7) = 4 8 ) and t'(8) = t(7). Write out the table of irreducible representations of M and separate them into self-conjugate and mutually conjugate sets, 14. Each twelfth root of unity o determines a one-dimensional irreducible representation t of M such that t ( x ) = con where x = 7" is any element of M . Show that the representation t is self-conjugate if and only if o is a fourth root of unity. Show that each such self-conjugate representation t of M prolongs to two irreducible representations f , and t of G. The representations tl and t - l have the same values on M , but r,(13) = 1 and t-,(13) = - 1.

208

3. Computation Representations and Characters CAYLEY TABLE'C4 0SB

1 2 3 4 5 6 7 8 9101112131415161718192021222324 1 1 2 3 4 5 6 7 8 91011121314I5161718192021222324 2 2 3 1 5 6 4 8 9 7 1 1 1 2 1 0 1 5 1 3 1418 1 6 1 7 2 1 1 9 2 0 2 4 2 2 2 3 3 3 1 2 6 4 5 9 7 8121011141513171816202119232422 4 4 5 6 2 3 1 1 0 1 1 1 2 8 9 7 18 1 6 1 7 1 3 1 4 1 5 2 4 2 2 2 3 1 9 2 0 2 1 5 5 6 4 3 1 2 1 1 1 2 1 0 9 7 8171816151314232422211920 6 6 4 5 1 2 3 12 10 11 7 8 9 16 17 18 14 15 13 2223 24 2021 19 7 7 8 9 10 11 12 4 5 6 2 3 1 2 0 21 19 23 24 22 17 18 16 15 13 14 8 8 9 7 1 1 12 10 5 6 4 3 1 2 19 20 21 22 23 24 16 17 18 14 15 13 9 9 7 8121Ol1 6 4 5 I 2 3211920242223181617131415 10 10 11 12 8 9 7 2 3 1 5 6 4 2 2 23 24 20 21 19 14 15 13 17 18 16 11111210 9 7 8 3 1 2 6 4 5 2 4 2 2 2 3 1 9 2 0 2 1 1 3 1 4 1 5 1 6 1 7 1 8 1 2 1 2 1 0 1 1 7 8 9 1 2 3 4 5 6 2 3 2 4 2 2 2 1 1 9 2 0 1 5 1 3 1418 1617 13131415161718192021222324 1 2 3 4 5 6 7 8 9101112 14 14 15 13 17 18 162021 1923 2422 3 1 2 6 4 5 9 7 8 12 1011 1515 1 3 1 4 1 8 1 6 1 7 2 1 1 9 2 0 2 4 2 2 2 3 2 3 1 5 6 4 8 9 7 1 1 1 2 1 0 16161718141513222324202119 6 4 5 1 2 3121011 7 8 9 17 17 18 16 15 13 14 23 2422 21 19 20 5 6 4 3 1 2 11 12 10 9 7 8 18181617131415242223192021 4 5 6 2 3 1101112 8 9 7 19 192021 222324 16 17 18 14 15 13 8 9 7 11 12 10 5 6 4 3 1 2 20 20 21 19 23 24 22 17 18 16 15 13 14 7 8 9 10 11 12 4 5 6 2 3 1 21 21 19 2024 22 23 18 16 17 13 14 15 9 7 8 12 10 11 6 4 5 1 2 3 22222324202119 141513171816101112 8 9 7 2 3 1 5 6 4 232324222119201513141816171210ll 7 8 9 1 2 3 4 5 6 24242223192021131415161718111210 9 7 8 3 1 2 6 4 5 a K1 = {l}, K z = {2, 3j, K3 = (4, 6}, K4 = IS), K s j7, 8}, Kcj = {9}, K , = (lo), Ks = (11, 12}, K9 = (13, 14, 15}, K I o = {16, 17, IS}, K I 1= {l9,20, 21}, and K l z = (22, 23, 24) are its conjugacy classes. 1

15. Construct the table of irreducible representations of G by means of the information in Problems 13 and 14.

16. The quotient G/H of the group G of Problem 12 by its normal subgroup H = { l , 5) is the dihedral group D, of order twelve. Find six irreducible representations of G by means of those of D, . 17. Compute the character table of G from the information obtained in Problems 13 and 16. Check the results with those of Problem 15.

18. The group G has four classes of inequivalent two-dimensional representations. Determine the characters of their tensor products. 19. The group G with the following Cayley table has five classes of complex irreducible representations. These arise from the representations of the

209

Problems

1 2 3 4 5 6 7 8 91011121314 2 3 4 5 6 7 114 8 910111213 3 4 5 6 7 1 2 1 3 1 4 8 9101112 4 5 6 7 1 2 3121314 8 91011 5 6 7 1 2 3 411121314 8 910 6 7 1 2 3 4 51011121314 8 9 7 1 2 3 4 5 6 91011121314 8 8 91011121314 1 2 3 4 5 6 7 91011121314 8 7 1 2 3 4 5 6 1011121314 8 9 6 7 1 2 3 4 5 11121314 8 910 5 6 7 1 2 3 4 121314 8 91011 4 5 6 7 1 2 3 1314 8 9 1 0 1 1 1 2 3 4 5 6 7 1 2 14 8 9 1 0 1 1 1 2 1 3 2 3 4 5 6 7 1

subgroup H = (1, 2, 3, 4, 5 , 6, 7) in two ways. (i) The 1-representation of H prolongs to two irreducible representations of G, namely, the following:

x, xz

1 1 1

2 1 1

3 1 1

4 1 1

5 1 1

6 1 1

7 8 9 1 0 1 1 1 2 1 3 1 4 1 1 1 1 1 1 1 1 1-1-1-1-1-1-1-1

(ii) The remaining six irreducible representations of H split up into three sets of mutually conjugate pairs. (a) Determine these conjugate pairs of representations of H . (b) Use induction to find three classes of inequivalent complex irreducible representations of G. 20. Using the results of Problem 19. (a) Determine the tensor products of the given irreducible representations. (b) Determine their reduction into irreducible components. 21. According to Problem 12, the group G defined there is the direct product H 0 K. Determine the irreducible representations of G by means of those of H and K. 22. Check the results of Problems 15 and 21 by means of their character tables. Although there are an infinite number of subgroups of the group of rigid motions which leave a point fixed, there are only a finite number of distinct classes of such groups. These classes are well-known and a fuller discussion of this can be found in Wolf (1967). One such class is that of the group of rotational symmetries of a regular orthogonal prism. 23. Show that the group P of rotational symmetries of a regular five-sided prism with base a regular pentagon contains ten elements.

210

3. Computation Representations and Characters

Work out the classes of conjugate elements of the group P described in Problem 23.

24.

The elements of a finite group G of rotations about a point 0 can be defined by means of vectors originating at 0. The vector ji determines a rotation g whose axis is that of %, whose angle of rotation equals the magnitude of %, and whose direction is that of a right-handed screw advancing in the direction of %. Find a geometric condition which determines that elements g and g' are mutually conjugate in G. 25.

26. Find an element g E P [Problem 231 of order five which generates a subgroup C, c P. Let x be any element of P not in C, . Let g = 2, g2 = 3, g 3 = 4, g4 = 5, and x = 6. These definitions together with those implied in the figure determine the Cayley table of P. Complete the remainder of this Cayley table. 1 2 3 4 5

2 3 4 5 1

3 4 5 1 2

4 5 1 2 3

5 6 1 7 2 8 3 9 410

For most groups of low order, there exists a special version of induction discovered by Talburt (1971). For a solvable or a nilpotent group G, see Rotman (1969, there exists an ascending chain

(1) c G, c G,-l

c

. . . c Go = G

such that G i is normal and of prime index in G i - l . For nilpotent G, it can be assumed that the G, are normal in G itself. Each G i - l is generated by Gi and s , - ~ .Talburt's algorithm depends on determining such a chain for G. The dihedral group D , of order twelve is a good example for the application of the algorithm. The table of D, is presented below. ~

~

I 2 3 4 5 6 7 8 9101112 2 3 1 5 6 4 9 7 8121011 3 1 2 6 4 5 8 9 7111210 4 5 6 2 3 11210ll 7 8 9 5 6 4 3 1 2111210 9 7 8 6 4 5 1 2 3101112 8 9 7 7 8 9101112 1 2 3 4 5 6 8 9 7 1 1 1210 3 1 2 6 4 5 9 7 81210l1 2 3 1 5 6 4 101112 8 9 7 6 4 5 I 2 3 l11210 9 7 8 5 6 4 3 1 2 121011 7 8 9 4 5 6 2 3 I

211

Problems

Let w denote a primitive twelfth root of 1. The number 2 denotes w 2 , 3 denotes w 3 , . . . , n denotes w". One begins with the series G,

c

G, c Go = G

with G, = {1,2, 3) and G, = (1, 2, 3, 4, 5,6}. The first step is to write down the character table of the cyclic group G, . 12 12 12 12 4 8 12 8 4

One now prolongs each of the one-dimensional characters as far as possible in as many ways as possible. For instance, the 1-representation 12 12 12 can be prolonged to G, in two ways: (a) 12 12 12 12 12 12 as well as by (b) 12 I2 12 6 6 6 where we use the fact (i) 4' = 2 and (ii) 4 commutes elementwise with G, . Note that (i) implies that 4 must map either to 12 or to 6 and (ii) that the initial representation is self-conjugate so that it prolongs to GI. The representation (a) 12 12 12 12 12 12 of G, is self-conjugate and prolongs to two distinct representations of G: T(1)12 12 12 12 12 12 12 12 12 12 12 12 T(2) 12 12 12 12 12 12 6 6 6 6 6 6

The representation (b) of GI is also self-conjugate, having the same values for the conjugate pairs (2, 3) and (4, 6). Thus it prolongs to two distinct onedimensional representations of Go , namely, T(3) 12 12 12 6 6 6 12 12 12 6 6 6 T(4) 12 12 12 6 6 6 6 6 6 12 12 12

Both of the remaining representations of G, , (c) 12 4 8 and (d) 12 8 4, are also self-conjugate. They prolmg to (c) 1 2 4 8 8 1 2 4

(d)

12844128

12 4 8 2 6 10

(d')

12 8 4 10 6 2

and (c')

so that one has all six of the distinct one-dimensional representations of G, itself cyclic of order six. Conjugation by seven interchanges members of the pairs (2, 3) and (4, 6) so that none of the four are self-conjugate representations, but rather form two sets of conjugate pairs of representations, namely, {(c), (d)} and {(c'), (d')}. Conjugation under seven interchanges the represen-

212

3. Computation Representations and Characters

tations (c) and (d). Consequently, one can write down an irreducible twodimensional representation of G immediately: T ( 6 ) , , (c) 12 4 8 T(6)12 0 0 0 T(6121 0 0 0 T(6)22 (d) 12 8 4

8 12 4 0 0 0 0 0 0 0 0 012 8 4 4 1 2 8 (d)

0 0 012 4 8 812 4 (c) 4 12 8 0 0 0 0 0 0

The other two-dimensional irreducible representation arises from the pair {(c'), (d')} in a similar manner. 27. Let A be the wave vector (1/8a)(I, 1, 1) contained within the first Brillouin zone Si of the diamond crystal. (a) Find the point.group $'3 of the group 2B of the wave vector A. (b) Work out the Cayley table of '@. (c) Determine the irreducible representations of 2B associated with its wave vector A.

Cbapter 4

The Representution The09 of SeveruZ Specid Grozips

This chapter is devoted to the representation theory of the symmetric group S,, of all permutations on a set of n objects and to that of the complex general linear group GL(V) of all nonsingular linear transformations on an rn-dimensional complex space V as well as certain of its subgroups. A complete treatment is a hard task, well beyond the scope of this introductory book. Consequently, our discussion is sometimes incomplete and without proofs. However, there is a general interest in the representations of these groups among people outside of mathematics for many of whom the standard treatments are either too protracted or too difficult. This last chapter is an attempt to present a readable account of various topics for such an audience. Even among students of mathematics there is doubtless a sizable group who would like to know something of the general situation before attempting the arduous program required for a rigorous and detailed understanding. The first section of the chapter is devoted LO the development of the ordinary representation theory of S,,. The presentation is based on an approach discovered by A. Young. His principal tools are the Young’s tableau and the Young’s frame. The ideas of these constructs are introduced and applied to deducing the irreducible representations of S,, . Since the methods prove to be lengthy, certain easier methods are considered for special cases. The second section is concerned with symmetric algebras and modules over symmetric algebras. The discussion of these kinds of algebras is largely 213

214

4. Represetitation Theory of Special Groups

limited to the case of the group algebra of a finite group. They are introduced to provide a framework in which to discuss the relation between the representations of S, and certain special representations of GL(V). Section 3 i s an application of the results of Section 2 to the calculation of the integral representations of GL(V). This connection is established by introducing the group S,, as left operators on the n-fold tensor product . 0V and the group GL(V) as right operators. There proves to exist a V 0.. natural duality between right GL(V)-submodules of the tensor product and right ideals of the group algebra A of S,. In the last section we treat various details about the representation theory of what are called the standard matrix groups. They are the complex general linear group GL,,(K) of all nonsingular, complex rn x i x matrices and certain of its subgroups. An effort is made to use these groups as a n introduction to the subject of Lie groups without benefit of the standard definitions. A bare minimum of topology and differential geometry is introduced. The concept of Lie algebra is developed by means of the example of the three-dimensional rotation group SO(3). A sketch is made of the classification of semisimple Lie algebras by means of their roots. Then irreducible modules and their weight diagrams are discussed. Examples are supplied in the cases of SO(3), SU(2), a n d SU(m). 1. THE REPRESENTATION THEORY OF THE SYMMETRIC GROUP

The most recent detailed treatment of the representation theory of the symmetric group i s that of Robinson (1961) to whom we must refer the reader interested in more than a n outline. Our goal is merely to present the high 1 ights. The ordinary representation theory of the symmetric groups was first worked out by Frobenius, but our presentation follows a n approach discovered by Alfred Young independently of Frobenius. The method of Young is based on a detailed analysis of the group algebra A of the symmetric group S,. Young invented a procedure, not using the theory of characters, for determining the primitive idempotents of A. The fundamental ideas hinge on the concept of a Young’s tableau which is described below. The section contains a discussion of the relations between partitions of an integer n , the Young’s frames and Young’s tableaux belonging to n, and the irreducible representations of the symmetric group S,. A method is explained of determining the primitive idempotents of the simple components of the group algebra A = KS,, of the symmetric group S,, over the complex numbers. A useful method of computing the value of the character of a n irreducible representation associated with a frame is given. A procedure for finding the matrix of a transposition ( r , r 1) in the Young’s semirational

+

1. The Representation Theory of the Symmetric Group

21s

irreducible representation is described. Finally, a sketch of the method of computation of the matrices of Young’s integral representation is included.

(1.1) DEFINITION. A partition of the positive integer n is a sequence {mi}, 1I i i k , of positive integers such that m i + l 2 mi, 1 2 i k - 1, and m , + . . . + mk = n. The sequence (3, 2, l} is a partition of 6. It is customary to order the partitions of n such that {mi} > {m,’}if, for the first j such that mi # mj‘, one has mj > mj‘. The ordered partitions of 5 are (5) > (4, l} > {3,2) > (3, 1, 1) > (252, 11 > ( 2 , 1, 1, 1) > (1, 1, 1, 1, 1). (1.2) DEFINITION. A frame F corresponding to the partition {m,}, 1 < i I k , of the positive integer n is a diagram consisting of k left-justified rows of empty square boxes, the ith row of which contains m iboxes. Such a frame is said to belong to the integer n. For example, the frame F corresponding to the partition (3, 2, I} of 6 is as shown in Fig. (1.3). There is no known

function J’ such that f (n) is the number of frames belonging to the integer n. However, tables are available which list this number for fairly extensive ranges of n. Let F and F‘ belong to the partitions {mi} and {m,’},respectively, of n. Then F > F‘ if and only if {mi} > {mi’}.

(1.4) DEFINITION. A tableau T corresponding to the frame F belonging to n is a diagram in which the distinct integers from 1 through n have been inserted into the frame F. Such a tableau is said to belong to F and also to the integer n. The canonical tableau T belonging to the frame F of Fig. (1.3) is as shown. There are 719 = 6! - 1 other tableaux associated with this frame F. These are obtained by performing all possible permutations on the entries of the tableau T of Fig. (1.5). They are mutually congruent in the sense that

given any two of them, say T’and T , there exists an s E S , such that s T = T . The precise action of s on T‘ is defined below. The notion of a canonical tableau for any frame should be clear to the reader. Note that the canonical tableau serves to label the squares of the associated frame F.

216

4. Representation Theory of Special Groups

It is convenient to simplify the notation for frames and tableaux. Either the symbol

*** ** *

or the symbol

... ..

can be used t o denote the frame F of Fig. (1.3) and a similar scheme used for any frame. The marks * or . replacing the squares are called nodes. A symbol such as 456 31 2

is used to denote a tableau belonging to the frame F. An element r of the symmetric group S, acts on any tableau T belonging to n. To illustrate this, let s = (123)(45) be an element of S, and let T denote the tableau 645 31 . 2

Then the tableau sT generated by the action of s on T is 654 12 . 3

The action of r E S , on the canonical tableau is defined by Fig. (1.6).

The effect of r E S,, on a tableau T belonging to n can be thought of as changing the names of the squares of T. Square i of T becomes square r(i) of rT. According to this viewpoint, r effects an alias transformation on T. However, one can also consider the geometric result of applying r to T whose entries are considered to move under the action of r. For example, the permutation s = (123)(45) transforms the tableau

123 45 6

into the tableau

23 1 54 6

1. The Representation Theory of the Symmetric Group

217

Observe that the entry 1 moves from its first position to the third, the entry 2 from its second position to the first, etc. One notes that 1 moves to the square occupied by s - l ( l ) , 2 moves to the square occupied by s-'(2), and, in general, the entry i to the square occupied by s-'(i). Thus if r E S,, is an alibi transformation moving the elements of the tableau T belonging to n, then the element i of Tmoves to the square of Tcontaining r-'(i). Given any tableau T with n elements, one can make the following definition.

(1.7) DEFINITION. The row-group P ( T ) of Tis the set of allp E S,, such that p does not transform any element i of Tout of its row. The column-group Q(T) of T is the set of q E S,, ,no member of which moves any element of T out of its column. It should be clear that P ( T ) and Q(T) are subgroups of S,, . The row-group P ( T ) of the tableau T of Fig. (1.5) is generated by the set {(12), (13), (23), (45)) of transpositions and the column group Q ( T )by the set W4),

(46), (25)).

The subgroups P ( T ) and Q(T) play an important role in the analysis of the representations of S,, . There is associated with P ( T ) the element P = p , p E P ( T ) , of the group algebra A = KS,, of the symmetric group S,, and with Q(T) the element Q &(q)q,q E Q(T), where c(q) is 1 for even and - 1 for odd q. Each tableau T belonging to n determines a unique element e(T) = PQ of the group algebra A according to the above definitions. Let p , p' E P ( T ) and q, q' E Q(T) with pq = p'q'. Then one has that p ' - l p = 4'q-l belongs to P ( T ) n Q ( T ) which implies that it is 1. Thus p = p ' and q = q' so that the element e(T) can be written in the form

=c

1

(1.9) REMARK.For each choice of the tableau T belonging to the frame F, the element e(T) is essentially idempgtent, that is, it differs from an idempotent in A by a scalar multiplication. As a, matter of fact, e(T) is essentially a primitive idempotent so that the left ideal Ae(T) is a minimal left ideal in the group algebra A. Furthermore, Ae(T) and Ae(T') are A-isomorphic minimal left ideals of A if and only if T and T' belong to the same frame F. It is a wellknown fact, which we discuss in slightly more detail below, that the number of classes of conjugate elements of S,, is equal to the number of distinct partitions of n, that is, to the number of distinct frames F which belong to n. Consequently, the set of minimal left ideals {Ae(T)), one T to each frame, is a full set of representative elements from the classes of isomorphic irreducible S,,-modules of the symmetric group S,, over the complex numbers. The proof of these facts is highly computational and intimately connected with the properties of the tableaux and the action of the symmetric group on them.

218

4. Representation Theory of Special Groups

Our treatment relies strongly on the works of Boerner (1963) and of Curtis and Reiner (1962) where various details are treated in greater depth.

I t is necessary to establish a relation between the row and column groups of a tableau T and those of a tableau T' = s T congruent to it. Consider an example to illustrate the desired definition. (1.10) EXAMPLE. The two tableaux T and T', given by

132 54 6

312 64 , 5

and

are congruent under the permutation s = (13)(56) of S, . The permutation r = (123)(45) effects an alibi transformation on the first of them such that

132 213 54 4 4 5 . 6 6 On the other hand, the permutation r' transformation on T ' such that

= srs-'

= (132)(46)

effects an alibi

312 231 64 +46 . 5 5 Note that the element in the first position of either T o r T' goes into the second position: the element in the second position goes into the third, etc. T h u s the permutation r' transforms the tableau T' in a manner parallel to that by which the permutation r transforms T.

(1.1 1) DEFINITION. The permutation r' E S,, is said to be congruent with respect to s to the permutation r E S,, if and only if r' effects the same alibi transformation on T' = sT that r effects on T. = s T b e conkruent to Tunder the permutation s. If r is an alibi transformation from T to rT, then the transformation r' = srs-l is an alibi transformation from T' to r ' T ' congruent to r.

(I.12) LEMMA.Let T'

Proof: Note that any element a E T moves to the square occupied by r - ' ( u ) under the alibi transformation r. The element a' of T' occupying the same square as u is s(a), while the element of T' occupying the same square as

r - ' ( a ) is s r - ' ( a ) . The action of r' on a' is to move it to the square occupied by r ' - ' ( u ' ) = sr-lLY-'(sa)= s r - ' ( u ) which establishes the result. This Lemma has a useful corollary.

1. The Representation Theory of the Symmetric Group

219

(1.13) COROLLARY. For any tableau T belonging to n and any s E S,,, we have P(sT) = sP(T)s-’, Q(sT) = sQ(T)s-’, and e(sT) = se(T)s-’. Proof. The group P ( T ) consists of all elements of S,, which preserve the rows of T under alibi transformations. By Lemma (1.12), sP(T)s-’ must be a subgroup of S,, which preserves the rows of T’ = sT. Thus sP(T)s-’ c P(sT). Starting with sT, one finds that s-’P(sT)s c P ( T ) , which gives the inclusion P(sT) c sP(T)s-’, so that P(sT) = sP(T)s-’. The arguments for Q(sT) and e(sT) are similar. (1.14) LEMMA. Let T be a tableau belonging to n. An element s E S,, is of the form s = pq, p E P(T), q E Q(T), if no two collinear symbols of T are cocolumnar in sT.

Proof. Suppose that s = pq, p E P(T), q E Q(T),and T‘ = pqT = (pqp-‘)pT. If a, b are in r o w j of T, they are in r o w j ofpTsince p is a row transformation on T. Since pqp-’ is a column transformation of p T by Lemma (1.12), it follows that a, b are in different columns of ( p q p - ’ ) p T = p q T . Now suppose that no collinear pair a, b of Tis cocolumnar in sT. Then no two elements of the first column of sT occur in the same row of T. Consequently, there exists p 1 E P ( T ) such that p,Ta nd sT have the same elements in their first columns. Furthermore, no pair a, b collinear in p1Tis cocolumnar in sT. Thus the elements of the second column of sT occur in different rows of p , Tand not in the first column ofp, 7’. Sincep, Tand Thave the same elements in each row, there exists a p 2 E P ( T ) such that p z does not move the first column of p , T and such that p 2 p ,T and sT have the same elements in each of their first two columns. After a finite number of repetitions of this argument, one finds a p E P ( T ) such that each column of p T and sT contain the same elements. Hence there exists a q’ E Q(pT)such that sT = q‘pT. Since Q(pT) = pQ(T)p-’, one has q’ = p q p - l , q E Q(T), and s T = q’pT= p q T , so that s = pq, P E W), E Q(T).

(1.15) LEMMA. Let T and T‘ be two tableaux associated with the partitions { m l , . . . , m,} and {m,’, ..., mt’}, respectively, of n, where the partition { m l ,..., m,) is greater than the partition {m,’, ..., m,’}. Then one has e(T’)e(T)= 0. Proof. The basic claim is that there are two elements a, b which are collinear in T and co-columnar in T’. If this be false, then m,’ 2 m l , otherwise some pair a, b of integers that occur in the first row of Twould occur in the same column of T’. Consequently, Q(T‘) contains an element q’ such that each column of q‘T’ contains the same elements as the corresponding column of T’ while the first row of q’T’ contains the same integers as the first row of T. Furthermore, no collinear pair of Tis co-columnar in q‘T‘. One now considers the second row of T and compares it with the second row of q’T’. This gives

220

4. Representation Theory of Special Groups

m2‘ 2 m z . One repeats the argument to discover that m i = mi‘,1 5 i I r, contradicting the assumption that { m l , . . . , m,} is greater than { m l ’ , . . . , m,’}. Thus there exists a pair a, b collinear in T and co-columnar in T’. The transposition t = (ab) belongs both to P ( T ) and to Q(T’).Thus one has e(T’)t = -e(T’) and t e ( T ) = e ( T ) from which it follows that e( T’)e( T ) = e( T’)tte( T ) = - e(T’)e(T ) .

Consequently, e(T’)e(T)= 0, as was to be shown.

(1.16) LEMMA. Let x be an element of the group algebra A of the symmetric group S,. Suppose there exists a tableau T, belonging to n, such that pxq = &(q)xfor all p E P ( T ) , q E Q(T). Then there is a complex number CI such that x = .*e(T). Proof. Let T be such a tableau for the element x = one has, for p E P(T), q E Q(T),

c &(q)x(s)s

= E(q)x =

c x(r)r, r

E

S,, . Then

c x(r)p-’rq-’ c x(psq)s. =

Thus one obtains

x(psq) = E ( ~ ) X ( S ) ,

(1.17)

For s

=

P EPV), 4 E Q W -

1 , this gives

(1.18)

x(pq) = & ( 4 ) ~ ( 1 ) , P E P ( T ) , 4

E

QV).

Equation (1.18) gives the desired result providing x(s) = 0 when s is not of the form pq. When s is not a pq, Lemma (1.14) implies there exists a pair a, b collinear in T and co-columnar in sT. The transposition t = (ab) is an element both of P ( T ) and also of Q(sT) = sQ(T)s-l. Thus there exists a transposition q-’ in Q(T) such that t = sq-ls-l which implies that s = tsq,

t E P(T),

q E Q(T).

It follows from (1.17) that

x(s) = x(tsq) = &(q)X(s)= -x(s). Thus x ( s ) = 0 unless s is a pq. This gives x

=

c

=

c x(Pq)Pq c &(q)x(l)Pq

as was to be shown.

=

=X U )

c &(4)P4

= x(l)e(T),

(1.19) LEMMA.Let T be a tableau belonging to n. The element e ( T ) is essentially idempotent in the group algebra A of the symmetric group S,, .

Proof. Note that (e(T))’ = PQPQ so that p(e(T))2q= pPQPQq = &(q)(c(T))’for p E P ( T ) , q E Q(T). It follows from Lemma (1.16), that

1. The Representation Theory of the Symmetric Group

221

(e(T))' = Ae(T). To show that 3, is not 0, consider the linear transformation L on the group algebra A = KS, defined by right-translation by e(T). Since &(q)pq,it follows that the matrix of L with respect to the e(T) = 1 natural basis {I = gl, . . . , go), a = n ! , has all 1's along the principal diagonal. Thus one has tr(L) = n!. On the other hand, let the vectors vi = a,e(T), 1 5 i _< p, spanning the nonzero left ideal Ae(T), be the first p elements of a basis B = {vi}, 1 5 i I n ! , of A. Then the matrix of L with respect to the basis B has the form

+ c,,+l

p+ 1 column

since Lvi = ai(eT)(eT) = Aai(eT) = Avi, 1 5 i I p, and L maps A onto Ae(T). Thus one has t r ( t ) = E$', where p is the complex dimension of Ae(T). Consequently, l p = n ! so that E. and p are each positive integers dividing the order of S, . The element e(T)//? is an idempotent in the group algebra A of S,, . (1.20) LEMMA.The left ideal Ae(T) is a minimal left ideal in the group algebra A of the symmetric group S, for any tableau T belonging to n . Proof. If Ae(T) is not minimal, it is the direct sum M @ N of proper left ideals of A. There exists a decomposition of the idempotent e = e(T)/l given by e=f+g, where f and g are nonzero, orthogonal idempotents with f = efe, g = ege. Thus pfq = &(q)f,pgq = &(q)g for p E P ( T ) , q E Q(T). It follows that f and g are nonzero multiples of e(T). This implies in turn that fg is nonzero, a contradiction. Hence Ae(T) is a minimal left ideal of A. This completes the argument. Let T and T' belong to the frame F with T' = sT for s E S, . Then e(T') = se(T)s-', so that e(T')s = se(T) # 0. Consequently, right-translation by s is a nontrivial A-homomorphism of Ae(T') into Ae(T). Since each of these left ideals is minimal, they are isomorphic. On the other hand, let T and T' belong to the frames F and F', respectively, with F > F'. Any isomorphism h of Ae(T') onto Ae(T) is a right-translation according to Theorem (3.13), Chapter 2. In particular, e(T) = ae(T')x b(t)t, where b = 1 b(t)t generates h.

222

4. Representation Theory of Special Groups

Hence,

e ( T ) = ae(T’)x b(t)t = ae(T’)x b(t)te(T)/A = b(t)at[ t - ‘e(T’)t]e(T)/E.. However. [t-le(T’)t]e(T) = e(T”)e(T) = 0, t E S,, , TI‘ = t-’T’, according to Lemma (1.15). This implies that e ( T ) = 0, which is a contradiction. Thus e ( T ) and e(T’) generate nonisomorphic minimal left ideals of the group algebra KS,, whenever T and T‘ belong to different frames associated with n. ( I .21) REMARK. The observation was made in Remark (1.9) that the number of conjugacy classes of S,, is equal to the number of distinct partitions of n. This comes about in the following manner. Each permutation in S, can be written in a unique way,

(1.22)

s = (n,1

. . . n1kJ . . . (%,I

. . . %k,,,),

as the product of n? disjoint cycles (nil ... niki), 1 5 i 5 m. This cycle structure of s is characterized numerically by giving the number a, of cycles of length 1, the number a2 of cycles of length 2, . . . , the number cx, of cycles of length r . Two permutations s and t of S, are conjugate if and only if they have the same cycle structure. The cycle structure of a conjugacy class X of clements of S, can be associated with a partition of n or with a frame F i n a natural fashion. For example, the frame F given by Fig. (1.23) specifies the (1.23)

clas< K of S , , consisting of all permutations with cycle structure, two cycles of length 2, one cycle of length 3, and one cycle of length 4. These results are read from the frame F by starting with the last row of two boxes indicating a ‘-cycle, the next to last row of two boxes indicating a second 2-cycle, the second from last row of three boxes indicating a 3-cycle, and finally the first row of four boxes indicating a 4-cycle. Many writers specify this class by either the symbol 2’3’4’ or 2234. More generally, the symbol

.

1 1 1 , ” i i 7 ~ ~.~. ~7:

with

51,177,

+ . . . + arm, = n

denotes the cla\s K of all permutations of S, with cycle structure: cxl cycles of length l l i l , . . . , x, cycles of length m,.

223

1. The Representation Theory of the Symmetric Group

(1.24) REMARK.The standard formula for the number of elements of the class K is n! m l a l a ,! * . . mP,ar! The number of distinct classes of isomorphic irreducible representations of S, over the field of complex numbers is equal to the number of distinct classes of conjugate elements and thereby to the number of different partitions or frames. As a specific example of the preceding discussion, consider the case of the group S, whose seven distinct classes of equivalent irreducible representations arise from the frames shown in Fig. (1.25). These are arranged in descending (1.25)

order according to the standard ordering convention. Each gives rise to an irreducible matrix representation called a Young’s ir~tegrairepresentarion. Two such integral representations belonging to different frames are not equivalent so that these representations contain a representative element from each class of equivalent irreducible representations of S, . Extending the notation to the corresponding representations, the seven frames of ( I .25) determine the irreducible representations shown in Fig. (1.26), whose ordering (1.26)

T ....., T ...., T . . . , T ..., T . . , T . . , T . .. ..

is taken to be that of the frames. This correspondence between frames and representations and the induced ordering is tacitly understood in the sequel, although it is frequently established by means of an enumeration such as

TI, TZ 3

T3

T,,

T5, T s , T7.

According to Chapter 2, each irreducible representation q,, is acsociated with a simple component J‘.’ of the group algebra KS, . Consequently, the group algebra KS, decomposes as the direct sum

J””’@J’

....

...

... @J” @ J ’

..

.. .. B.7‘ @ J ‘ @ J ’ ,

224

4. Representation Theory of Special Groups

where the notation is introduced in the obvious manner. It is usually more convenient to write the decomposition as

J’ @ J 2 @ J 3 @ J 4 @ J s @ J 6 @ J’.

(1.27) REMARK.The calculation of the dimension of the Young’s integral representation belonging to a given frame is illustrated in the case of the frame

E F

shown in the diagram which belongs to the partition (5, 3, 1, l} of 10. One substitutes for the frame its node diagram comprising the array

***** ***

(1.28)

*

* The principal nodes of a node diagram are those lying in the first column. To each node there corresponds a hook which is the set consisting of the given node together with all other nodes lying in the same row to the right and in the same column below. The length of a hook is the number of nodes it contains. The hook graph is obtained by replacing each node of a diagram by its hook length. The hook graph of the node diagram (1.28)is

85421 521 . 2 1 The dimension f of the irreducible representation T of S, belonging to the frame F is the quotient of n ! by the product of all the hook lengths of the hook graph of F,

(1.29)

f

= n!/(product

of the hook lengths).

In the particular instance of (1.28),one finds

for the dimension of the associated irreducible representation. The concept of hook plays an important role in the representation theory of the symmetric

225

1. The Representation Theory of the Symmetric Group

groups, but it will be beyond the scope of this book to discuss these matters at greater length. The set {f i , . . . ,f,} of the dimensions of the irreducible representations of S5 corresponding to the set of frames (1.25) is {I, 4, 5, 6, 5 , 4, I}, respectively, according to Eq. (1.29). The irreducible representation T . . . corresponding to the frame

..

: :. has dimension five. It follows from the general theory of Chapter 2 ...

that the simple component J

(1.30)

J

... "

'

'

of the group algebra KS, is the direct sum

= L' @ Lz Q L3 Q L4 @ Ls

of minimal left ideals L', 1 5 i < 5. The minimal left ideals in a particular decomposition of J"

can be determined by means of the set of standard

. These are the tableaux

tableaux associated with the frame

(1.31)

123 45

124 35

125 34

134 25

135 24 .

(1.32) DEFINITION. A standard tableau T belonging to the frame F is any tableau belonging to F i n which the numbers in each row increase from left to right and those in each column increase from top to bottom. (1.32') THEOREM.The number of standard tableaux associated with a frame F is the number of minimal left ideals in any direct sum decomposition of the simple component of KS, associated with F. This is the dimension of the irreducible representation of S, belonging to F. Equation (1.30), for example, can be written ... 123 124 125 134 135 J " = L 4 5 @ L 3 5 Q L 3 4 @ L2 5 @ L 2 4 , where each of the minimal left ideals in the decomposition is characterized by the standard tableau with which it is labeled. In any group algebra K S , , a similar decomposition exists for each simple component J. The standard minimal left ideals of J are those corresponding to the standard tableaux of the frame F of J .

(1.33) REMARK. We have accumulated sufficient terminology and results to give a full description of the standard decomposition of the group algebra KS, into its irreducible parts. First, there exists an ordered set {F(l),. . . , F(r)} of frames in one-to-one correspondence with the partitions of n. Second,

226

4. Representation Theory of Special Groups

there exists an ordered set {T(i,I), . . . , T(i,fi)}of standard tableaux associated with each frame F ( i ) , 1 5 i 5 r. The group algebra KS,, is the direct sum KS,

=J F ( I )

@

. . . @JF(*),

where each of the simple components is itself a direct sum

(I 24)

JF") = L T ( i , 1 )

@

...@ Ln', fr)

of standard minimal left idmls. Each standard minimal left ideal of (1.34) is generated by a standard primitive idmipotent arising from a standard tableau. The standard tableaux are ordered according to a scheme deducible from (1.31). The tableau Tiprecedes the tableau T, if, reading from left t o right and top to bottom along the boxes of the frame, the entry in Tiis smaller than the corresponding entry of T, at the first difference. The basic task remaining is a description of the process of computing the matrix of s in the Young's integral representation. Before attempting this, we give two useful schemes for calculating characters and representations of the symmetric groups. (1.34') REMARK. Let K denote any class of conjugate elements of the symmetric group S,, . We outline a computational procedure for determining the value at K of the character of the irreducible representation of S,, determined by the partition {ml, . . . , n i k ) of n. The scheme is highly procedural and involves what we call a box for want of a standard terminology. A box is a symbol of the form [nl, . . . , n k ] ,where the entries n i , 1 i 2 k , are integers. A box has the value zero if any of its entries are negative. The standard box containing the intcgers ( 1 2 ~ ) . I 5 i 5 k , is that one in which the entries appear in nonincreasing order. If p is a permutation converting a nonstandard box into a standard one by permutation of the entries, then the value of the nonstandard box is plus o r minus that of the standard one accordingly a s p is a n even or odd permutation. In particular, the value of a box is zero whenever there is a repetition among its entries. The value of a standard box whose entries are nonnegative integers without repetitions is 1 . The calculation proceeds by allowing cycles to act on boxes. To keep notation to a minimum while still illustrating the general idea, consider the action of a cycle of length 1 1 1 on the box [n I , n2 , n 3 , n4, ns 1. The rule is ???[/7,,

n2.n2.n4.

n,l

=

+

[n1 - m. n 2 , n 3 , n 4 . n,] [ n , ,n2 -m, n 3 , n4 > nsl + [nl. / * 2 . 1 7 3 - m , n4, n,l + [n,,n 2 , n 3 , n4-m, nsl [ n , ,n2 , n 3 1 1 4 , ns - m].

+

.

To be specific, Ict a cycle of length 3 act on the box [6, 4, 21.

3[6.4,2]=[3.4,2]+[6,1,2]+[6,4,-l]=-[4,3,2]-[6,2,1],

227

1. The Representation Theory of the Symmetric Group

where nonstandard boxes are replaced by standard ones and the box [6,4, - 11 of value zero is suppressed. One proceeds according to the following steps, each illustrated by the calculation of the value of the character of S, determined by the partition (4, 3, 2) of 9 for the class K with cycle structure 1 2 32. out the node diagram associated with the given partition, (1) Write ....

namely, : . in the example. (2) Determine the lengths of the principal hooks in the hook diagram, namely,

6*** 4** 2* in the example. (3) Associate with the representation the standard box whose entries are the principal hook lengths, namely, [6,4, 21 in our example. (4) Let the cycles of the cycle structure of K act consecutively on the standard box, beginning with the cycles of greatest length. In our example, this process assumes the form

1 2 32[6, 4, 21 -+

= -+

= +

= -+

+ [6, 1,2] + [6,4, - 11) + [6, 1, 21) 1 2{[0,4,2] + [3, 1,21 + [3,4, -11

1 2 3{[3,4,2] 1 2 3{[3, 4, 21

+ 13, 1,21 + [6, -2,21 + [6, 1, -11)

+

1 2{[0, 4, 21 2[3, 1, 21) ~ 2 , 4 , 2 1 +LO, 2,21+ LO, 4,oi 2[1, 1, 21 2[3, - 1, 21 2[3, I , 01)

+

+

1{2[3, 1,OI) 2[2, 1, 01 2[3,0, 01

+

+

+ 2[3, I , -I].

( 5 ) The character of this representation has the value on K given by the final expression when each box is replaced by its numerical value. In the example, one obtains 2(1) 2(0) 2(0) = 2.

+

+

(I .35) REMARK. The development of the representing matrices in the case of

Young’s integral representations is a tedious process which we sketch below. However, there is a simpler process for obtaining the matrices of Young’s

228

4. Representation Theory of Special Groups

rational seminormal form. We give a summary of the results. For additional details, the reader may consult Robinson (1961). Since every element s E S, can be written as the product of transpositions, it proves sufficient to determine the matrices which represent the transposition (r, r l), 1 I r < n, in Young's rational serninormal form. One has the following fundamental theorem.

+

(1.36) THEOREM. Let F be the frame belonging to the partition {ml, . . . , mk} of n. Let T , , ... , Tfbe the standard tableaux associated with F given in their natural order. The irreducible representation of S, corresponding to F is an f x f'matrix representation. The rows and columns of the matrix representing the transposition (r, r 1) can be labeled by means o f the standard tableaux. The resulting matrix o f t = (r, r 1 ) has the general form shown in Fig. (1.37).

+

+

(1.37)

Tf

The matrix o f t is found by advancing along the principal diagonal. Having reached the position (u, u ) or (T,, T,), entries are determined by the rules:

( I ) r,,, = 1, when r and r + 1 occur in the same row of T, . (2) t,, = - I , when r and r + 1 occur in the same column of Tu. (3) When rand r + 1 occur neither in the same row nor in the same column, the procedure is more complicated and produces off-diagonal as well as diagonal elements. Assume that r and r + 1 are in positions (m, n) and (p, q), respectively, of T, . One examines the standard tableaux following T, for a T, which coincides with T, when the locations o f r and r + 1 are interchanged. Since To follows T,, note that m < p and q < n. When T, is discovered, one takes t,, = -2,

where I/].

= (n

-

t,"

=

1 - I?,

t""

=

1,

t,, = A,

nz) - (q - p ) .

(4) Zeros occur in all other positions of the matrix.

...

As an application of the ideas of (1.36), consider the representation T ' of S , belonging to the partition (3, 1,

... l} of 5. We find the matrix {tij} = T' (t)

229

1. The Representation Theory of the Symmetric Group

where t is the transposition (34). The standard tableaux of the frame F belonging to (3, 1, 1) are 123 Tl=4 ,

124 T2=3 ,

125 T3=3 , 4

134 T4=2 , 5

135 T5=2 , 4

145 T6=2 . 3

5

5

The final result is presented in Fig. (1.38). We begin the evaluation of the element t , , by noting that 3 and 4 occur neither in the same row nor the same column of TI. However, T, results from the transposition of the elements 3 and 4 in T, . The element 3 is in the position (1, 3) = (m, n) of TI, while the element 4 is in the position (2, 1) = ( p , 4). Thus one finds I/A = (3 - 1) - (1 - 2 ) = 3. Hence, we have t , , = -1/3, t I 2 = 819, t , , = 1, and t,, = 113 which completes the first two rows and columns of the matrix. Continuing with t,, , we see that 3 and 4 are in the same column of T3which gives t 3 3 = - 1 and completes the third row and third column. Now, for t,, , we note that 3 and 4 are in the same row of T, which gives t44 = 1, completing the fourth row and fourth column. Finally, T5is obtained from T6 by the interchange of 3 and 4. The element 3 is in position (1, 2) = ( m , n), and the element 4 is in the position (3, 1) = ( p , 4 ) . This gives l/A = (2 - 1) - (1 - 3 ) = 3. Consequently, the nonzero elements of the last two rows and columns are: t,, = - 1/3, t,, = 8/9, t , , = 1, and t66 = 1/3. This completes the work and gives the result shown in Fig. 1.38. (1.38)

I -1/3

S/9

1 1 / 3 0 0 0 0 0 0 0 0

1

0 0

0 0 0 1 0 0

0 0

0

0 0 0 0 0 0 0 0 -1/3 8/9 1 1 / 3

We turn now to an outline of the calculation of the matrix of s E S, in the Young’s integral representation determined by some frame F belonging to n. We observed without full proof that the set {TJ,1 < i sf, of standard tableaux belonging to F determine a corresponding set {e(T,)) of essential idempotents such that the simple component J F of the group algebra A of S, is the direct sum of minimal left ideals

J F = Ae(T,)

+

*

A

-

+ Ae(Tf).

230

4. Representation Theory of’Special Groups

Difficulties arise from the fact that the idempotents corresponding to the set te(Ti))do not form a complete set of orthogonal primitive idempotents of J r . One has only the following lemma. (1.39) LEMMA.The product e(Ti)e(Tj)= 0 whenever Tiand < Ti. diagrams belonging to a frame F with

TJ

are standard

The proof is omitted. Unfortunately. the product e(Tj)e(Ti)need not be zero under the conditions of the lemma. Since a straightforward calculation of the matrix corresponding to ;I group element .r requires a complete set of orthogonal primitive idempotents, it becomes necessary to make a troublesome transformation on the set {(>(Ti)]. I I i if.Notwithstanding, it is possible to efrectively define a subset CZ’ of the group algebra A consisting of quantities {N,~, , wr} such that the set E of elements ( e l , . . . , e,} defined by (1.40)

ei = e(Ti)wi,

1I i cJ;

is a complete set of orthogonal idempotents for the minimal two-sided ideal J F. The matrix { s i j ] ,I <: i, , j Cf, associated with s by means of the set E is determined by (1.41)

eisej= s i j e i j ,

1I i,,j

is a set of matrix units associated with E. where {eij},1 i i , j Boerner establishes all of the details of an algorithm for the calculation of the . Y ~ We ~ restrict . ourselves to an outline of the algorithm and t o an illustration of some of the details. Unfortunately, a really good example needs to be of such high dimension that we are forced to use some less satisfactory ones of lower dimensions. The principal steps are as follows: (1) Write down the sequence of standard tableaux TI, ..., Tf corresponding to a given frame F in increasing standard order. This has been carried out in Fig. (1.42) for the frame F whose hook diagram is

53 1 31 1 One checks that the dimension f of the corresponding irreducible representation is 6!/(1(3(3(5)))) = 16. ( 2 ) Beginning with T , and continuing through T,-,, one determines thoae T, greater than Tisuch that some pair collinear in T, is cocolumnar in T, . This process has been carried out for the Lableaux T , , T, , TI I , T,2 , T,3 , arid T,, with the results tabulated in rows TI through T14of the Fig. (1.42).

231

1. The Representation Theory of the Symmetric Group

In the case of TI, one finds that each of the diagrams TI, TI], and T12has no collinear pair (a, b) which is cocolumnar in TI.A collinear pair (a, b) in lj greater than TI which is cocolumnar in TI is listed in row TI under column T i . The remaining rows supply the same information for T, through T14. Given a pair T i , 7', with Ti < q-and no pair (a, b) cocolumnar in 17;. and collinear in T j , there exists a p i j E P(T,.) such that each column of Ti contains (1.42) Ti 123 45 6 Ti 123 Ti 45 6 T2 123 45 46 5 TI1 135 12 24 6 Tlz 135 12 26 4 T13 136 12 24 5 TI4 136 12 25 4

MATRIXPREPARATION Tz 123 46 5 46

T3 124 35 6 14

T4 124 36 5 14

Ts 125 34 6 25

Ts 125 36 4 25

Tz

14

14

15

15

26

26

14

14

15

12

12

12

12

12

12

12

34

26

Ti1 26

12

12

12

12

12

12

12

14

14

24

Tlz 24

12

12

12

12

12

12

12

25

34

12

12

12

12

12

12

12

14

14

T7 Ts Ts Ti, Tii Ti, 126 126 134 134 135 135 34 35 25 26 24 26 5 4 6 5 6 4 16 16 14 14 Ti1 Ti2

Ti, Ti4 Ti5

Ti6

136 24 5 16

136 25 4 16

145 26 3 14

146 25 3 14

Ti3

7'14

14

14

16

16 26

16

36

14

14

15

15 TI3 25

15

25

35

35

T14 14

14

15

24

the same elements as the corresponding column of p i j T , . One proceeds through the list of standard tableaux constructing a list of such p i j . In the case of Fig. (1.42), the list begins with p l , l l and p l , l z ,where

(3) Using this list, one constructs for each standard tableau TI a list of all possible permutations {ri,in}of the form ri,i" ' p .[ , t.i p i. i . , .~ . . Pi,,-I,in, (1.43)

where i < i, < . . . < (1.44)

< in. With each such r i , i nthere is associated a sign Ei'" = (-

1>",

where n is the number of factors in r i , i n .

232

4. Representation Theory of Special Groups

(4) Let s E S,, . T o determine the elements of the ith row of the matrix of s under the given representation, one constructs from the set {Ti, . . ., ri,i,} of the permutations r with initial index i, a second set {,,

(1.45)

{s, r i , i

. . ., r i , iks)

, ~ ,

of permutations depending on s. Then one computes the tableaux T'O'

= $-I

T i , . . ., Fk'= ( T i , i k s ) - l T i .

As an illustration, consider the element s = (234) E S , . In order t o compute the matrix element s l m , 1 5 rn i 16, one selects the permutations p l , l = (24)(35) a n d p , , , , = (26)(35). Then one calculatesp,,,,s = (253) a n d p , , , , s = (25346). The list of permutations forming the set (1.45) is (1.45')

f(234), (253), (25346)).

One obtains TCO) =

123 142 (243)45 = 35 , 6 6

T ( ' )=

123 135 (235)45 = 42 , 6 6

T"'

123

= (26435)45

6

165

= 32

4

for the associated tableaux. (5) Each of the tableaux T'O', T ( ' ) , ..., T ( k )makes a contribution E(". 0 5 j . k , to each s i n l ,1 5 rn If,whose value is given by $. I

", =

+ c(l) + ... + E(k),

where the c's are understood to depend on the column index rn of s i m . Each T('.' is related to an r i , with an associated sign q(',)= (- 1)"" where n, is the number of factors of r i ,,. The r corresponding to T'O) is the identity and the associated sign plus. m <,f,if T(') has a cocolumnar pair (a, 6)which (5a) For a given s i m , I I is collinear in T,, then

,

( I .46)

p)= 0.

(5b) If T(" has no cocolumnar pair (a, b ) which is collinear in T,, then there exists q E Q(T('))such that each row of qT('.)contains the same elements

233

1. The Representation Theory of the Symmetric Group

as the corresponding row of T, . Let cq = I for q even and one has

-

1 for q odd. Then

p)= Eq Fi(L).

(1.47)

The application of these rules to obtain s,,,, , 1 I m 5 16, is carried out in Table (1.48). The tableaux T(O),T(’),and T(’) are listed at the side of the

(1.48) TI

MATRIX ENTRIES FROM TABLEAUX

Tz

T3

T4

Ts

T6

T7

T8

T,

Tlo Ti1 Tiz

Ti3

Ti4 Ti,

Ti6

123 123 124 124 125 125 126 126 134 134 135 135 136 136 145 146 45 46 35 36 34 36 34 35 25 26 24 26 24 25 26 25 6 5 6 5 6 4 5 4 6 5 6 4 5 4 3 3 142 (13) (13) 1 35 1 6

(36) -1 -1

(36) (16) (16) (13) (13) (13) (13) (13) (13) (45) (16)

135 (23) (23) (14) (14) -1 1 (16) (16) (14) (14) 1 -1 (16) (16) (14) (14) 42 1 -1 -1 1 6 165 (13) (13) (14) (14) (34) -1 (26) (26) (14) (14) (13) (13) (13) (13) (14) (14) 32 1 4 s1,=0

0

1

0

0

0

0

0

0

0

-11

0

0

0

0

table. When T, contains a collinear pair (a, b) which is co-columnar in one of the tableaux T(’), T(’),or T(’), this pair is listed under T,, in the corresponding row. Otherwise, the value of c4 is listed followed by that of E ( ” ) = E ~ E ~ ( ’ )This . second type of listing first occurs in row

142 T(O)= 35 6 under T, = T 3 ,where cq involved is q = ( I ) since

=

I and

=

I , so that do)= 1 for m

I42 35 6 has the same rows as

124 T3 = 35 6

= 3.

The q

234

4. Representation Theory of Special Groups

One notes in general that = - 1, and E , ( ~ = ) - 1 in our example. = 1, The values of I 5 nz 5 16, are listed in the last row of Table (1.48). (1.49) EXAMPLE. The reader may wish to check his understanding of the procedure by determining the matrix of (234) in the irreducible representation of S , given by the frame : : '. The result should be as shown in Table (1.50).

( I .50)

123 124 125 134 135 45 35 34 25 24 123 45

0

1

0

0

-1

124 35

0

0

0

1

-1

125 34

0

0

0

0

-1

134 25

1

0

0

0

-1

135

0

0

1

0

-1

I t is clear that the calculations involved are rather tedious for even small values of n. P. D. Swardstroni has developed a program operating on the 1 BM 7094 to make these computations for the representations of S,, , 3 1.17 5 7. The method works in general but limitations of machine storage make it impractical to deal with larger values of n or even with some of the higherdimensional representations of S , and S , . We turn now to a discussion of the relations between the representations of the symmetric group S,, and those of the general linear groups GL(V) of nonsingular, linear transformations on an rn-dimensional vector space.

2. MODULES OVER SYMMETRIC ALGEBRAS

This section defines the concept of a symmetric algebra and investigates the special properties of a' module M over a symmetric algebra A. The word algebra always refers to an algebra A with identity 1 which is finite-dimensional over the complex numbers. The treatment is confined largely t o the case where the symmetric algebra A is the group algebra KS, over the complex

2. Modules over Symmetric Algebras

235

field of the symmetric group S,. These considerations lead naturally to a presentation of the elegant method of Curtis (1956, 1958) for establishing the relationships between the ideal theory of KS, and the integral representations of the general linear group GL(V) of nonsingular linear transformations on a complex, finite-dimensional space V. The treatment of this section admits of a substantial generalization, but we must refer the reader to the papers of Curtis or to the monograph of Curtis and Reiner for additional information. We state the definition of semisimplicity in the case of a finite-dimensional algebra A with identity. Then give the version of Wedderburn's theorem which holds in this special case. Every semisimple algebra is shown to be a symmetric algebra. Given any module M over the symmetric algebra A, there exists a particular two-sided ideal A, of A which is called the nucleus of M. Let D denote the algebra HomA(M, M) acting on M as right-multiplication. Whenever the nucleus AM has an identity, there exists a natural duality between the right-ideal structure of AM and the right D-components of M. Let M denote the n-fold tensor product V 0 . .. @ V of the m-dimensional, complex space V with itself. Then M can be made into a KS,-module. Furthermore, M affords a tensor representation T of the general linear group CL(V). Let A denote the complex group algebra KS, of the symmetric group S, and b the enveloping algebra of the group T(GL(V))of linear transformations on M. Then A is a symmetric algebra on M and b is Hom,(M, M). Thus one can apply the general theory to deduce the relations between the ideal theory of KS, and the tensor representations of GL(V). We recall some definitions from Chapter 2. Let A be a finite-dimensional algebra over the field K of complex numbers. An ideal J of A is said to be nilpotent if there exists a positive integer n such that the product of n or more factors from J is always 0. The sum of all the nilpotent left ideals of the algebra A is a nilpotent, two-sided ideal R called the radical of A. A finitedimensional algebra A with identity 1 is said to be semisimple if and only if A contains no nonzero, nilpotent two-sided ideals. A semisimple algebra A which contains no nontrivial two-sided ideals is called simple. There is a famous theorem of Wedderburn which can be stated in the case at hand as follows. (2.1) THEOREM (Wedderburn). Let A be an algebra with identity such that A is finite-dimensional over the complex numbers K. Suppose that A is semisimple. Then A is the direct sum (2.2)

A = J1@...@J'

of minimal two-sided ideals J', I i i r , such that each J' is a simple algebra over K . Furthermore, any minimal two-sided ideal J of A coincides with one of the J'. Finally, there exist integers n, such that each J', 1 i i i r, is iso-

236

4. Representation Theory of Special Groups

morphic to the full matrix algebra of all n, x n , matrices over the complex numbers. As in Chapter 2, we use the symbol

B=B(l)@*..@B(r) for the set of all quasi-diagonal matrices of the form

where each B(i) denotes the algebra of all n , x n i complex matrices, 1 5 i 5 r. The set B is a subalgebra of the algebra of all LY x LY complex matrices where LY = n,

+ ... + n,.

By Wedderburn’s theorem, there is a family of homomorphisms hi : A such that Im h i = B(i). The map h : A B defined by

-+

B(i)

--f

(2.4)

h(x) = h,(x)

+ . . + hr(x) ’

is an isomorphism of A onto B.

(2.5) DEFINITION. A finite-dimensional algebra A over the complex numbers K is said to be symmetric if and only if there exists a nondegenerate bilinear formf: A x A -+ K such that f(a, b)

(2.6)

=f@, a)

and

(2.7) for all a, b, c in A. A bilinear form satisfying (2.6) is symmetric and one satisfying (2.7) is associative. (2.8) REMARK.Every finite-dimensional, semisimple algebra A over the complex numbers K js a symmetric algebra. To see this, let h : A -+ B be the isomorphism (2.4) defined previously. For a, b E A, one has

h(a)

= x1

+ ... + xr,

xi € B i ,

and h(b)=y, + * - * + y , , Then define f(a, b)

= tr(x,y,)

yi€Bi.

+ . . + tr(x, y J *

237

2. Modules over Symmetric Algebras

+

+

One has tr((x y)z) = tr(xz yz) = tr(xz) + tr(yz) for matrices x, y, and z , so thatf(a b, c) =f(a, c) +f(b, c). Similarly,f(a, b c) =f(a, b) +f(a, c). Also tr((crx)y) = tr(x(ay)) = a tr(xy) for matrices x and y and complex numbers a so that f is a bilinear form on A. It follows also from tr(xy) = tr(yx) thatf(a, b) =f(b, a). Since tr((xy)z) = tr(x(yz)) for matrices x, y , and z, f(ab, c) =f(a, bc), which proves thatfis associative. Furthermore. given any a E A, there exists an a* E A such that h(a*) = x,* + ... + x,*, where xi* denotes the matrix Hilbert adjoint to x i . Thus

+

f(a, a*) = tr(x,x,*)

+

+ . . . + tr(x,x,*).

Recall that tr(xixi*) is positive unless xi = xi* = 0. It follows that either f(a, A) = 0 orf(A, a) = 0 implies that a = 0, so thatfis nondegenerate. Consequently, A is a symmetric algebra. We see, i n particular, that KG is a symmetric algebra for every finite group G. Every choice of a basis {a,}, 1 5 i I n, of a complex n-dimensional algebra A gives rise to two matrix representations of A. The first of them is defined by x -+ A(x), where xa, = 2 ,I(X)~,aj,

and the second by x

+ A(x),

a,x

=

1I iI n,

where

1A(x)ijaj,

1I iI n.

In the case of a symmetric algebra .4, there exists a nice relationship between these representations of A determined by the basis {a,} and those similarly defined by the dual basis (b,} of (a,} with respect to the associative formf. (2.9) LEMMA. Let {a,}, 1 5 i 5 n, be a basis of the symmetric algebra A and let {bi}, 1 5 i 5 n , be its dual with respect to the formf of A, that is,

f ( a i , bj) =f(bj, a,) = d i j . Then, for 1 5 i 5 n , one has the equations, xai =

A(x)~, a j ,

xb, = Proof. First, note that

1A(X)~,bj ,

(2.10)

a, x

=

iff (2.1 1)

bi x =

A(x),~a j ,

c A ( x ) ~bj~.

c , I ( x ) ~aj~ x , f ( x a , , bj)aj, so that A(x)~,=f(xai, bj). Moreover, one has b, x c f(bi x, aj)bj c f(bi, xaj)b, c f(xaj, bi)bj xai =

=

=

The remainder is analogous.

=

=

=

2(x),, bj .

238

4. Representution The0 ry of' Special Groups

Let M be a finite-dimensional complex vector space which is a left A-module of the algebra A.

(2.12) INiwITiox. The dual space M" of M is a right A-module with multiplication of,/'€ M" by a E .I defined by ( fa)x =f(ax),

x E M.

7 he pr,mf that R.1" is ii right A-module with this definition of multiplication is

i t 3 to the reader. Our goal is to introduce a certain two-sided ideal A, of A, called the nucleus of M. which plays a significant role i n developing the integral representation theory of the general linear group GL(V). The nucleus is defined with the aid of ii map 7 : M x M" -+A which we now introduce. Let {a,} and {bi]. denote a pair of bases of the symmetric algebra A dual with respect to the associative bilinear formf'of A. Given (m,f) E M x iM*, define y by y(nl7.f) =

C biJ(ai m).

I t is easy to verify that y is bilinear over K. In addition, y is bilinear over A in the sense that

?(am1

+ bm, ? . f=) ay(m,,.f) + by(m2 , f )

and y(m. /;a +.fz b)

where m. m , , m2 E M,.f: ./;. ,f2 argument i s to show that y(am,f') m E M , ~ M". E and a aJld (2.11).

E

=

E

=

IJ(m.f1)a + Y(m?.fz)b,

M* and a, b

a:;(m, f ) .

E

y(m,fa)

A. The difficult part of the =

r(m,.f)a,

A . We demonstrate the first of these using Eqs. (2.10)

~ ( a m - f )= C bi.f(ai(am)) = bi./((aia>m) = b ; , f ( ( CA(a);jaj)m) = 1 A(a),j b,f'(ajm) = 1 abjf(ajm) = a bjf(aj m) = ay(m,f).

C

c

I3

The argument that y(m,./a) = y(m.f')a is similar. The form y is nondegenerate. For suppose that y(m,f) = 0 for every f ' M*. ~ Then 0 = bif'(a,m) implies that f(aim) = 0 for all i. Since the identity of A is a complex linear combination of the basis {a,), this means that.f(m) == 0 for a l 1 . f ' ~M* so that m itself must be the zero of M. We leave to the reader the argument that y(m.,f)= 0 for all m E M implies t h a t f = 0. The bilincarity of 7 in this second sense implies that the linear span of all elements of A of the form y ( r n . f ) , m E M, f E M*, is a two-sided ideal J of A.

1

239

2. Modules over Symmetric Algebras

The ideal J defined above is called the nucleus of M and (2.13) DEFINITION. denoted by AM. Since our analysis is restricted to the case in which A is a semisimple algebra, one has

A=A,oA,

(2.14)

where A is a two-sided ideal complementary to A,. According to (2.14), the identity 1 of A decomposes as the sum 1’ + I”, 1’ E AM, 1” E A, of orthogonal central idempotents. In particular, A, is generated by the central idempotent 1’ which acts as the identity of AM. The group algebra FG of a finite group G over a field F of characteristic p , p dividing [G : I J, is an exarnple of a symmetric algebra which is not semisimple. In the case of such more general symmetric algebras, a left A-module M is called regular exactly in those cases where A, has an identity. The identity 1’ of A, can be written in the form

1’ = C y(mi

(2.15)

where mi E M, f , E M*, 1 5 i 2 k . For f E M*, one has that y(m,f-fl’) y(m,f) - y(m,f)l’ = 0 for all m in M which implies that

=

f l , ’ =fl’ = f :

(2.16)

Thus the right-multiplication 1,’ acts as the identity on M*. Similarly, the left-multiplication 1,’ acts as the identity on M, that is,

1L ’m = l’m = m,

(2.17)

m E M.

(2.18) LEMMA.The ideal A is the kernel of the representation of A afforded by M while AM is faithfully represented on M. Proof. Let x be an element of the kernel K of the representation of A afforded by M. Then XI’ = x

C ?(mi ,fi)= C y(xmi ,fi) = 0

so that K c A.Conversely, let x E

A.Then

xm = x(1’m)

= (x1’)m = 0

for every m E M so that A c K. Since AM n A = 0, AM is faithfully represented on M. The set D = Hom,(M, M) is a complex algebra playing a central role in our later analysis. We do not verify that D is an algebra, but observe that if u, v E D, then u v is a homomorphism of M which clearly commutes with multiplication by elements of A. A similar statement can be made about the product uv. Details are left to the reader as an exercise. Since A has an identity

+

240

4. Representation Theory of Special Groups

1 and therefore contains K , D is a subalgebra of Hom,(M, M). We have the following useful theorem.

(2.19) THEOREM. Let M be a left A-module for the semisimple algebra A over the complex numbers K . Then D = Hom,(M, M) is a finite-dimensional, semisimple algebra over the complex numbers. Proof First observe that HomA(M, M) = HomAM(M,M) since A is the kernel of the representation. The nucleus AM is a semisimple K-algebra faithfully represented on M. Hence there exists a K-basis B = {ml, . . . , m,} unitary with respect to a suitable inner product of M, such that each lefttranslation aL, a E AM, has a quasi-diagonal matrix with respect to B. Denote it by

where each of the Ai’s is a square matrix of some fixed dimension n, x n i , independent of the choice of a E AM.Given any matrix M(a,), there exists a* in AMsuch that M(aL*)= M(a,)*. From a,*d = daL*,one obtains

or (2.20)

M(a,)*M(d)

=

M(d)M(aL)*

M(d)*M(a,)

= M(a,)M(d)*.

The linear transformation d* with matrix M(d)* satisfies

d*aL= a,d*, which implies that d* E D. Thus D is closed under the adjoint operation. Let d belong to a nilpotent, two-sided ideal of D. Since d*d is nilpotent, tr(d*d) = 0, which implies that d = 0. Therefore D is semisimple since it contains no nontrivial, nilpotent, two-sided ideals. Finally, D is finite-dimensional over K as it is a subalgebra of the finite-dimensional algebra Hom,(M, M). The study of the various relationships between D and AM is facilitated by the introduction of a map f l x I y : M + M , corresponding to each pair, J’E M*, y E M, which is defined for m E M by

m(fl x I Y> = YhS)Y. The functionfl x 1 y is an element of D = Hom,(M, M). First note that

(am>(Slx I Y> = r(amJ)Y = [ar(m,f)lu = a[r(mJ>yl= “(fl x I Y)1 ~ y, m E M, and a E A. Consequently, f ]x y commutes with for , j ’ M*, multiplication by a. The remainder of the argument is left to the reader.

241

2. Modules over Symmetric Algebras

(2.21) THEOREM.For each t E Hom,(M, M) there exists a t = aL. Proof. Given t E Hom,(M, M), it follows that

Wfl x I Y)1 Let a = y(tmi,fi), where 1' = m E M, one has

=r(mf)y,

E

A such that

m E M.

y(mi,f,) is given by Eq. (2.15). Then, for

aLm = am = [C ~(tmi,fi)]m = C r(fmi,fi>m = C t[y(mi ,fi>mI = t[C y(mi >fi>m]= tm.

Hence, t

= aL,as

was to be shown.

The custom is t o write the elements of A to the left of the elements of M on which they operate and those of D to the right. The commutativity between A and D = Hom,(M, M ) is then expressed by (am)d = a(md) for m E M, a E A, and d E D. It is indispensible to our methods to establish a one-to-one correspondence between the right ideals of the form eA, with e an idempotent of A,, and the right D-components of M. Since D is semisimple, every right D-submodule N of M is a D-component of M. Let J denote the set of right ideals of A, and '3 the set of right D-components of M. Define the map A: J % by A(eA) = eM = eAM. Then eM is a right D-submodule since (eM)d = e(Md) for d E D. If e and f a r e idempotents of A, with e A = fA, then fe = e and ef = f so that e M c f M and f M c e M . Thus A is well defined. Define the map p : % + J for any right D-submodule N of M by p(N) is the subspace of A, generated by all elements of the form y(n, m*), n E N, m* EM*. Since y(n, m*)a = y(n, m*a) for a E A, p(N) is a right ideal of A,. --f

(2.22) LEMMA. The map A: J + '3 is a bijection of J onto '3 with inverse the map p : '3+ J, a bijection of % onto 3. Proof. Let the right ideal fA of A, contain the right ideal eA, with f and e idempotents of A,, so that fe = e. Consequently, A(eA) = eM c f M = A(fA). Let the right D-submodule N of M contain the right D-submodule P. Then each element y(p, m*) generated by P is also generated by N. Therefore, p(P) c p(N). Thus both A and p are inclusion preserving maps. lf x E eM, then y(x, m*) = y(ex, m*) = ey(x, m*),

m* E M*, is an element of eA which implies that p(eM) c eA. Conversely, let x E e A c A M .Then

x = XI'

=x

C r ( m i > L >= C y(xmi,fi) = 1 y(exm, ,si)

242

4. Representation Theory of Special Groups

is an element of p(eM). This result shows that eA c p(eM) Hence, ,d.(eA)

= p(A(eA)) ==

= p(I.(eA))

c eA.

eA and p?is the identity map on J.

Let N be a right D-component of M. There exists a projection 71 E

Hom,(M, M)

of Nl onto N since D is semisimple. By Theorem (2.21), there exists an e E A, with mn = em. m E M. I t follows that (e2 - e)M = 0, which means that e2 = e since A, is faithfully represented on M. Note also that y(n, pi*) = y(en, m*)= ey(n, m*),

n

E

N, m*

E

M", so that p(N) is contained in eA, a right ideal of A,. However, e

= el' = e

2y(mi ,f;)= 2 y(emi ,.fi)

is an element of p(N) so that eA c p(N). Furthermore, 2(,u(N))= 3.(eA)

= eM = N.

Thus j.p is the identity map on %. We conclude that both ;Iand ,D are bijections which are the inverses of each other. We are able to show that 1 and p also preserve algebraic structure. The first result in this direction is the following theorem. (2.23) THEOREM. Let the right ideal eA of A, be A-isomorphic to the right ideal fA of A,. Then the right D-submodule R(eA) is D-isomorphic to the right D-submodule i(fA). Proof: Let / I be an A-isomorphism of the right ideal eA onto the right ideal fA. Then, for x E eA,

h(x) where a

=

h(e)

E

= h(ex) = h(e)x = ax,

fA. Also, for x

E fA,

K'(X) = h-'(fx) where b

=

K'(f)

E

= h-'(f)x = bx,

eA. I t follows that

(ba)x = x, x E ~ A ,

and

(ab)x

=

x,

xE~A.

We define two maps, cr: eM + f M (i(eA) -+ A(fA)) and z: f M - + e M (;I(fA) + i(eA)), by o(x)

ax

E

fM.

x

E

1(eA) = eM,

s(x) = bx

E

eM,

x

E

A(fA)

=

and = fM.

243

2. Modules ouer Symmetric Algebras

These two maps are easily seen to be D-homomorphisms, A(eA) + A(fA) and A(fA) + 1(eA), respectively. Note that (za)(em) = (ba)em = ((ba)e)m

= em

and (az)(fm) = (ab)fm so that zu = l l ( e A ) and az = which completes the argument.

= ((ab)f)m = fm,

Thus a: A(eA) + A(fA) is a D-isomorphism

The companion theorem is also valid. (2.24) THEOREM. Let the right D-submodule N of M be D-isomorphic to the right D-submodule P of M. Then the right ideal p(N) of AM is A-isomorphic to the right ideal p(P) of AM. Proof. Let h : N + P be a D-isomorphism of N onto P and let y(ni', f i ' ) be the zero element of p(N). Then one has, for m E M,

C

(C y(hni',fi'))m

= =

C

C r(hni',fi')m = 1 (hn,Xfi' I x I m)

C h(ni'(fi' I x I m)) C h(y(ni',fi')m)

= h(C

=

y(ni',.fi')m) = h((C y(ni',fi'))m) = 0,

so that y(hni',fi') = 0 since A,is faithfully represented on M. Consequently, there exists a well-defined map h : p(N) + p(P) such that

h"(1 y(ni , f i > ) = C y(hni , f i ) .

The map

h" is easily seen to be additive and, for a E A,

h"((CY(ni >fi>)a)= h"(CY(ni f i a)) = y(hni f i a> = C y(hni L>a= (1 y(hni A'>). =(h"(Cy(ni,fi)))a, >

9

>

3

h" is an A-homomorphism. Similarly, there exists an A-homomorphism : p(P) -+ p(N) defined by

so that

h"-'

h"-'(C?(Pi >A>)= C y(h-'pi

One can see without difficulty that h"-'h" is an A-isomorphism of p(N) onto p(P).

=

--

lp(N)and hh-'

=

so that h

We summarize our results before passing to an important application. We have shown that if A is a complex, semisimple algebra acting on a left Amodule M, then D = HomA(M,M) is a complex semisimple algebra acting on M as a right D-module. Furthermore, there exists a bijection 1from the right ideals of the nucleus AM of M onto the right D-submodules of M such

244

4. Representation Theory of'Special Groups

that 3, maps minimal right ideals of A, onto irreducible right D-submodules of M. Also that eA and fA are A-isomorphic right ideals of A, if and only if /I(eA) and 2(fA) are D-isomorphic right D-submodules of M. These facts provide a powerful tool for analyzing a large class of irreducible representations of the full linear group GL(V) of nonsingular linear transformations on an nz-dimensional complex vector space V. The role of A is played by the group algebra K S , of the symmetric group S, and that of the . @ V of the vector left A-module M by the n-fold tensor product V 0.. space V with itself. We must introduce the group algebra KS, as left operators on the tensor product space M. It is sufficient, of course, to specify how the elements of S, are to act on the basic tensors of M. The action of s E S, is defined by permutation on the positions of the factors of a basic tensor of M rather than by permutation on the elements of a basis of V. Our problem is complicated by the necessity of employing a heavy burden of indices and notational conventions in the sequel. In order to clarify the fundamental idea without introducing superfluous details, we consider first the case of M the four-fold tensor product of V with itself, s = ( 1 4 2 3 ) a n d r = ( 1 4 3 ) , a n d m = x @ w @ z @ y a basic t e n s o r o f M . T h e successive factors of m have been placed out of their natural alphabetical order deliberately. The action of s = (1423) on m to produce sm is given by (1423)[x o

w oz@ y ] = z @ y o w o

x

according to the rules : 3 -+1 under s so that the third factor z of m goes into the first position (a) of sm; (b) 4 -+ 2 under s so that the fourth factor y of m goes into the second position of sm; (c) 2 + 3 under s so that the second factor w of m goes into the third position of sm; (d) 1 -+ 4 under s so that the 1st factor x of m goes into the fourth position of sm. The permutation s = (1423) can be written as

.=( 1

)=(

3 4 2 1 2 3 4

s - q l ) sF'(2) 1 2

s-'(3)

3

s-y))2

which indicates the action of s on m in a manner usefully employed in our general definition below. To check that he has the procedure clearly in mind, the reader may wish to verify that (143) and rs = (1342) have the actions (143)[2 0y 0w

x] = w 0y @ x 0z

245

2. Modules over Symmetric Algebras

and (1342)[x O

w O z Oy] = w O y Ox Oz.

These results show that the action of (1423) followed by that of (143) is the same as the action of (143)(1423). Now we give the general definition in the case where M is the n-fold tensor product of V with itself and s is an arbitrary element of the symmetric group S, . The element s can be written in either of two equivalent forms I (SU)

2 42)

...

n s(n)

or

(s-i(I)

s-’(2) 2

...

s - ’n( n ) ) .

The second of these is more convenient for our immediate purposes, as we have seen previously. Let m denote the basic tensor v, . . . 0v, of M. Then one defines sm by s[v* 0.. . OV,] = vs-l(l) 0 . .. OV,- l ( , ) .

Observe that this is merely a continuation of our initial procedure. We must verify that if r and s are elements of S,, then the action o f s followed by that of r coincides with that of rs. Note that r(sm) is (2.25)

r[v,-l(l) O . . . Ovs-l(,,l

o... Ovs-l(,-I,,),

- vs-l(r-l(l))

) O . . . 0V(rs)- ~ ( n ) 0.. . 0 v,] = (rs)m.

= v(rs)- ‘ ( 1 = rs[vl

One stares at Eq. (2.25) until it can be believed, or else reasons that the entry in the first position of the result r(sm) comes from the r - ’ ( l ) position of sm so that it is v s - l ( r - l ( l ) )the ; entry in the second position of r(sm) comes from the r-’(2) position of sm so that it is v ~ - ~ ( ~ - ~and ( ~ )so ) ; on. This argument establishes the desired fact that the action of S, on a basic tensor satisfies the module requirement, r(sm) = (rs)m. Given any basis { v ~ ., . ., vm} of V, there exists a corresponding extended basis of the n-fold tensor product M consisting of the set {vj, 0. . . 8 vj,f of all basic tensors formed from the basis {vl, . . . , v,} with each indexji, 1 2 i n, ranging independently through the integers from 1 through m. Since we must deal repeatedly with index sets ( j l , . . . ,j,), it becomes convenient to adopt several different shorthand conventions for them. We sometimes use the single symbol (J) to denote such an index set. A general tensor m of M can be expanded in terms of the extended basis as (2.26)

m=

C m ( j l , . . . ,i,)[vj, o ... @vj,],

246

4. Representation Theory of Special Groups

where we maintain our policy of omitting ranges of summation unless there is acute danger of misunderstanding. Equation (2.26) can also be written

m

(2.27)

=

C m(J)v(J).

Thus we use either of the symbols m ( j , , . . . , j n )or m ( J ) to denote the coefficient of a basic tensor itself denoted either by vjl @ . . . @vj,, or v ( J ) . The first o f these, of course, is more specific, but the second more convenient. Each element s E S,, determines a linear transformation T(s) on M whose action on the elements of the extended basis is defined by T(S)[Vjl0. . . O Vj,]

= SIVjl

O . . . 0Vj,I

- ~ j ~ - 1 ( ~ )

0.'. Ovjs-l(,,).

One observes that since the element s acts as a permutation on the elements of the extended basis, the linear transformation T(s) maps a basis onto a basis and is therefore a nonsingular linear transformation on M. One notes as well that if v ( J ) is any element of the extended basis, then T ( ~ s ) v ( J= ) ( ~ s ) v ( J=) r(sv(J))= T ( s ) [ T ( r ) v ( J )= ] [T(r)T(s)]v(J). This equality implies that T(rs) = T(r)T(s) and that the correspondence s + T ( s )is a representation of the symmetric group S, on the tensor product space M. One must always keep in mind that this particular representation is not s 4s 0. . . 0s, n-factors, where s acts as a permutation on the basis vectors of V. To the contrary, we repeat that s acts like a permutation on the set of elements of the extended basis. It follows that the matrix of T(s) with respect to this particular basis is a permutation matrix on the coefficients of a given tensor m of M. We adopt the usual convention and refer to the coefficients of m with respect to the extended basis as the components or tensor components of m. The theory which we elaborate is really nothing more than a systematic analysis of the permutation effected by T ( s )on these components. We begin this analysis in the next paragraph. Let the tensor m of M have the expansion m = m(J)v(J).Then T(s)m is given by T ( s ) C m ( j , ,..',j,,,"j] O . . . O V j " 1

1m(.i,, . . . ,.i,,)s[vj o . . . o vjnl = C m(.j,,. . . ,,jn)[vjs- o . . . o vj,- ](")I. =

I

This expansion can be written in a more suitable form by making a change of indices: k i= j s - l ( i or ) k,(;)=,js-l(s(i)) =ji.This substitution leads to

1

Vs)m = m(k,(,) , . . . , ks(n))[vkl O ... O vk,,I. T o summarize. let us agree that when ( J ) denotes ( , j , , . . . , j n ) ,(sJ) denotes ( j S ( , ).,. . , ,is(,,)). For example, let ( J ) = ( j , , j , , j , , j,) and s = (1342). Then

(2.28)

247

2. Modules over Symmetric Algebras

(sJ)denotes ( j , , j l ,j , ,j2).Equation (2.28) asserts that given a tensor m of M, the components of the tensor T(s)m are related to the components of m by the formula: (2.29)

[T(s)rn](J) = rn(SJ).

Naturally, one usually considers M an S,,-module and frequently prefers to write sm rather than T(s)m. In such cases, Formula (2.29) becomes

sm(J) = m(sJ),

(2.30)

with the interpretation of the symbol (sJ)introduced previously. One may question whether the first definition of the action of an element S E S ,on a basic tensor m=x,O...@x,,, x i = c a j i v jfor 1 < j < n , is consistent with the second one. According to the second definition, sm is given by

sm=x:j,c,,l . . ~ a j , c n , , , [ v j l ~ . . . ~ v j , l . However, since u j s c s - l ~ ,=~ ~ s - l ~ ithis ~ equation may be rewritten, by rearrangement of the order of the a's, as

o

I(n)[vj,o . . . vj,l sm = 1 ails- . . . ajnS= [Cajls-l(I)vjt0.'. OCaj,s-l(n)vjn] = Xs-1(1)O . . . OXs-l(,,),

which agrees with the first definition. Thus the two definitions are equivalent for basic tensors. We turn now to a connection between S,, and GL(V) where V is the rn-dimensional space introduced previously. The n-fold tensor product M of the space V with itself is a natural representation space or module for the general linear group GL(V). Each g of GL(V) is represented by the n-fold tensor product g @ . . . @g of g with itself. If vj, 0... @vj, is an element of the extended basis of M, then one recalls that (Vjl

0 .* . @ vj,)g

= vj,g 0. . .

vj*g.

The elements of GL(V) are taken to act on the right so that our results will conform to the notation introduced earlier in this section. Denote by G the group of linear transformations on M which constitute the image of GL(V) under the tensor product representation. Let D be the linear envelope of G consisting of all complex linear combinations of elements from G. Then d is a complex algebra of linear transformations acting on the space M which becomes a right d-module. Our intention is to prove that fi coincides with HomKSn(iM,M) so that the results of the first part of the section

248

4. Representation Theory of Special Groups

can be applied to this speciai case. In order to see that 0 is contained in Hom,,,?(M, M), let s E S, and g E GL(V).Then one has [ 4 v j I 0.'. Ovjn)lg =(vj,-,,,,O...Ovj,-,,,,,)g = ( y j 7 - I ( , ) g O...Ovj,-t,,)g)

= s(vjlg@ = S[(VjI @

. . . 0V j n g ) . . . 0V j , ) S ] .

Thus one finds that (sm>s = 4 m s )

(2.31)

for s E S,, , m E M, and g E GL(V). Since S, and G constitute a set of Kgenerators of KS,, and D, respectively, it follows that D is contained in HomKS,(M, MI. The fact that HomK,,(M, M) is contained in D will be demonstrated after considering a special subspace W of M. If D = HomK,,(M, M), then KS, and 0 satisfy the conditions of the previous theorems. Consequently, all the irreducible D-subspaces of M are of the form e M where e is a primitive idempotent of KS,. The most obvious primitive idempotent of KS, is the idempotent e = ( l / n ! ) s, s E S,. For any m E M, the element em has the property that t(em) = em, t E S , . since (2.31')

t(em)

=

1

t[(l/n!) sm]

=(

~ / nC ! ) tsm = em.

The subspace W of symmetric tensors consists of all w E M such that sw = w for s E S,. It follows from (2.31') that the range eM of e is contained in W. Conversely, one notes that W is contained in eM. Each basic tensor which is the product v @ . . . @ v of equal factors is clearly an element of W. More generally, if w = M(J)v(J)is any element of M, then w is an element of W iff w ( J ) = u.(sJ) for every index diagram ( J ) . This idea will be elaborated in a number of ways in later discussions so that it is worth considering in more detail. For example, let ( J ) = ( j l , j 2, j , ,j4)= (3, 1, 2, 4). Given any tensor m, m(J) is the coefficient of the basic tensor v3 @ vl @ v2 0v4 in the expansion of m. The assertion that n7(J) = m(sJ) for all s E S4, means exactly that the twenty-four basis elements v1 Ov, @ v 3 @ v 4 , . . . , v4 @ v , OV, 0v, have the same coefficient. However, if ( J ) = ( j l , j 2, j 3 ,J4)= (1, 1, 1, 2), then the implication is much less inclusive. It means only that the tensors v, Ov, @ v l @ v , , v1 Ov, @ v 2 Ov,, v, Ov, Ov, O v , , and v2 Ov, Ovl Ov, have the same coefficient. These observations suggest the introduction of an equivalence relation R on the elements of the extended basis of M. Two tensors vjl @ - - -@vj, and vk, @ . . . @ vk, of the extended basis are said to be in the relation R if and only if the sets {vj,. . . . , vjn} and {vkl,. . . , vk,} contain the same elements from

1

249

2. Modules over Symmetric Algebras

the basis {vI, . . . , v,} of V with the same multiplicity. An element m of M belongs to W if and only if its components are constant on each equivalence class of R. This point of view allows us to describe a basis and to determine the dimension of W in a natural way. First, one can select a representative element p from each equivalence class of R by choosing that element p for which the factors from {v,, . . . , v,} appear with nondecreasing subscripts. For example, the representative element from the class C consisting of v, o v , o v , O v , , v, o v 2 o v l ov,, v1 ov, o v , , v2 O v , o v , v2 ov, @ v l ov,, and v1 @ v , o v , O v , is the vector vI @ v , @ v , o v , . Second, one can assert that the dimension of W is equal to the number of classes of R which is seen to be equal to the number of unordered samples of size n taken with repetition from a population of rn elements. This number is well-known to be ( m + : - ' ) . An extension of this idea is introduced below to discuss the case of more complicated symmetry subspaces of M. We want to observe that W has a more intuitive set of generators. To do so, denote by B the set of all basic symmetric tensors of the form v 0 . . . v, v E V. The set B is a set of K-generators of the space W of symmetric tensors if every f ' E W* whirh vanishes on B also vanishes on W. Note that every linear functional f ' E W* is determined by a set { f ( J ) } of components such that if w = w(J)v(J), then

ov,,

ov,

2

f ' ( W ) = C f ( J ) w ( J )= E f ( J ) W ( S P J )

= Cf(sJ)w(J).

Consequently, one can assume without loss of generality that (2.32)

c

f(SJ)

=f(J),

S E

s,,.

Let v = Aivi denote a linear combination of the basis elements (vi} of V with the coefficients A i regarded as indeterminates. Then the basic symmetric tensor v 8.. . 8 v has the form 1Ajl . . . Ajn(vjl 0. . . 0 vj,). Let f ' be an element of W* which vanishes on B. Then (2.33)

0 =f'(v 0. . . 0v)

=Cf(ji,

. . . ,.j,,)Aj, . . . Aj,.

The terms of Eq. (2.33) can be collected into partial sums consisting of all terms with a common factor A l a l ... A,,"",CI, ... CI, = n. Two coefficients , f ( J l , , . . ,j,,) and f ( k , , . . . , k,) o f f ' multiply the same monomial A i a l . . . Anan if and only if k i = 1, 1 5 i _< n, exactly as many times as ji= 1, 1 i ~ n k i= 2 exactly as many a s j i = 2 and so forth and so on. These conditions are realized if and only if

+ +

f(ki,

. . ., k n )

= f ( j . s ( ~.).,. , j s ( n J >

that is, (2.34)

f ( W =f(sJ)=f(J>,

,

250

4. Representaion Theory of Special Groups

for some s E S , . Thus A l d l .. . An(l" appears with a coefficient C(a,, . . ., a,) equal to rf'(J) for some integer r and some index diagram J. Equation (2.33) can be rewritten (2.35)

0=

c C(Cc,,. . . ,

...

C(,)Alal

Anam,

where the 2's are indeterrninates. It follows that all of the C(CY,, . . . ,CY,)'sare zero and. consequently. all of theJ'(J)'s are zero. Thus every linear functional ,f' of W* which vanishes on B also vanishes on W. Therefore B is a set of K-generators of W. We summarize this conclusion in the form of a lemma. The set B of basic symmetric tensors of the form v 0 . .. 0v is (2.36) LEMMA. a set of K-generators of the subspace W of symmetric tensors. This result can be used to establish that the enveloping algebra b coincides with HomK,,l(M, M ) . Let Eji be the linear transformation on V whose action on the basis {,Y,. . . . , v,} is given by 1I i,,j, k I m.

(vk)Eji= a k j v i r

The set { E j , ) , I 5 i, j I m. is a K-basis of Hom,(V, V). Given two elements v I , 8.. . @ v,, and vil 8.. . 0v;, of the extended basis of M , one has (2.37)

(vk,

8. . . 0vk,)(Ej,i, 0. . . 0 Ej,, in) = (Sk,,l . . . 6," j"(vil 8.. . 0v;,,).

It follows from (2.37) in a quite straightforward manner that the set

{ E j li l 8. . . 0Ej, ,J, 1 I i, ,j , I m, is a K-basis of Hom,(M, M). We denote the tensor Ej, i , 0. ' . 0E j , by the symbol E(J, I ) and observe that

v(K)E(J. I ) = 6(K, J ) v ( I ) , where 6(K,J ) = I if ( K ) = ( J ) and otherwise is 0. We have asserted that every T E Hom,(M, M) can be written

T=

t ( J , I ) E ( J ,I ) ,

t ( J , I ) E K.

This leads to (2.38) and (2.39)

1v(s-IK)t(J, I ) E ( J , I ) =2 t(F'K, I)v(/)

[ s ( v ( K ) ) ] T =[ v ( s - I K ) ] T =

c t(K, I ) v ( I ) c t(K, I ) v ( s - l l ) c t(K, JI)V(l).

s [ v ( K ) T ]= s 1 v(K)t(J, I ) E ( J , I ) =

t(K, I ) s v ( I ) =

=s

=

25I

3. The Integral Representations of the General Linear Groups

Such an element T belongs to HomKsn(M,M) if and only if (2.40)

(sm)T = s(mT),

m

E

M , s E S,.

It follows from (2.38), (2.39), and (2.40) that T belongs to IAJmKsn(iM, I) if and only if t(s-'K, Z) = t(K, SZ)

or t ( J , I ) = t(sJ, SZ)

(2.41)

for all index diagrams I and J. Equation (2.4i) implies that T is an element of HomKSn(M,M) if and only if T i s a symmetric tensor of Hom,(M, M) regarded as the tensor product Hom,(V, v ) 8 . .. @ Hom,(V,

v),

n factors.

It follows from Lemma (2.36) that each Tbelonging to Hom,,"(M, written in the form (2.42)

T=

M) can be

c ( i ) [ t i@ . . . @ ti],

where ti E Horn#, V). The enveloping algebra is a linear subspace of HomKs,(M, M) and is therefore closed. However, every linear combination of the form (2.42) can be approximated arbitrarily near by an element (2.43)

T'

=

1 c'(i)[g,@ . . . @ g i ]

belonging to b. It follows that b is a closed dense subset of HomKSn(M,M) and therefore must coincide with it. We have the following theorem. (2.44) THEOREM. The enveloping algebra H o m K S p , M).

b coincides with the algebra

We now denote b by D to conform to the notation introduced in the earlier part of the chapter. 3. THE INTEGRAL REPRESENTATIONS OF THE GENERAL LINEAR GROUPS

This section is concerned with the application of the results of Section 2 to the determination of the integral representations of the full linear group GL(V) on a complex m-dimensional space V. An integral representation T of GL(V) is a matrix representation in which the entries of the representing matrix T(g), g E GL(V), are polynomials in the elements of the matrix of g with respect to any basis of V. All integral representations of GL(V) are

252

4. Representation Theory of’Special Groups

completely reducible. They decompose into irreducible components equivalent t o the canonical irreducible CL(V)-submodules N(F, V) obtained by reducing the tensor representations of GL(V) on the tensor product M = V @ . . . @ V. Such irreducible modules are generated by the primitive idempotents of KS,, acting on M. The primitive idempotent ( ] / t i ! ) s, s E S , generates the irreducible GL(V)-submodule W of symmetric tensors. Usually. it is sufficient to determine the primitive idempotents up to a nonzero scale factor, so that we deal mostly with essential idempotents to avoid unnecessary factors. However, one must insert the scale factor in special instances. The analysis parallels that for the symmetric subspace W, but requires an elaborate notational scheme yet t o be established. Given a primitive idempotent e E K S , , an element m belongs to the irreducible right D-submodule eM if and only if (3.1)

em

= m.

Since the analysis of Eq. (3.1) is complicated, we introduce the basic ideas for a special case. Let M denote the sixfold tensor product of an rn-dimensional vector space V with itself. The irreducible right D-submodules of M are determined by the primitive right ideals of s6. Given a frame associated with the partition 3 + 2 + 1 = 6, consider the Young tableau

123 T=45. 6 The group P(T) of row transformations of this tableau is generated by the set {(12), (13), (23), (45)) of transpositions and the group Q(T) of column transformations by the set {(14), (16), (46), (25)). It is unnecessary for our purposes t o write out the full expansion of the essential idempotent e(T) = &(y)qp,sometimes denoted merely by e. An element m E M has the general form

C

(34

C n?(jl,. . .,j 6 ) ( v j ,o . . . @ vj6).

This equation is frequently replaced by one in which the Young’s tableau defining e is more explicitly displayed. Thus one uses (3.2‘)

0 vj2 0

3. The Integral Representation of the General Linear Groups

253

The symbols (3.3) are referred to as index diagrams. Again we find it useful to denote such diagrams by a single symbol as (3). An element s E S, acts on h e subscripts of such a diagram as (3.3) to produce a new diagram (3.3') These diagrams clarify the relationships between the components of an element m of M and those of sm, s an element of s6. One has jl j2 J3

A tensor m is said to be symmetric in its rows, with respect to the tableau 123 45 6

5

if and only if m

=p m ,

that is,

for every p belonging to the group P(T) of row transformations of 123 45 6 I

A tensor m is antisymmetric in its columns if and only if &(q)m= qm, that is,

254

4. Representation Theory of Special Groups

for every q belonging to the group Q(T) of column transformations of the tableau 123 T=45

6

ithiclz is taken to heJixetl in the following discussion of our special case.

The usual treatment, according t o Boerner and others, of the subspace U = eM determined by Eq. (3.1) is through an intermediate subspace W = PM of the tensor product space M. We retain the symbol W for this space since W is a symmetric space in the following sense. Let w = Pm. Note that p w = pPm = P m = w, p E P ( T ) , so that w is symmetric (invariant) under any permutation p from the row group P(T). One sometimes says that W is symmetric under the r 0 ~1 -of s the given tableau. Every element u E U = QPM is the image Qw of an element w of W. The equation u = Qw can be used t o determine a set of equations for the components of the image u in terms of those of w, namely, one has

u

= = =

C ~01,. . . ,j6)(vj10 * . .@ v j 6 )= QW Q 1 Lc(j1, . .. v.j6)(vj, O

(C & ) i i . ( j y ( l )

9

Ovj,) .. jq(6)))(yj, O - * '

' '

. 7

. O vj6).

These lead t o the component equations: (3.6)

4 j l , . . . ,id = C & ) 4 j y ( l ) . . .

jq(6)h

where q runs through the elements of the column group Q(T). Since the components of the elements of W = P M are symmetric in their rows, one has (3.7)

.il

j, j3

for p E P ( T ) . In contrast, the components of the elements of U = QW are antisymmetric in their columns. If u = Qw, then gu = qQw = E(q)Qw = E ( ~ ) uso , that (3.8)

jl j 2 j 3 4 q h j4 j 5

li6

I lJq,6)

jy(l)

jq(2)

= u 1q(4)j q ( 5 )

jq(3)

I

for q E Q(T). The symmetry of the components of the elements of W in their rows and the antisymmetry of the components of elements of U in their columns play a key role in our considerations.

3. The Integral Representations of the General Linear Groups

255

According t o Eq. (3.6), the components of u belonging to U = QPM are linear forms in the components of tensors w lying in W = P M . Our problem is to determine a basic set of these forms. We sort out the components of an element w of W into equivalence classes. An index diagram J is equivalent t o an index diagram J’ if and only if J ’ = pJ for some p E P(T). This is easily seen to be an equivalence relation A partitioning the set of all index diagrams associated with the components of tensors of W into equivalence classes C. One notes that two index diagrams J and J‘ are equivalent if and only if each row of J contains the same integers with the same multiplicities as the corresponding row of J’. For example,

112 J=23 4

and

121 J1 = 32 4

are equivalent index diagrams which are not equivalent to

122 J2=23 . 4 The index diagram J2contains the integer 2 twice in the first row while the index diagrams J and J1 contain it only once. Observe that a tensor m of M is a n element of W if and only if the components of m are constant on the equivalence classes of the relation A. This follows since Eq. (3.7) shows that if w E W, then w(J) = w(pJ). Conversely, if m(J) = m(pJ) for all p E P ( T ) and index diagrams J, then p m = m, which implies that m = ( l / I P ) ) P mis an element of W, where ]PI denotes the order of P(T). One selects a representative diagram J R from each equivalence class C of A according t o the rule that the entries do not decrease in any row of J R . For example,

112 JR= 12 3 is the representative element of the class

121 121 211 112 112 211 21 , 12 , 21 , 12 , 21 , 12 3 3 3 3 3 3 The representative element from each equivalence class is called a rowordered index diagram. The components corresponding t o the row-ordered index diagrams can be taken as the independent components determining

256

4. Representation Theory of Special Groups

the subspace W = P M . The equations defining this subspace by means of its components assert that each component with index diagram belonging to the class C is equal t o the component with row-ordered index diagram from that class. The determination of the independent components of tensors belonging to the space U = QW is a good deal more complicated, although the final result is equally easy to describe. The independent components of tensors in U can be chosen to be those with index diagrams in which the entries d o not decrease from left to right along the rows and increase as one proceeds down each column. Such index diagrams are called standard index diagrams. The discussion proceeds in three steps. First, one shows that components with standard index diagrams in which all the indices are distinct are linearly independent. Then one shows that components with nonstandard diagrams containing these same indices depend linearly on those components with standard index diagrams. Finally, one considers the case in which the indices are not necessarily distinct. Assuming that m 2 6. it is no special restriction to consider first the case in which the set { j , . . . . ,j o ] consists of the integers 1 through 6 so that the standard index diagrams have the indices

123 123 124 124 125 125 126 126 45 46 35 36 34 36 34 35 6 5 6 5 6 4 5 4 and

134 134 135 135 136 136 145 146 25 26 24 26 24 25 26 25 , 6 5 6 4 5 4 3 3 each of which corresponds t o one of the standard tableaux for this frame. These are listed as an increasing sequence with respect to their usual ordering. Each component u ( 0 , ) corresponding t o a standard index diagram D,, 1 5 i 5 16, can be expressed as

(3.9)

u(Di) =

1& ( 4 ) ~ ( 4 D i ) .

Since the effect of any q E Q(T) is t o permute the integers of D i , those integers which appear in 4 0 , are exactly those belonging t o D,.One must keep in mind that 4 acts on the subscripts of the entries of D i , that is, (3.10)

jL

j 2

j 3

25 7

3. The Integral Representations of the General Linear Groups

Note, moreover, if q

= (146)

is preceded b y p

= (123),

then

so that q does not act as a column transformation on the index diagram at the second step. The elements from P(7‘) and Q(T) act as YO,, and column transformations on the basic tensors and not necessarily as row and column transformations on the index diagrams. Given a basic tensor of the form

the action of p E P(T) is always t o permute elements which belong t o fixed rows in the basic tensor and that of q E Q(7’) is to permute elements which belong to fixed columns. In the case at hand, where p = (123) and q = (146),

If

a

@

b

(4P) d 0 e

@

c

I If

c @ a @ b

=qd

@

e

However, when one considers the action of q p on an index diagram E, the action of q, as we have seen, may not be t o permute the indices which belong t o a fixed column ofpE. To return to an example of (3.9), consider u ( D 1 ) where

123

D,= 45 . 6

258

4. Representation Theory of Special Groups

Since the ii>(Ei)are symmetric in their rows, each component occurring on the right-hand side of (3.11) with an index diagram E i which is not rowordered can be replaced by the corresponding row-ordered component without altering the value of the component iv(Ej). The right-hand side of (3.11) appears with the row-ordered index diagrams

123 123 234 234 236 236 45 56 15 56 15 45 6 4 6 1 4 1 and

135 135 345 345 356 356 24 26 12 26 12 24 6 4 6 1 4 1

1 1 I 1( 1

after these substitutions. Thus one finds that 141

I23 ;t5

= it1

123 135 ;t5 - it.1 f

135

+

+ R1,

where R , denotes the sum of those components w(EJ with E , a row-ordered but not a standard index diagram. The next step is t o show that the components u ( D l ) , . . , , u(D,,) are linearly independent. The process of row-ordering the index diagrams according t o this scheme is repeated for the remaining standard components of u to obtain sixteen equations:

+ ... + U I , I ~ ~ ~ (+DR,I ~ )

u(D1) = a , , ~ ( D l )

+

u(D2) = u ~ I I v ( D I ) . . .

(3.12) u

~

D

l

~

~

~

a

l

~

,

+ l

+ Rz

a2,16 l ~ ( D 1 6 )

~

~

~

~

D

~

~ R~1 5 ~

~

'

~

a

l

~

z4(D16)=a1~,~rc'(D1)+' " + a 1 6 , 1 6 ~ 1 ' ( D ~ 6 ) + R ~ 6 ~

Consider the system of Equations (3.12) as defining sixteen linear forms,

u ( D , ) , . . . , u ( D I 6 ) ,in the linearly independent, row-ordered components of tensors of W. These sixteen forms are linearly independent if the matrix A = {trijj, I I i, j I 16, has a nonzero determinant. One demonstrates this by showing that A is an upper-triangular matrix with all 1's along the main

diagonal. (3.13) REMARK.Each element a i i . 1 < is 16, arises from the action of q = 1. e(q) = 1 , since, as we shall see, only the identity permutation followed by no alteration of the rows leaves any D iunchanged. Thus one has a,, = 1,

,

1

6

M

'

259

3. The Integral Representations of the General Linear Groups

1 5 i 5 16, as stated. A component w(E)occurs on the right-hand side of the equation defining u(Di) if and only if the index diagram E arises by rowordering q D i for some q E Q(T). For example, let q = (146) E Q(T) act on

il j4

J2

h

j3

.i,

to give

j4

h

j, j,

j3

jl

In particular, consider 125 D , = 36 ; 4

then

325 qD6 = 46 . 1

The row-ordered diagram E eventually obtained from D , is

235 46 1 which results from interchanging the indices j , = 3 and j z = 2 by means of the permutation p’ = (24). Note that p’ is not a row transformation on

123 45 . 6 However, one can write the result as p‘qD, = q(q-’p’q)D6 p = (164)(24)(146) = (12) is a row transformation of

= qpD,

, where

123 45 . 6 More generally, the row-ordered index diagrams which occur on the righthand side of each equation of u(Di) in the system (3.12) arise from D i by the application of a qp, p E P(T),q E Q(T), where P ( T ) and Q(T) denote the row and column groups on the Young’s tableau

123 45 . 6 No row-ordered diagram Ei appears on the right-hand side of the equation for u(Di) more than once. Otherwise, one has qpDi = @PDi for distinct pairs (q,p ) and (ij, p”). We have seen in Section 1 that this is impossible. Thus the a i j are all equal to 1, 0, or - 1.

260

4. Representation Theory of Special Groups

(3.14) REMARK.Suppose now that p ' q D i = D k , where D, is a standard index diagram different from D i. We assert that D i < Dk in the usual ordering of the diagrams. To see this, denote by r the first row (counting down from the top) of D ialtered by the permutation q. An element of r changed by q is replaced by a larger element from below. After row-ordering q D i t o obtain D,, the first element of the new row r' corresponding to r must be at least as large as the first element of r , the second element of r' must be at least as large as the second element of r , and so on. Since r' differs from r, one must eventually reach an element of r' which is larger than the corresponding element of Y. This proves that D i< L f , . These observations establish our claim that A is upper-triangular with 1's on the main diagonal. Thus det A # 0 and the forms u(D,), . . . , u( Dl,) are linearly independent. There remains the task of demonstrating that the linear form u(E), E a nonstandard diagram whose indices make up the set {I, . . . , 6}, is a linear combination of the forms u(D,), . . . , u(D,,). This argument depends on of KS,, which was introduced by H. Weyl. Make correspond an involution to each x = x(s)s, s E S, , of KS,, an element I defined by

-

1

(3.15)

2

-

=

c x(s)s-I.

G=

I t is not difficult to see that is an involution on KS , such that YI. In particular, E is a primitive idempotent if and only if e is a primitive idempotent. To be specific, the primitive idempotent e = c(q)qp corresponds to the primitive idempotent E = c(q)p-'q-' = &(q)pq. The second equality holds since P(T) and Q ( T ) are closed under inverses and ~ ( q=) e(q-'). The right ideal eA defining the subspace U = eM corresponds t o the left ideal AE under this involution. In particular, eA and A&have the same dimension over K. There is a close relation between AE and eM discovered by Weyl. For any u E U, let C(E)E KS,, be defined by

2

E(E) =

(3.16)

1 2

1 u(sE)s

for some fixed index diagram E. The element E(E) of KS, is called a ring tensor component. (3.17) THEOREM. The set of all ring tensor components arising from components of tensors u contained in the irreducible right D-submodule U = eM, e denoting a true idempotent @)/A, belongs to the minimal left ideal AE generated by the involute E of e. Proof. An element u of M belongs to U if and only if eu = u, that is, if and only if seu = su for every s E S,, . Hence for any diagram E (su)(E)= ( s 4 ( E ) =

c

& ( q ) ( s q p w ) l A=

c 4q)u(sqpE)/A.

3. The Integral Representations of the General Linear Groups

Also

u"(E)&= = =

c u(sE)s1&(q)pq/I. cc

=

26 I

1 2 &(q)u(sE)spq/i,

&(4)u(t4-'P-1E)tlA =

c c 4q)+?pE)r/A

2 (tll)(E)t= 1 u(tE)t = G(E).

Thus C(E) E (KS,)C for every u E U and every index diagram E. This result allows us to prove that components u(E) determined by nonstandard index diagrams depend linearly on the standard components. Let E be a nonstandard index diagram with indices the set { 1, . . . , 6). The ring tensor component G(E) = u(sE)s is an element of A&which is sixteendimensional. There exists a linear relation among the components u ( E ) , u(D,), . . . , "(D,,). Since fu(D,), . . . , u(D,,)} is a linearly independent set, this relation can be expressed in the form

c

(3.18)

u(E) = a,u(D,)

+ . . . + a,,u(D,,).

The coefficients in Eq. (3.18) can be evaluated by elimination between Eqs. (3.6) and (3.12) which we repeat for convenience: (3.6)

u(E)

=

c &)w(qE)

and

(3.12)

t i ( D i )=

aijw(D,)

+ Ri,

Note that Eqs. (3.12) can be solved for (3.19)

1

~ ~ (= o j )bj,u(D,)

it,(Dl),

+ Kj,

1I i I 16.

. . . , it(D,,) to obtain 1 ij I 16.

Here B = {b,J, I I j , s I 16, is an upper-triangular matrix and K, denotes the sum of the iv(F) with F row-ordered, but not a standard index diagram. It is sufficient t o consider Eq. (3.6) for the case of components with columnordered index diagrams E since the components of a tensor u in the space U are antisymmetric in their columns. Then one notes that when E is a columnordered index diagram, the process of row-ordering can be carried out without disturbing the fact that the final diagram is column-ordered. To see this, let E be a column-ordered index diagram with m rows, all of \c,hose indices a r e distinct. Clearly, one can row-order the mth row by interchanging whole columns to obtain a new index diagram Em which is column-ordered and whose rnth row is row-ordered such that w(E,,,) = w(E). Suppose one has reached an index diagram E,,, which is column-ordered with rows r + 1 through m row-ordered and IV(E,+~) = ~ i t ( E )If . there exists i < k such that row r contains b < a with a in column i and b in column k , then one has a

262

4. Representation Theory of Special Groups

situation similar t o the index diagram E,,, sketched as follows: i

k

XI

Yl

X,-I

Y,-I

a c

with the significant columns

h rowr d rowr'l

with b < (1 < c < ti. We permute E,,, t o a new index diagram E,' by means of a rowz permutation 17' interchanging the first I' elements of column i with the corresponding first Y elements of column k . This gives a diagram with the columns: i

k

which is still column-ordered since h < c and a < (f. After a finite number of such permutations, one arrives at an index diagram E, which is not only column-ordered. but also row-ordered in rows r through m and for which it.(E) = ii.(Er).By repeating this process, one arrives at a standard index diagram F with iifE) = i i . ( F ) . Since ir.(E) = it(qE). q = I . it follows that the equation u ( E ) = E(q)it.(qE) contains components corresponding to standard index diagrams on the right. Thus one can rewrite the above equations as u ( E )=

(3.20)

c j \i.(Dj) + K ,

where K denotes the sum of those components of 14' belonging t o row-ordered, but not standard index diagrams. Replacing each component it.( D j ) of (3.20) by its value from (3.19), one obtains (3.21)

u(E)= =

1 c j [ C hj,u(D,) + K j ] + K [Icjhjs]u(D,)+ 2 c j + K. Kj

I

When we compare Eq. (3.18) and (3.21), we discover that cj K j + K = 0. Otherwise. there exists a nontrivial linear combination of u ( D , ) , . . . , u(D,,) which is linear in the nonstandard, row-ordered components of w. This con-

3. The Integral Representations of the General Linear Groups

263

tradicts the fact that the coefficient matrix {ajj} of Eq. (3.12) is nonsingular. Thus one finds (3.22)

u(E) =

1 1 cjbjsu(Ds)>

where the {cj} are determined by Eq. (3.20) and the {bjs}by Eqs. (3.12) and (3.19). The actual determination of these coefficients is straightforward for small values of rn and n, but we defer specific examples for the moment. See Examples (3.43) and (3.44) below. We have completed two-thirds of the program for our special case, namely, that the forms u ( D j ) , 1 I iI 16, are linearly independent and that any form u(E) where E is a diagram whose indices are the set { I , 2, 3. 4, 5, 6) can be expressed linearly in terms of these. The forms u(E) where E contains distinct indices, not necessarily belonging to the set { 1,2, 3, 4, 5, 6}, can be treated in a similar manner. Such a u(E) is the coefficient of vj, vj,

0 vj2 0 vj3 0 vj3

'j,

where the component vectors {vji}, 1 5 i 6 , are all distinct. By renaming the basic vectors of V, such components u(E) coincide with those just discussed. There remains the question of components u(E) with repetition among the indices of the index diagrams E. To fix the idea, consider diagrams for which the indices are 1, I , 2. 2, 2, 3. These diagrams determine coefficients which are multipliers of basis tensors of the general form v1 v, v3

0 v1 0 v2 0 v2

including all basic tensors obtainable from it by permutation of its factors. There exists a linear transformation T on V such that Tv, = v l , Tv, = vl, Tv, = v 2 , Tv, = v 2 , Tv, = v 2 . and Tv6 = v 3 . The linear transformation T 0. . . 0T, six factors, converts any linear relation among the basic tensor v1 v4

0 v, 0 v3 0 v5

'6

and its permutations into a linear relation among the basic tensors v1 v2 v3

0 v1 0 0 v2

v2

and its permutations. It follows that Eqs. (3.12). (3.19), (3.20), and (3.22)

264

4. Representation Theory of Special Groups

remain valid for such index diagrams. Some equations among this set become redundant and some trivial, but they remain valid. The discussion which we have given of our special case extends with little change to the general situation. We must refer the reader t o more specialized treatises for the details. I n summary, the action of elements of S, on elements of M = V 0 . . . 0V, n factors, has been defined. Each canonical tableau T defines a n essential idempotent e = e(T) = QP. This essential idempotent P(T) determines an irreducible GL(V)-submodule of M according t o Eq. (3.1). The expansion of Eq. (3.2’) holds with the components m(J) of m designated by means of index diagrams ( J ) corresponding t o the frame F of T. The concepts of symmetric in tlir rows and antisymmetric in the colt.lmns remain valid with the obvious modifications. The space W of “symmetric” tensors is introduced, and Eq. (3.6) assumes the form (3.23)

44 =

1 4q)l4qJ),

for the components of a tensor u E U analog of Eq. (3.12), namely. (3.24)

=

4

QPM

+

u(Di) = ~ a j j w ( D j )R , ,

QV), =

QW, where u = Qw. An

1 I i
holds for the components designated by standard index diagrams{ D,}, 1 5 i I is the dimension of the minimal right ideal e(T)A. Again, one finds in general that these components constitute linearly independent forms with variables in the independent components of the symmetric tensors of W. Furthermore, they turn out t o constitute a basis for components with index diagrams whose indices are the set { 1, . . . , n}. Results analogous t o those of our special case hold for components u(E) when E has distinct indices not necessarily belonging t o the set { 1, . . . , n) and also for cases where E contains repeated indices. We formulate, without proofs. some of the principal results as theorems. Let M denote the GL(V)-module V 0.. . 0V, n-factors, where V has dimension 111. Let {m,, . . . . n7,] be a partition of n determining the frame F with k 2 m, and let T be the canonical tableau belonging t o F. Then e(T)M = N(F, V), c ( T ) the essential idempotent determined by T, is a n irreducible GL(V)-module. One defines a new sequence {ml’, . . . , n?,’} whenever k < m by taking mi’ = 1 1 1 , . 1 I i c; k. and mi‘ = 0, k < i I n7. Let the set {A,] be defined by

A corresponding t o standard tableaux. Heref

(3.25)

ibi= mi’+ m

-

i,

I I i 5 m.

(3.26) DEFINITION. The GL(V)-module N(F, V) is called the canonical GL(V)-177odule dcrerrnined hy F and the representation T(F, V) it affords the canonical representation.

265

3. The Integral Representutions of the General Linear Groups

THEOREM. The dimension d of the canonical GL(V)-module N(F, V) is equal t o the number of standard index diagrams which can be put into the frame F.

(3.27)

THEOREM. The dimension d of the canonical GL(V)-module N(F, V) is given by

(3.28)

d=

A@,, . . . ,%,) A(m-1, ..., 1,O)'

where Ai, 1 5 i I m, is given by Eq. (3.25) and A(zl, . . . , z,) is the difference product of the integers z1 through z, . The difference product A(z,, . . . , z,) is defined to be ( ~ 1-

z2)(z1 - ~ (22

(3.29)

... (

~ 1- Zm)

3 )

- z3) . . . (z2 - z,)

(zm - 1 - zm>. (3.30) EXAMPLE. Let V be a four-dimensional vector space and M be the three-fold tensor product V @ V 0 V. There are three partitions of 3, namely, {3}, (2, l}, and (1, 1, 1) to which correspond the frames Fl = '.. , F2 =

..

. , and F3 =

: . We consider

each of these frames in turn.

Let TI = 123 be the canonical tableau of Fl

=

R l . The stan-

dard index diagrams of F,, in ascending order, are 111

112 113

222 223

114 122 123

224 233

124

234 244 333

133

134 144

334 344 444,

which shows, by Theorem (3.27), that the dimension of N ( F , . V) is 20. Now, one has nz,' = 3, m2' = 0, m3' = 0, and m4' = 0, so that

+ 4 - 1 = 6, A2 = 0 + 4 - 2 = 2, % =0 ,+ 4 - 3 = 1, A4 = 0 + 4 - 4 = 0. EL, = 3

Consequently, d = A(6, 2, 1,0)/6(3,2, 1,O)

=

266

4. Representation Theory of Special Groups

Thus the results of Theorem (3.28) agree with those o f Theorem (3.27). (b)

Let T ,

=

l 2 be the canonical tableau of F2 = 3

. The stanU

dard index diagrams of F 2 , listed in ascending order, are

11 1 1 1 1 12 12 12 13 13 13 14 2 3 4 2 3 4 2 3 4 2 14 14 22 22 23 23 24 24 33 34 3 4 3 4 3 4 3 4 4 4

so that the dimension of N(F,, V) = 20. I n confirmation, ,Il = 5, ,Iz = 3, 2, = 1 , and A4 = 0, so that, by Theorem (3.28), d = A(5, 3, l,O)/A(3, 2, 1,0)= ( 5 - 3)(5 - 1)(5 - 0)(3 - 1)(3 - 0)(1 - 0) (3 - 2)(3 - 1)(3 - 0)(2 - 1)(2 - 0)(1 - 0)

(c)

Let T ,

1

=2

3

be the canonical tableau of F, =

= 20.

U. The

H

standard

U

index diagrams of F, are

1 1 1 2 2 2 3 3 3 4 4 4 In this case, one has 1, = 4, ,Iz = 3, 2,

= 2,

and

A4

= 0.

This gives

d = A(4, 3, 2, O)/A(3, 2, 1, 0) = 4. The canonical GL(V)-modules play a fundamental role in the integral representation theory of the general linear group. For fixed V, GL(V), and M, the isomorphism class of the GL(V)-module N ( F , V) depends only on F. Given a tableau T‘ different from the canonical tableau T of F, the GL(V)module r(T’)M will generally be distinct from the GL(V)-module c(T)M. However. there exists s E S, such that T‘ = sT, so that e(T’) = se(T)s-l. This implies that Ae(T’)s = Ae(T). which shows that Ae(7’) and Ac(T) are A-isomorphic right ideals. Consequently. e(T)M and e(T’)M are GL(V)isomorphic right GL(V)-modules. As a result, there are at most CI distinct classes of isomorphic, irreducible canonical GL(V)-modules where x is the number of partitions of n. the number of times that V occurs as a factor of $1. Tlirrtj are less than this nutnbrr when V has dimension m less than n.

3. The Integral Representations

of

26 7

the General Linear Groups

(3.3 1) THEOREM. Every integral representation of the general linear group GL(V) is completely reducible and decomposes into the direct sum of irreducible GL(V)-modules, each of which is isomorphic to one of the various canonical GL(V)-modules N ( F , V) determined by different choices of the frame F. (3.32) REMARK. There are other classes of representations of the general linear group GL(V) for which the theorem of complete reducibility fails. The reader may have noticed a crucial omission in our previous remarks. Suppose that V is a vector space of low dimension. For instance, let V have dimension one. Then M has dimension one for every n 2 1 and can serve as a representation space only of the identity representation or the alternating representation of S,. Note that the alternating representation of S, is a one-dimensional representation in which the even elements of S, map into the identity and the odd elements into its negative. Thus one suspects that for such low dimensional V, most of the idempotents e(T), T belonging to an integer n greater than 1, act as the zero transformation on M and produce only the trivial GL(V)-module (0). The precise theorem is as follows. (3.33) THEOREM. A canonical tableau T belonging to the frame F determines a nontrivial canonical GL(V)-submodule N(F, V) of M = V 0 . . . 0V, n factors, if and only if the number of rows of F does not exceed the dimension in of V. There is another interesting duality between the decompositions of M = V 0. . . 0V as a left KS,,-module and its decompositions as a right GL(V)module. By complete reducibility, one can write (3.34)

M = L, @ . . . OL,,

where each L, , 1 5 i 5 k , is an irreducible left S,-module, or as (3.35)

M = R, @ . . . OR,,

where each R i , 1 I i I s, is an irreducible right GL(V)-module. One can reorder the elements of (3.34) so that the first 2 members contain exactly one element from each of the distinct classes of isomorphic S,-modules represented in this decomposition. Let N i be the direct sum of the n , summands of (3.34) which are isomorphic t o the mi-dimensional So-module L , 1 2 i i i. Thus one finds that ~

(3.36)

M

=

N, @ . . . O N , ,

each N i , 1 I iI 2, the direct sum of n, irreducible summands, all mutually S,-isomorphic and of common dimension m i .In a similar fashion, one finds that M = P, @ . . . @P,, (3.37)

268

4. Representation Theory of Special Groups

each P i , 1 5 i 5 rn, the direct sum of r , irreducible summands, all mutually GL(V)-isomorphic and of common dimension t i The irreducible GL(V)submodules which are the direct summands of Pi are not isomorphic to the direct summands of Pj , i # j . (3.38) THEOREM. Each N i , 1 I i 5 %,of (3.36) is a right GL(V)-submodule of M. Furthermore, N, is the direct sum of m imutually isomorphic irreducible right GL(V)-submodules, each of which is of dimension n i ,1 5 i 5 A. The duality, of course, is in the interchange of number and dimension in passing from irreducible S,-submodules to irreducible GL(V)-submodules. There is a companion theorem, which follows. (3.39) THEOREM. Each P i , 1 < i < m, of (3.37) is a left S,-module of M. Furthermore, P i is the direct sum of t i mutually isomorphic left &-submodules, each of which is of dimension r i , 1 < i < m. (3.40) REMARK. Given M = V 0. . . @ V (n factors), the canonical tableau T of a frame F belonging t o n determines a nontrivial canonical GL(V)module N(F, V) if and only if the number of rows of F does not exceed the dimension rn of V. One can compute the number of standard tableaux which belong to the frame F thereby determining the dimension f of the minimal right ideal e(T)A. The number f is the number of times the canonical right module N ( F , V) occurs in M. Conversely, one can determine the dimension d of N ( F , V) by Theorems (3.27) or (3.28). The number d is the number of times the irreducible left S,-module AZ(T) occurs in M. The module N ( F , V) occurs in M if and only if e(T)M # {0}, where T is the canonical tableau corresponding to F. (3.41) EXAMPLE. The previous theorems and remarks can be applied to Example (3.30). The frames in question are Fl = ... , F2 = , and F3 = . The standard tableau corresponding to Fl is 123; the standard tableaux ’

corresponding to F2 are l 2 and 3 1

1’;

and the standard tableau corresponding

t o F3 is 2. Thus the corresponding minimal right ideals e(T,)A, e(T,)A, and 3 e(TJ.4 have dimensions one, two, and one, respectively. These dimensions reveal that N ( F l , V) occurs in M once, N ( F 2 , V) occurs in M twice, and N ( F , , V) occurs in M once. Thus (3.42)

M

= N(F1, V)

0 2N(FZ1 V) 0N ( F 3 , V),

3. The Integral Representations of the General Linear Groups

269

so that the various dimensions are seen t o balance. Conversely, one sees that AZ(Tl) occurs twenty times in M, AZ(T,) occurs twenty times in M, and AZ(T,) occurs four times. Again, the dimensions check. (3.43) EXAMPLE. Consider the three-fold tensor product M = V 0V 0V, where V is a two-dimensional vector space. Then M is an eight-dimensional vector space affording representations of both S , and GL(V). The partitions of 3 are {3}, (2, l), and (1, 1, l}, which determine the frames Fl = . . . , F2 = '

, and F3 = : . The canonical tableau Tl = 123 of Fl has the row group

P(Tl) = ((l), (12), (13), (23), (123), (132)) and the column group Q(Tl)= (( 1)). The essential idempotent of Tl is e(Tl) = (1) + (12) + (13) + (23) + (123) + (132).

The invariant subspace W = c(Tl)M is the space of symmetric tensors. This is typical. The frame corresponding to the trivial partition {n} of n > 1 always defines an idempotent determining the subspace W of symmetric tensors of M . The dimension of W, by an earlier observation, is ("+:-') which gives ($) = 4, in our particular example. For such a simple case, one can determine a basis of W by observation. It consists of

0v1 0 v1, w2 = v1 0v1 0v2 + v1 0v2 0v1 + v2 0v1 0vl, w3 = v1 0 v2 0 vz + v2 0v1 0 v2 + v2 0vz 0v1. w4 = v2 0 v2 0v 2 . It is easy to see that Wl = (wJ, W, = (w2), W, = (w3),and W, w1

= v1

= (w4)are one-dimensional subspaces of W, each of which affords the identity representation of S,. Since the column group Q(T)= {(l)}, it follows that U = QW = W in this particular instance. Our previous remarks on duality imply that U affords a four-dimensional, irreducible representation of GL(V). The subspace U can also be described by standard index diagrams which determine a full set of linearly independent components of any u E U. These standard index diagrams are J1 = (1, 1, I), J , = (1, 1,2), J , = ( I , 2, 2), and J4 = (2,2,2). A general element u of U is determined by the component equations:

u(1,2, 1) = u(l, 1,2), 4 2 , 1, 1) = 4 , 1,2), 4 2 , 1,2) = 4 1 , 2 , 2 ) , @,2, 1) = u(l,2,2), where the components u(1, I , I), u(1, 1,2), u(1, 2,2), and u(2,2,2) are arbitrary.

270

4. Representation Theory of Special Groups

- l 2 of F2 = : , has a row groupP(T,) 2-3 column group Q(T2)= {(l), (13)). Thus

The tableau T

'

=

= {(l),

(12)) and a

QP = [(I) - (13)1[(1) + (1211

which expands to ( I ) + (12) - (13) - (123). The frame : corresponds to the partition (2, 1 ) so that, by Eq. (3.25), one has rn, = 2, rn, = 1, 2, = 2 + 2 - 1 = 3, and i., = 1 + 2 - 2 = 1. Thus the dimension of the canonical module N(F2 , V) is given by d = A(3, l)/A( I , 0) = 2, according to Theorem '

(3.28). The standard index diagrams of F2 are J

- I and J2 = :2. The '-2 only nonzero components of elements u of QPM are u(:'), u(:'), u(:'), and u(:'). The component equations are

21 4, 1 = -u(:'>,

( u y ) = -u(:')

with 4;') and u(:') arbitrary. Since the dimension of N ( F 2 ,V) is two, it follows that the dual representation of S , occurs twice in M. The dual representation is also two-dimensional so that N(F, , V) is of multiplicity two as an irreducible right GL(V)-submodule of M. Thus

M

N(F i V) 02N(F2 > V),

1

3

so that F , determines only the zero GL(V)-submodule of M . This result checks with Theorem (3.33). In confirmation, one notes that the components of tensors transforming according to F, must be simultaneously skewsymmetric in all indices and have a repeated index-thus must be zero. The foregoing example does not employ our machinery in much depth, so that we introduce an additional example. (3.44) EXAMPLE. Consider the four-fold tensor product M of a four-dimensional vector space V with itself. This space M has dimension 256. The number 4 has the five partitions: (41, (3, l}, ( 2 , 2) (2, I , l}, and { I , 1, 1, I}, listed in descending order. These determine five frames:

Fl

=

.... ,

F2

- .

" '

F , --. . ,

"

7

F 4 = : ,

and

F,=:.

According t o Theorem (3.33), each of these determines a canonical, irreducible GL(V)-submodule N ( F , , V), 1 i 5 5. of M. Using Theorem (3.28), one finds that dim N(F,. V) = 35, dim N ( F 2 , V) = 45, dim N ( F 3 , V) = 20, dim N(F,, V) = 15, and dim N ( F , , V) = I . According to Theorem (2.31'), the dimensions of the irreducible representations of the symmetric group S , belonging to the frames F,, F 2 , F 3 , F 4 , and F, are the number of stan-

3. The Integral Representations of the General Linear Groups

271

dard tableaux, 1, 3, 2, 3, I , respectively, associated with them. This means that

by our duality theorems. We examine the canonical, irreducible GL(V)-submodule N ( F 2 . V) in more detail, The partition (3, 1) gives the sequence (3, 1,0,0) which determines I , = 6, I , = 3, I , = I , and I., = 0. Thus one has dim N ( F 2 , V) is d = A(6, 3, I , O)/A(3, 2, 1, 0) = 45. We verify this number by writing down the full list, in ascending order, of standard index diagrams for the frame F 2 . 111 111 I l l 2 3 4

112 112 112 113 113 113 2 3 4 2 3 4

114 114 114 122 122 122 123 123 123 2 3 4 2 3 4 2 3 4 (3.45)

124 124 124 133 133 133 134 134 134 2 3 4 2 3 4 2 3 4 144 144 144 222 222 223 223 224 224 2 3 4 3 4 3 4 3 4 233 233 234 234 244 244 333 334 344 3 4 3 4 3 4 4 4 4

The canonical tableau T2 = T = 123 of the frame F2 has for its column 4 group Q(T) = {(l), (14)) which makes the applications of Eqs. (3.6). (3.12), (3.18), (3.19), and (3.21) rather simple. There are five classes of index diagrams associated with F, , namely, (a) all indices the same. (b) three the same and one different, (c) two of one value and two of another, (d) two of one value 111 and two which are distinct, and (e) all distinct. Examples of these are 1 11' '12 123, respectively. Since elements u belonging to U = QW 2 2 3 4 have components skew-symmetric in the columns of the index diagrams, it follows that components u(jii) of type (a) are always zero. Components of type (b) are zero for the same reason, except in the case of u(;ji) or u(j"), where i # j . Thus the last three types are the ones of most interest. We recall that Eq. (3.6) relates components of u = Qw, w E P M , only when their index diagrams contain exactly the same indices with the same multiplicities

272

4. Representation Theory of Special Groups

although, of course, not in the same arrangement. This means, for instance, that the components of u with index diagrams from (3.46)

112 121 122 211 212 221 2 2 1 2 1 1

are linearly related, but they are not related to components of u whose index diagrams have indices belonging to a different set. Since there is only one standard index diagram in (3.46), each component of u with an index diagram from (3.46) is a multiple of u(i”). Let an element u of QW be of the form Qw. Then one has, by Eq. (3.6), u ( J ) = H ~ J-) ~ ( ( 1 4 ) J ) for every index diagram J . In particular,

A typical example of type (d) is given by the set of index diagrams

(3.47)

112 113 121 123 131 132 3 2 3 1 2 1 211 213 231 311 312 321 3 1 1 2 1 1

The standard index diagrams of (3.47) are I l 2 and I 13. The modified version 3 2 of (3.12) which arises is - w(i’3)>,

u(:”)

=

4 1 3 )

= w ( i I 3 ) - w(:’3),

w(:”)

from which one obtains the modified version of (3.19), namely, w(;’2)

= u(:”)

w(;~= ~ )u(;I3)

+ + w(iZ3). w(:’3),

273

3. The Integral Representations of the General Linear Groups

Expressions for some of the components with nonstandard index diagrams are U ( y ) = W(i”’>

- w ( y 1 ) = w(:”)

4;31) =,431)

- W ( ; 3 1 ) = ,+,(i23) - 4

U(?”) = w ( ? 1 2 ) - W ( y ) = $423)

- W ( i 2 3 ) = u(:’”, i 1 3 )

-W

( y )

=

-u(;*~),

= -u(:”).

The situation for components with index diagrams of type (e) is illustrated by means of the set of index diagrams

123 4 213 4 312 4 412 3

(3.48)

124 3 214 3 314 2 413 2

132 4 231 4 321 4 421 3

134 2 234 1 324 1 423 1

142 3 241 3 341 2 431 2

143 2 243 1 342 1 432 1

listed in ascending order. There are three standard index diagrams, namely, 123 124 , and i34, in the set. Equation (3.12) assumes the form 4 ’3

- W ( t 2 3 ) = W ( i 2 3 ) - W(?34), u(;’4) = ,4:24> - 3 2 4 ) = W ( y 4 ) - w ( f 3 4 ) , 4;”)= W ( i 3 4 ) - 4 3 3 4 ) = M’(i34) - W ( 3 3 4 ) . U(i23) = W(i23)

(3.49)

While for (3.19), one has W(i23)

(3.50)

W(i24)

w(:”)

+ w(;34), = u<:24> + w ( f 3 4 ) , = 4;”) + 1 4 : ~ ~ ) .

=4

2 3 )

Using Eqs. (3.49) and (3.50) in (3.6), one obtains expressions for some of the components corresponding t o nonstandard index diagrams of the form U(:14)

= wl(:14) - W(;14) = U(:23)

u(2”) = W =

( y )

= W(:24) - M 9 ( ; 3 4 )

+ w ( : ~ -~ )4;”) - 4:”)= u(:23) - w(3 412) - W ( i 2 3 )

4:”)+ W ( f 3 4 ) - u(:’4)

- W(i24)

- U(;34),

- w ( f 3 4 ) = U ( i 2 3 ) - u(i’4).

The reader will observe that even in very simple cases the dimension of the problem gets rapidly out of hand. Nevertheless, the basic procedures are

2 74

4. Representation Theory of Special Groups

quite straightforward. We turn now to the consideration of the matrix representations of GL(V)afforded by one of the standard canonical GL(V) modules. We recall the details of the linear transformation induced on the n-fold tensor product M = V 0 . . . @ V by an element g of GK(V). Let {vl, . . . , vm} be a basis of V and let g E GL(V) be defined by

vig

=

cvja;.

Then the action of g on the extended basis of M is given by (Vil @

. . . 0vi")g

=

1Vj1Uj,il 0. .

=

c

(Vjl

'

0

c vj,

Uj,i"

0. . . 0Vj,)UjlL' . . .

Uj,l".

Let w = (vii0 . . . @ vi,)w(il,. . . , in) be any element of M. Let u = (vjl 0 . . vj,)u(jl, . . . ,j,,) be the image of w under g. Then one has

1

wg = = =

C (vil@ . . . 0vi,)gw(il,

..., in) C ( 1 (vjl 8 . . . 0vj,)ujlil . . . ujnin)w(il,. . . , in) [ 1 \t.(i,, . . . , in)ajlil. . .ujni-](vjl 8 . . .0 vj,,),

which implies that (3.51)

u( j , , . . . ,j,,) =

1 w(i,, . . . , in)ujlil. . . ajnin.

Equation (3.51) can also be written

u(J) =

(3.52)

1 w(1)ug;

in a more compact notation. Both of the equations (3.51) and (3.52) describe the action of g on M in terms of the components of the vector w and those of its transform u = wg. Our problem is to describe the action of g restricted t o a canonical right GL(V)-submodule N ( F , V) of M. This is effected by considering on the lefthand side of Eq. (3.52) only those components u ( J ) for which ( J ) is a standard index diagram of F and by replacing those components il(Z), ( I ) not a standard index diagram, on the right-hand side by their values in terms of components with standard index diagrams. We work out some of the details, starting with Example (3.43) to illustrate the basic idea. (3.53) EXAMPLE. The space M of Example (3.43) is the three-fold tensor product of a space V with basis {vl, v2}. Let g E GL(V) be defined by

v l g = v,u:

+ v2 a:,

v2g

= vlu:

+ v:

u: .

Then one has the action of g on v i 0vj 0vk given by (Vi

0vj 0vJg

=

1(v, 0 v, 0vJaf a i 4,

275

3. The Integral Representations of the General Linear Groups

which leads t o the component equations : u(rst) =

(3.54)

1 w(ijk)afa: a:.

The specific form of (3.54) depends upon the frame F and the corresponding canonical GL(V)-submodule N(F, V). For the case of F1 = . .. , there are four standard index diagrams: 1 1 1 , 112, 122, and 222. The canonical module N ( F l , V) coincides with the space W of symmetric tensors so that the component relations are especially simple. They are

4121) = 4211) = u(l12),

4212) = 4221) = u(122),

with u(l1 I), u(l12), u(122), and 4222) arbitrary. We tabulate these results in a form suitable for later comparisons. See Table (3.55). The columns of Table (3.55) give the expansions of the com(3.55) 111 I12 122 222

111 1

112 121 211

1

1

1

122 212 221

1

1

1

222

1

ponent with index diagram labeling the columns in terms of the components with standard index diagrams labeling the rows. The table is read

u(ll1) = u ( l l l ) , 4121) = u(112), 4221) = u(122),

w(ll1) = w(lll), lC(121) = w(l12), 14221) = w(122),

etc. Making these substitutions into Eq. (3.54), one obtains

u(l11) = "(1 1 I)a:a;a: + w(l12)a:a:a: w(l2l)a:a:a: + w(211)a:a~a~ w(I22)a:a:a: w(212)aIaia: + w(221)a:a:a: + w(222)a:a:a: = w(l1 l)(a:a:a:) w(ll2)(a:a:a: a:a:a: a:a:a:) + w(l22)(a:a:a: + u ~ u : u :+ a:a:a:) w(222)(a:a:a:).

+

+ +

+

+

+

+

276

4. Representation Theory of Special Groups

We now assert that if u(rst) is any component of u corresponding to a standard index diagram rst, then u(rst) = w(I I l)a,!a,la:

+ U;afa:) + w(122)(ara,a, + urasat+ ara,at) + w(222)(ara, a, ). + w(l lz)(afa,la; + a:.: 1 2 2

2 1 2

2 2 1

2 2 2

The reason for this “duplication” is that the index diagram rst on the lefthand side of (3.54) appears as a constant on the right-hand side of (3.54), while the substitutions made for the w(ijk) are obtained from (3.55) and are entirely independent of the index diagram rst. Naturally, the great simplicity of the result depends partly on the fact that we have worked out the case of the symmetric tensors. We turn now to the canonical module N(Fz, V). For the case of Fz = : ’ , there are only two standard index diagrams, namely, l 1 and 12. The analog of Table (3.55) is shown in Table (3.56), where 2 2

(3.56)

11 1

2

11

11 1

12 21 1 1

12 21 22 2 2 1

22 2

-1

12 2

1

- 1

one notes that w(:’) = w(:’) = W($’)= w(;’) = 0 by antisymmetry of the components of w in the columns. One finds that u ( i ’ ) = w(;’)a:a:a$ + w(~1)afa:a: w(:2)alula2 1 2 2 w(122)a,a,a, 2 2 1

+

1 1 2

2 1 1

= w(:’)(a1alaz - a,a,az)

+ w(:2)(a:a:a:

+

2 2 1

- a,a,a,).

In general, one has for u(Y) where rs is a standard index diagram, t

1 1 2

2 1 1

u(:S>= w(:’)(arasat - arasat)

1 2 2 + w(h2)(a,asat

2 2 1

arasat)

for the same reason as before. With these examples in mind, let us indicate a procedure suitable for finding the matrices of the representation afforded by a canonical GL(V)module N(F, V) in the general case. Let M be the n-fold tensor product of an m-dimensional vector space V with itself. Let F be a frame of not more than m rows belonging to n. The canonical irreducible CL(V)-submodule N(F, V)

3. The Integral Representations of the General Linear Groups

277

of M belonging t o F has dimension k equal to the number of standard index diagrams which can be built on F. This number can be calculated by means of Theorem (3.28). The matrix representing an element T of GL(V) can be determined in the following manner. (a) Write down, in ascending order, the complete list D,, . . . , D, of standard index diagrams which can be built on F. (b) Then write down the complete list L,, in ascending order, of all index diagrams, standard or otherwise including D,, which have the same set of indices as D,. Omit from the list all index diagrams which correspond to zero components by antisymmetry. (c) Select the first standard index diagram D,' of F not appearing in the list L,. Write down a second list L,, in ascending order, consisting of all index diagrams including D,' which have the same set of indices as D,'. Again omit any index diagrams which correspond to zero components by anti symmetry. (d) Proceed by induction. If the list Li is completed, select the first standard index diagram Di+,' from those still remaining, if such exists. Write down the list L,+, in ascending order of all index diagrams including Di+,' which have the same set of indices as Di+l'. Omit all index diagrams which correspond to zero components by antisymmetry. (e) Eventually one exhausts the list of standard index diagrams on F. At this time, one has a complete ordered list L, u ... u L , of all nonzero components of tensors which belong to N(F, V). (f) Each list Li , 1 i i i t , contains one or more standard index diagrams, D , < D , < ... < D, , where for simplicity we have omitted a second index on the symbols indicating that they belong to Li . These standard index diagrams are used to label the rows of a table of the form of Table (3.57). The (3.57)

columns of Table (3.57) are labeled with the complete set, El < E, < ... < E, , of all index diagrams belonging t o the list Li . Observe that the standard

index diagrams appear in the columns as well as the rows, but that nonstandard index diagrams appear only in the columns. The entries in the column headed by E j , 1 <j 5 s, are the coefficients from Eq. (3.22) which are obtained by those applications of Eqs. (3.6), (3.12), and (3.19) pertinent to the frame F. Such tables can be constructed easily from Tables (3.55) and (3.56).

278

4. Representation Theory of Special Croups

The standard index diagrams are D, = (1 1 l), D, = ( I 12), D, = (122), and D, = (222). The first list I., consists of D, alone. The second list L, consists of (112), (121), and (211). The other two lists L , and L, we leave t o the reader. Note that Table (3.57) assumes the form of Table (3.57") for list L , . (3.57*) 112

112 1

121 211 1 1

(g) The matrix of any linear transformation T on N(F, V) induced by g 0 . . . @ g,g E GL(V), has the general form {T(D j , Di}), where D j and D i range independently through the standard index diagrams of F. The coordinate transformation induced by T has the form involves only index diagrams Each matrix element T ( D j . 0;) to the list of D j . To be specific, one has

Ek belonging

(3.58) where EL occurs in the same list as D j , and the a j n , 1 5 /z 5 s, are found in the j t h row of Table (3.57). The general form of (3.58) is independent of the index diagram D , and depends entirely on the ,jth row of (3.57). The symbol A:?; is formed from the elements of the matrix {t,"}, 1 5 u, L' m, of T with respect to the basis {vl, . . . , v,j of V used t o determine the extended basis of M. It is defined by where { P ~ .. . . , e,,)and ( j , , . . . ,j,J are the index sets of (Ek)and (Di), respectively, obtained by reading from left t o right and top t o bottom in the diagrams. 4. GENERAL REMARKS ABOUT THE REPRESENTATION THEORY OF

CERTAIN MATRIX GROUPS

In this section we discuss various details about what we call the stundurd

matrix groups, a nonstandard terminology introduced in the interest of

later brevity. The standard matrix groups are:

(i) GL,(K) the coniplex generul linear group of all nonsingular, m x m matrices with entries from the field of complex numbers.

4. The Represenfation Theory of Certain Matrix Groups

279

(ii) SL,(K) the complex special linear group which is the subgroup of GL,(K) consisting of all elements with determinant 1. Also called the complex unimodular group. (iii) GL,(R) the real general linear group consisting of all nonsingular, m x m matrices with entries from the field of real numbers. (iv) SL,(R) the real special linear group which is the subgroup of GL,(R) consisting of all elements with determinant 1. Also called the real unimodular group. (v) U(m) the unitary group consisting of all complex unitary m x m matrices. (vi) SU(m) the subgroup of U(m) consisting of all elements of determinant 1. Sometimes called the unitary unimodufar group. (vii) SO(m) the group of all real orthogonal m x m matrices with determinant 1. This is the group of rotations of a n m-dimensional, real Euclidean space X. The subgroups of the complex general linear group introduced above will be referred to as the standard subgroups. Our initial discussion is concerned with the subduced representations of the standard subgroups obtained from the irreducible integral representations of GL,(K). Then we introduce the families of rational, semirational, and semirational integral representations of the standard groups. These families of representations not only play an important role in the general theory, but also exhaust the list of continuous irreducible representation for certain of the standard subgroups. We follow these specific matters with a discussion of a wide variety of facts about matrix groups, Lie groups, and Lie algebras of considerable interest in the applications of representation theory outside of pure mathematics. These facts are discussed more thoroughly and accurately in a number of books from the standpoint of an experienced pure mathematician. Some standard references are : Chevalley (1946), Hausner and Schwartz (1968), Helgason (1962), Hochschild (1965), Jacobson (1962), and Wolf (1967). These books make hard reading for anyone outside of pure mathematics and none-too-easy for many people inside, so that we offer a rigorless presentation of some of the ideas. Since it is almost true that the most difficult thing about a Lie group is its definition, we do not give one. The required discussion of basic topology and geometry tend t o overwhelm many readers long before anything of even remote interest appears on the horizon. We offer a bare skeleton of topological definitions and facts and an equally sparse set from differential geometry. The reader is urged to adopt the attitude that Lie groups, especially matrix Lie groups, are groups which geometrically are very much like surfaces and curves in three-dimensional, real Euclidean space. Naturally, such a treatment will not please everyone. Furthermore, the lack of a precise foundation

280

4. Representation Theory of Special Groups

prevents the inclusion of proofs. Nevertheless, we are hopeful that some will find our discussion a useful entry to a rather difficult area. If they develop a desire for a rigorous presentation there is no shortage of sources. We use the Lie algebra of the three-dimensional rotation group SO(3) as an initial example of a Lie algebra g. The concepts of abelian, nilpotent, solvable, simple, and semisimple Lie algebras are explained. We discuss the Cartan subalgebras of a semisimple Lie algebra and give a very rough sketch of the structure theory. The notions of weights and highest weight of a representation or module of a Lie algebra g are introduced. Examples from the representation theory of S 0 ( 3 ) , S U ( 3 ) , and SU(m) are given to fix the ideas. A connection is established between certain representations of SU(rn) determined by Young’s tableau and the so-called fundamental representations of SU(m). The irreducible integral representations of GL(V) discussed in Section 3 provide integral representations of the group GL,(K) isomorphic to GL(V). Henceforth. we refer to these as integral representations of GL,(K). The irreducible integral representations of CL,,(K) subduce integral representations of the standard subgroups. It is generally true that an irreducible representation T of a group G does not remain irreducible when restricted to a proper subgroup H of G. Consider, for instance, the case of the subgroup H = {l}for any irreducible representation of degree greater than 1. In view of this, the following theorem is noteworthy. (4.1) THEOREM. The irreducible integral matrix representations of GL,(K) remain irreducible when restricted t o the complex special linear group SL,(K), the real general linear group GL,(R). the real special linear group SL,,(R), the unitary group U(m),and the special unitary or unitary unimodular group SU(m). The matrix representations afforded by different canonical modules N(F, V) sometimes give the same representation when restricted t o SL,(K), SL,(R), and SU(m). One obtains all distinct integral representations of these last three by considering frames with at most m - 1 rows.

(4.2) REMARK. One should note that the m-dimensional rotation group SO(m), sometimes called the real special unitary group, is not contained in the above list. This absence suggests the introduction of different methods for discussing the representation theory of SO(m). There is a broader class of irreducible representations of the general linear group which we mention quite briefly. The natural generalization of the integral representations of the group GL,(K) are the rational representations T i n which the elements of the matrix T(g) corresponding t o g E GL,(K) are rational functions in the elements of g. A still larger class of representa-

281

4. The Representation Theory of Certain Matrix Groups

tions are those in which the elements of T(g) are either rational or integral functions of the real and complex parts of the elements of g. To make this last idea quite definite, let o l l is,, oI2 it,, g = 021 it,, oZ2 it,,

It

+ +

+ +

l

denote an element of GL,( K ) . The semirational representations of GL,(K) are those in which the elements of T(g), g E GL,(K), assume the form A o i j tij)/q(oij> tij), where p and q denote polynomials in the eight variables {ol . . . , t,,}. The semirational integral representations are those in which the elements of T ( g ) are of the form p ( a i j ,s i j ) , where p is a polynomial in eight variables. The definition in the cases of GL,(K) and the standard subgroups should be clear to the reader. These kinds of representations of GL,(K) generally turn out t o be completely reducible and to have irreducible components which can be expressed in terms of the irreducible integral representations of Section 3 and the determinant function. We have the following two theorems about finite-dimensional representations. 9

(4.3) THEOREM. Every semirational representation of SL,n(K) is a semirational integral representation. All such representations are completely reducible. (4.4) THEOREM. Every continuous representation of the group SL,(R) is integral and completely reducible. We are particularly interested in the group SU(m) which plays an important role in the applications of representation theory to physics. The continuous representations of SU(m) are given by the following theorem.

(4.5) THEOREM. All continuous representations of the group SU(m) are integral. The continuous irreducible representations are obtained from the canonical GL(V)-submodules N(F, V) for frames with not more than m - 1 rows. One development of the representation theory of SU(m) and SO(m) sometimes proceeds along lines quite different from those discussed above. In particular, their representation theory as well as that of the other standard groups can be investigated with the methods of topology and differential geometry. A detailed study of such methods is beyond the scope of the present book. Nevertheless, we will make a brief survey of the application of topological and geometric ideas in representation theory. Unfortunately, our discussion must begin with a rather long list of unmotivated, but necessary definitions. [The reader who finds the list too tiresome can skip and return later if so inclined.]

282

4. Representation Theory of Special Groups

Let S be a nonempty set. A topology z on S is a dis(4.6) DEFINITION. tinguished family of subsets of S such that

uuEx

(i) the set S and the empty set @ are members of z; (ii) if each of {U,,}, G E E , belongs to z, then U, belongs t o T ; (iii) if U and V belong to 5 , then U n V belongs t o z. The subsets or elements of the topology z are called open sets of S. A family 8 of subsets of S is said to be a hasis of the topology z if every nonempty set L' of z is the union of sets from 8. The familiar rn-dimensional real Euclidean spaces RE(/??)are given a topology z by taking as a basis 8 of z the collection of all c-spheres S,(p). F: > 0, p E RE(rn), where S,(p)

= {x E

RE(rn): (1 x - pII < E } .

Here I(x - pII denotes the Euclidean distance from x t o p in any real Euclidean space RE(iii). A szthhusis of a topology z on a set S is a collection 3 of subsets of S such that every U E z is the union of sets, each of which is the intersection of a finite number of sets of I?. We give only examples of bases of a topoIogy since bases are more important for us than subbases. The set 8 of open circles C = {x E : IIx - xo 11 < E ] , xo E E > 0, together with the empty set @ is a basis for the standard topology of the two-dimensional real Euclidean plane 5Q. A subset 0 of the plane '1' is open in this topology if and only if each point x of 0 is contained in some open circle C which itself is contained in 0. The set 8 of open spheres S = {x E X : I/x - xo I/ < E } together with @ is a basis for the usual topology of three-dimensional real Euclidean space X. A set 0 of X is open if and only if either it is @ or the union of open spheres. The nonempty open sets of the standard topology of any /n-dimensional real Euclidean space RE(n7) are those containing an open sphere about each of their points. Most of our topological considerations deal with real Euclidean spaces. A topology is usually introduced on a nonempty set S in order to define notions of nearness and continuity. The basic concept is that of a continuous m a p f ' : S W of one topological space S into another 12.:

v,

--f

(4.7) DEFINITION. The map f : S + W is continuous if and only if f - ' ( O ) is open in S for every open set 0 of W. A bijection , f : S ---t W is a h'omeonzorpiiistii iff f and f are continuous.

-'

(4.8) REMARK.Definition (4.7) is completely equivalent t o the standard one of elementary calculus. This states that the map f :RE(n) -+ RE(rn) is continirous ut the point x E R E ( n ) if and only if given E > 0, there exists 6 > 0 such that 11 f ( y ) -f(x)II < c whenever / / y - x/I < (5. The functionf: RE(n) + RE(r77) is said to be continuous if and only if it is continuous at every point

4. Zhe Represenration Theory of Certain Matrix Groups

283

x E RE(n). Now let f(x), x E RE(n), belong t o any open set 0 of RE(m). Then some open sphere

is also contained in the open set 0. By continuity off at x, there exists 6 > 0 such that if jly - x/I < 6, then Ilf(y) -f(x)II < E. This shows that the sphere S,(x) = { y e RE(n) : I1y - xI1 < S } belongs tof-'(O). Consequently, f -'(O)is open since it contains a spherical neighborhood of each of its points. Thus the calculus definition implies (4.8). The converse is left to the reader. A nonempty subset V of a topological space S inherits a topology from the including space. This topology is equivalent to the one with the " smallest number of open sets" with respect to which the inclusion map i : V - + S is continuous where i(x) = x, x E V . (4.9) DEFINITION. The open sets of the relutioe topology of Y (as a subspace of the topological space S ) consist of all subsets of V , each of which is the intersection with V of an open set of S. (4.10) REMARK. The subset V with this topology is called a subspuce of S. There are various reasons why this rather strange-looking definition is adopted. Consider some familiar examples of topological spaces. Let '$ be any two-dimensional plane contained in the three-dimensional real space X. The basic open sets of X are open spheres S, each of which has either an open circle or the empty set for its intersection with 'Q. The induced or relative topology of '$ as a subspace of X agrees with its usual standard topology. Let V be the boundary

v = {x E x : llxll = 1)

of the closed unit sphere in X. The intersection of the open spheres in X with V are again the right objects to define the usual topology of V. Let V be the restriction of a continuous function f:S -+ W to some subspace V of S. If 0 any open set in W, one has ( f l V)-'( O) = f - ' ( O ) n V , an open set in V. Thus the restriction f l V of the continuous function f on S is a continuous function on V with its relative topology.

fl

Several other notions from topology play an important role in our subject. We state additional definitions, some of which are somewhat special (not the general definition), but suitable for our purposes. (4.1 1) DEFINITION. A neighborhood of a point p in the topological space S is any open set of S containing p.

(4.12) DEFINITION. An urc (closed arc) in a topological space S is the continuous image of an open (closed) interval of real numbers.

284

4. Representation Theory of Special Groups

(4.13) DEFINITION. A siniple closed curz’e in a topological space S is any homeomorphic image of the boundary of the unit disk in the two-dimensional Euclidean plane.

A topological space S is connected if any two distinct (4.14) DEFINITION. points a and b of S can be joined by a closed arc in S. (4.15) DEFINITION. A topological space S is locally connected if given any neighborhood N of a point p of S there exists a connected neighborhood C of p which is contained in N . (4.16) DEFINITION. Let S be a connected, locally connected topological space. Suppose that any simple closed curve C in S can be continuously deformed t o a point within S. Then S is called a simply connected space.

(4.17) EXAMPLE. The unit cube in RE(m) is a simply connected space for rn > 0. All the real Euclidean spaces are simply connected. The boundary of the unit circle in the plane is a connected, locally connected space which is not simply connected. We need several other topological notions which will not be precisely defined. A topological space S is compact means among other things that every continuous functionf: S -+ R from S into the real numbers R assumes its largest and smallest value at some points of S. A topological space S is locally compact means that every neighborhood N of a point p E S contains a compact set C which in turn contains a neighborhood W ofp. Letf: S + W be a continuous map from the topological space S onto the topological space W . There exists an equivalence relation R on S defined by xRy if and only if f(x) =f(y). This relation partitions S into equivalence classes where [XI denotes the class containing x. There exists a bijectionf’ from the set S’ of equivalence classes onto W definedf”~] =f(x). The set S’ can be given a quotient topology such that ,f’ is a homeomorphism. Let f:J + RE(m) be an arc in RE(nz) with domain the open interval J of real numbers. (4.18) DEFINITION. The arcf is dixerentiahle at thr point x E J if and only if limy+x[f(y) -f(x)]/[y - x] exists. The arc f is diflerentiable on J if it is differentiable at each point of J. Let {vl, . . . , v,} be a basis for RE@). Then for x E J,

+ . . . +f,(x)v,n where each element of the set (f,}, 1 i i 5 nz, is a map from f(x)

=f1(X)V1

?

J to the real numbers. It is a simple matter t o prove that the arc f is differentiable if and only if each of the components fi, 1 5 i 5 rn, is differentiable.

285

4. The Representation Theory of Certain Matrix Groups

These very general ideas become applicable t o the standard matrix groups by an embedding of them as subsets of real Euclidean space RE(n7) for a suitable value of m. The complex general linear group GL,(K) is a subset of the algebra K, of all complex matrices of the form { a j j } . 1 I i, j 5 m, where each a i j has the expression a j j = a i j + i.rij with a i j and s j j real numbers. There exists a bijection f :K,,, -+ RE(2m2)defined by

f [(aiJl

= (01

1 7

71 1,

.. ., ~ m r n rrnm), 7

In the case of K 2 , the map assumes the form

We identify K , with its image in RE(2m2),noting in particular, that f embeds GL,(K) as a subset of RE(2m2).Indeed, GL,(K) is all of RE(2m’) except for a hypersurface consisting of those elements of K , of zero determinant. We make a few remarks about hypersurfaces below. I n particular, GL,(K) is an open subset of RE(2m2) with every g E GL,(K) contained in an open 2m2-dimensional sphere consisting entirely of points of GL,,( K ) . Roughly speaking, GL,(K) is similar to the subset of RE(2m’) which remains after removing a plane, that is, a hyperplane. However, there is an important difference in that GL,(K) is a connected subset of RE(2m2).Locally GL,(K) is like RE(2m2)and most of the usual concepts of real Euclidean space are fully meaningful. We assume of the reader a good intuitive grasp of the concepts of arcs and tangent vectors, surfaces and tangent planes, and the like in three-dimensional space and a willingness to accept the extensions of these ideas t o higher dimensional spaces without benefit of full discussion and proofs. Fortunately, our present situation is different from that incurred in many places in modern analysis where frequently one is mainly concerned M ith how badly one’s intuition goes astray. The area in which we work is one in which the development has been along lines agreeing with intuitive notions. We consider geometric objects which are like surfaces and which we call h-surfaces, meaning higher-dimensional surfaces. Generally, such h-surfaces arise as the solution sets of one or more algebraic equations in 2m2 unknowns. To clarify the idea, consider the ordinary sphere in three-dimensions which is the solution set of the algebraic equation (4.19)

x2+y2+z2=1.

Other classical surfaces such as ellipsoids and hyperboloids are the solution sets of similar quadratic equations. Given a second equation such as (4.20)

(x

- 1)2

+ yz + z2 = 1,

286

4. Representation Theory of Special Groups

the reader notes that the simultaneous solution set of (4.19) and (4.20) is a circle. again a familiar geometric object We are interested, generally speaking. i n the solution set of a family of h- equations in 2rn2 unknowns of the form (4.21)

/ I ~ ( . Y ~. .. .

.

.xZrn2)

= 0.

1<j I k <2t~?~,

where the function I < , j 5 k . is algebraic. The solution set is generally ( 2 m 2 - /<)-dimensional when the set of functions {pi: is independent, a concept we d o not try t o make precise. The concepts of arc and of a tangent vector t o an arc pass from three t o 2/n2 dimensions without loss of intuition or meaning. A simple closed curve in RE(2/n2) is the homeomorphic image of a circle in the two-dimensional plane. The arcs which arise in our discussions have tangent vectors which turn continuously as one progresses along the arc. The real general linear group GL,,,(R ) is an h-surface which is the subset of the complex linear group GL,,(K) obtained as the solution set of ni2 real equations. namely. the set of equations asserting that the imaginary parts of all elements of a matrix g E GL,,( K ) are zero. Since GL,,,(K ) has real dimension 2m2. it follows that the h-surface GL,,(R) has dimension m 2 , a well-known fact. The unitary subgroup U(n7) of GL,,,(K)i s determined by a set of equations which assert that { u i , ! . 1 < i.,j 5 177. is an element of U(n7) if and only if (4.22)

uiiiiki = 0,

I Ij < I< I m.

N ~ ~= L I~ . , ~

I I j I 111.

and (4.23)

There are 1 4 1 7 7 - 1)/2 complex equations in the set (4.22) and. effectively. 177 real equations in the set (4.23). Both sets together determine i n 2 independent real equations that must be satisfied in order that a matrix { a j j )of GL,,(K) belong t o the subgroup L"(n7).Thus C'(n7) is an h-surface of dimension m 2 . One can also argue that the special unitary group SL'(n1) is an h-surface of dimension / n 2 - I . These facts about 11-surfaces have a familiar and useful analog. Recall that the solution set S of a family of k independent real linear equations in n real unknowns. (4.24)

2 u j j x j= 0,

1ii

k < n,

can be described in several ways. One such way is to state there exist n - k parameters ( i , .. . . . I., -kj such that each point {xi},I 5 i i n, of S is given by (4.25)

xi= ~bjj?.j, I I j l n - k ,

I
4. The Representation Theory of Certain Matrix Groups

28 7

The set S has dimension n - k in such cases. The parameters {;vi} assume all possible real values and determine each point on S exactly once. Similar statements can be made about the nature of solution sets which are h-surfaces of GL,(K). The principal differences are that the equations replacing (4.25) must be more complicated and that one can be sure of nice solutions only in a suitable neighborhood of a point on the h-surface. We illustrate the basic idea by considering an element g E U(m). There exists an open neighborhood N ( g ) of g in GL,,(K). 6 > 0. and a set ( f i j } , 1I i , j 5 m2, of functions such that each element ( a i j ) of C ! ( m ) contained in N ( g ) is determined exactly once by

(4.26) for those

(4.27)

a I-J .= f i j ( ) . , .

i=

.

. . . I .,,,2 ) .

. . . . I.,,.) with

l i , - Xio\

< 6,

I I i , j 5 tn2,

I I i i m2.

Here { f i j ( x o ) ] 1, 5 i. j 5 1 1 7 ~ .is the matrix of g. Furthermore. each of the functions f i j , 1 2 i, j 2 i n 2 , is an analytic function of the real variables { I b l . .... in the sense that it has a convergent power series expansion about Lo. In particular. each f i j is infinitely differentiable with respect to the set ( I v l , . . . , I.,,,,) of parameters. By fixing all except one of these. say of these with their value at one obtains a curve G j in the space GL,(K) which passes through the point g and which has the tangent vector H j at g = {Aj(%')}. The tangent vector H j can be regarded as an element of K,,, , but not always as one of GL,,(K), in the same way that the tangent vector to an arc G in the three-dimensional real Euclidean space X is usually taken to be an element of X.

x".

(4.28) EXAMPLE. The real ortitogonu1 group O(m) is the hubgroup of real matrices belonging to the unitary group L'(n2). The special real orthogonal group SO(m) consists of those elements with determinant 1 in O(n7). Geometrically, SO(m) is the group of linear transformations on the real Euclidean space R E ( m ) which leave invariant both the length and orientation of vectors. This group is commonly called the in-dimerisional rotntion group. The group SO(3) is the famiiiar group of rotations of three-dimensional space. Not only is it a good example of the ideas under consideration. but also its representation theory is important in the applications to quantum mechcnics. An element { a i j ) ,1 5 i , j 2 3, of GL,(K) belongs to SO(3) if and or:ly if it has real elements which satisfy the equations: (4.29) and

(4.30)

x:ajiaji= i ,

1 < , i s 3.

288

4. Representation Theory of Special Groups

Thus O(3) is a subgroup of the nine-dimensional real general linear group GL,(R), and its components satisfy six real equations. We conclude that O(3) is three-dimensional. Each element g E O(3) has det(g) either I or - 1. Thus SO(3) is the part of O(3) with det(g) equal to 1. Since this is not an independent relation, the dimension of SO(3) is also 3. This dimension can be obtained by geometric considerations. Every rotation R in the threedimensional space X is determined by specifying an oriented axis and a magnitude 4 of rotation, Consequently, the three-dimensional rotation group SO(3) can be placed in correspondence with the sphere 6of radius n: drawn in Fig. (4.31). Let f : G + SO(3) make correspond to each vector belonging (4.3i)

t

X

to 6 the rotation R whose directed axis is along the direction of and whose magnitude 4 i s the length of Any rotation R with magnitude 4 greater than i( coincides with a rotation R' with magnitude 4' not exceeding n about the same line, possibly differently orientated. This shows thatfmaps G onto SO(3). The niapfis also one-to-one except for 2 of length n. For these, f ( f ) and are the same rotation. The sphere 6 bears its relative topology as a subset of X and the group SO(3) bears its relative topology as a subset of GL,(K). Since a rotation changes little when its axis is barely tilted and its magnitude slightly altered. the map f is seen intuitively to be continuous. When one identifies antipodal points, such as a and a' of 6in Fig. (4.31), to obtain a new set E',the induced map f ' : S' + SO(3) arising from f is a homeomorphism of G' (in the quotient topology) onto SU(3). Thus SO(3) i~ topclogically like the sphere 6 with antipodal points identified. This identification doe\ not change the local nature of the sphere 6. Consequently, a small neighborhood in SO(3) is topologically like a small neighborhood of KE(3). However, the global nature of RE(3) is quite different from the global nature c f SO(3). To point out two differences: RE(3) is locally compact, but not compact while SO(3) is compact. The space RE(3) is simply connected.

x.

f(-x)

289

4. The Representation Theory of Certain Matrix Groups

Recall this means that RE(3) is connected, locally connected, and any simple closed curve C in RE(3) can be continuously deformed to a point. On the other hand, SU(3) is not simply connected. A simple closed curve in SU(3) which can not be continuously deformed into a point is the image underf’ of the segment aa‘ of Fig. (4.31). This segment is a simple closed curve in S’. If one attempts t o move the point a = a’ from its position on the boundary” the curve breaks. Otherwise, one can not deform it into a point. However, both RE(3) and SO(3) are connected. According to Eq. (3.9), Chapter 3, every rotation R of three-dimensional real space X has a matrix of the form “

0 -sin a cos a

0 cos a 0 sin a

with respect t o a suitably chosen basis. To obtain this form, the x-axis is taken to be the axis of rotation of R and c( the radian measure of the magnitude of the rotation. We occasionally allow ourselves the convenience of confusing the idea of a linear transformation and its matrix in the remainder of this section. This often makes the discussion less awkward, and lets the reader supply the required interpretation. In particular, we confuse GL,(K) and GL(V). The rotation R has a trace given by 1 + 2 cos a which suggests the following lemma. (4.32) LEMMA. Two rotations R , and R2 of the three-dimensional real space X are conjugate if and only if they have the same trace.

The proof is omitted. Note that the set of all rotations about any fixed line of the real threedimensional space X is a subgroup of SU(3). By proper choice of axes and notation, the group H , of rotations about the x-axis consists of all rotations of the form 1

0

0

sin 2,

cos lL1

This is a basic example of what is called a one-parameter subgroup of a Lie group. As such, it can be considered as a map H I : R SU(3) which is a homomorphism of the additive group R of real numbers into the group SU(3). The general definition is the obvious analogue. We see below that the one-parameter subgroups play an important role in the general theory. There --f

290

4. Representation Theory of Special Groups

are, of course. one-parameter subgroups H 2 and H , of rotations about the y-axis and the z-axis. These are described by

HZO”,) =

1

cos)i., -sin ?,.

siY2 0 cos 2,

and 0

I/

1

The one-parameter subgroups play a significant role since the elements in a neighborhood of the identity of many important groups can be written as the products of elements from a finite family of one-parameter subgroups. To see such a factorization in the case of SO(3), let R denote any element of SO(3). Let {v,, v 2 , v3} be an orthonormal basis of the real three-dimensional space X, and let R be defined by Rvi = u i , 1 i i 5 3. It follows that the set { u l , u 2 . ujl is also an orthonormal basis of X . (i) There exists an element H,(A1’) rotating u, into the (v,, v,)-plane with - n < i,’ 5 0. ( i i ) There exists an element H 2 ( A 2 ’ ) such that H2(?.2‘)Hl(Al’) rotates u3 into vj with -277 < A’ 5 0. (iii) There exists an element H3(A3’) such that H,(n3’)H2(~2’)H,(Alf) rotates u, into v, and u2 into v2 with -27t < A,’ 5 0.

One has H 3 ( I . , ’ ) H 2 ( ? ~ , ’ ) H ~ ~ A 1 ’= ) Uvii .

1I i I 3,

so that R - ’ = lf3()~3’)H2(A2‘)H,(A,’)r from which it follows that R can be written in the form R =~10”1)~2@2W3(~3)~

(4.33)

where 0 I i,, < n. 0 5 L2 < 271,O i A, < 27t with 2, = -Ai‘, 1 5 i i 3. The matrix o f fi with respect to a set of coordinate axes through v,, v2, and v3 is given in Fig. (4.34).

(4 34)

!I

co\

A,

XI

cos A,

\in A,

sin A, sin ‘lsiii A, $111 A, - coy A, sin i w

A2

cos A3 cos A,

A, sin A 3 cos A , cos A, sin A , sin A, sin A, sin A, cos 1 cos A, sin A, sin A,

-cos

-sin A, cos A,

cos A, cos A,

Thu\ n e find that every element of SO(3) can be written as the product of element\ from the subgroups H , , H,, and H,. Furthermore, given any neighborhood Ny,) of the element go = H1(A,’)H2().2O)H,(~3*) of SO(3),

291

4. The Representation Theory of Certain Matrix Groups

zio]

there exists a 6 > 0 such that the set of f with I f i < 6 determines, exactly once, each element g(x) of a neighborhood ” ( g o ) contained in N(go). Here

.dx)= H1(;il)H2(n2)H3(A3). Let g(x‘) be another element in ”(go). Then it is clear from the form of (4.34) that g ( f ) g ( z ’ )is a matrix whose elements are analytic functions of the components of and i’. Let J1 be the open interval (5 : lLlo - S < 5 < iIo + S}. Then there exists a differentiable arc f,: J1 + SO(3) defined by (4.35)

f i(4) =

(5>H2(n,0)H3(ju30>.

We refer to the a r c h as a A,-path through g o . One can define ;.,-paths through g o , 1 < i 2 3, in an analogous fashion. The tangent vector to the path fiis given by

4 /dt= Wl ( i ” ) / ~ ( 1 H 2 ( ~ 2 ° > ~ ~ ( ~ 3 0 ) , These concepts prove most useful in a neighborhood of the identity element of SO(3) which has the coordinates = 0. The expressions for the A-paths fi, f 2 , and f3 in a suitable neighborhood of the identity in SO(3) are 1 0 0 0

sin

5

cos 5 0 -sin( cos 5

cos

5

sin

t

0 cost -sin

5

0

At the identity of S0(3), these arcs have the tangent vectors

292

4. Representation Theory of Special Groups

One notes that each of these three tangent vectors is a skew-symmetric matrix. that is, its transpose is its negative. Furthermore, any real linear combination of the three. for example,

I

0

-a3

nz

Jj.

-a, -a, 0 is also skew-symmetric. Conversely, any real skew-symmetric matrix is a real linear combination of A , , A , , and A , . In this sense, the tangent plane eo(3) at the identity of SO(3) can be identified with the set of all skewsymmetric. 3 x 3 real matrices. This set of skew-symmetric rnatrices is a linear space over the real numbers, but it is not closed under ordinary matrix multiplication. It proves to be a most fruitful idea t o introduce a binary operation [ , ] on the real linear space eo(3). If A and B are two real, skewsymmetric matrices in 9 4 3 ). then one defines n,A,

f

02A2

f

a,A, =

0 a,

a3

[ A , B ] = A B - BA,

(4.36)

where A B and BA denote the standard products of the matrices A and B. Note. i n particular. that [ A , B ] is skew-symmetric whenever A and B are skew-symmetric. This product [ A , B ] i s called the Lie product of A and B. The real linear space 543) together with the Lie product [ , ] is called the Lie ulgc&a of the special orthogonal group SO(3). One finds by direct applications of the definition that (4.37)

[ A , B] = -[R. A].

[A. B

{ ' [ A . BJ = [ P A , B ] = [ A , p B ] .

+ C ] = [ A . B] + [ A , C],

[A

+ B. C ] = [ A , C ] t [ B , C],

where A . B. C E so(3) and p is any real number. Thus most of the familiar laws of algebras are valid with one special exception. The usual associative law fails and is replaced by a more complicated rule, (4.38)

[ [ A . B], C]

+ [ [ B ,C]. A ] + [[C. A ) , B] = 0.

which is usually referred t o as the Jacobi idcnritj.. (4.39) DEI-'INITION. A rcwl Lic algrhra A is a vector space over the real numbers for which rhere is defined a Lie product [ , ] such that the rules of (4.37) and (4.38) are satisfied. The theory of Lie algebras is highly developed. We refer the reader t o Kaplansky (1963) for an elegant introduction and t o Jacobson (1962) for details. Lie algebras admit of a very detailed classification in many instances. ification is of great interest in the study of Lie groups of which GL,,(K)and many of its subgroups are particular instances.

4. The Representation Theory of Certain Matrix Groups

293

The discussion of the group SO(3) is special only in the details. Each of the standard groups has associated with it a real Lie algebra. This Lie algebra is obtained by examining a neighborhood of the identity of the group in question and determining a family of one-dimensional subgroups which play the role of H I , H 2 , and H , in the case of SO(3). The results are as follows: (a) The Lie algebra of the complex general linear group GL,,,(K)is the set gI,(K) of all m x m complex matrices. (b) The Lie algebra of the real general linear group GL,(R) is the set gl,(R) of all in x m real matrices. (c) The Lie algebra of the complex special linear group SL,(K) is the set 51m(K)of all m x m complex matrices with trace zero. (d) The Lie algebra of the real special linear group SL,(R) is the set sI,(R) of all m x m real matrices with trace zero. (e) The Lie algebra of the unitary group U(m)is the set ~ ( mof) all skewHermitian complex m x m matrices. (f) The Lie algebra of the special unitary group SU(m) is the set sii(m) of all skew-Hermitian m x m complex matrices with trace zero. (g) The Lie algebra of the special orthogonal group SO(m) is the set so(m) of all skew-symmetric m x m real matrices. The Lie algebras discussed so far arise from h-surfaces in real Euclidean spaces. Consequently, the scalars involved are the real numbers. However, Lie algebras exist over any field. We first discuss the case of real Lie algebras and then turn t o complex Lie algebras where various problems are simpler. Most definitions given for the real case extend directly to any field and are not repeated. The determination of the one-dimensional subgroups of a matrix group may present difficulty for some of the various subgroups of the complex general linear groups. However, if the Lie algebra of a subgroup G of GL,(K) is known from other considerations, it is easy to specify the one-dimensional subgroups of H and G. The method employs several results on matrices and linear differential equations which we discuss briefly. Let A be an n x n complex matrix. Then exp(A) is (4.40) DEFINITION. defined by the infinite series (4.41)

exp(A) = 1 + A

+ A 2 / 2 !+ . . . + A " / n ! + . . . ,

which is the same series, of course, used to define exp(x) for x a real or complex number. The convergence of (4.41) is most easily proved by use of a norm I] I/ on

294

4. Representation Theory of Special Groups

gl,,(K) which determines a topology equivalent to the Euclidean topology on gi,,(K). The obvious candidate for the norm is given by

(4.42)

IlA/12 =

2 luijlz =

-y{cTij2

+

Zij2>,

so that the norm (IAlj of A is its Euclidean distance fron the zero matrix. The &-spheresof this norm S,(A) = ( M

E gI,(K)

: (1 M

-

A 11 < E )

are the spherical neighborhoods of the Euclidean topology so that the topology induced by the norm /I I/ is the same as the Euclidean topology on gi,,,(K). The norm 11 11 prokides a convenient working tool because of the fundamental inequalities IIA

+ Bll I!I4+ IIBII

and

IlABlI

s 114 IIBII.

These rules extend t o the cases of IZ summands and n factors by induction. The approximating sums of (4.41) are of the form

s, = 1 + A + . . . + A " / n ! , Then. for i < j , one has [ISi- S j / (= /ISj - Sill = I \ A i + l / ( i + I)!

- a i + l /(i+ < I)!

0 < n.

+ . - .+ A j b ! l (

+ ... + u j / j ! ,

a = I/AIl.

Since this last s u m tends to 0 as i tends t o infinity, it follows that (4.41) converges t o an ti x IZ matrix. Many of the usual rules for the numerical exponential function remain valid: exp(nA) where 1 is the

= (exp(A))" II

x

ti

and

exp(A) exp( - A )

identity matrix and 0 is the n x

(4.43)

exp(A

IZ

= exp(0) = 1,

zero matrix. The rule

+ B ) = exp(A) exp(B)

holds when A a i d B are commuting matrices; otherwise, certain difficulties arise which are considered in a more complete treatment. Let t

E(X E

then the series exp(tA)

=

1

R 1 --b < x < b},

+ tA -t. . . 3. t"A"/n! + . . .

converges uniformly as a function o f t , and d exp(tA)/dt = A exp(tA).

295

4. The Representation Theory of Certain Matrix Groups

One knows from the theory of linear differential equations with constant coefficients that any matrix equation of the general form dfldt

= Af

has a solution of the form

f = C exp(tA), where C is a constant matrix. One notes that given any A E gI,(K), thenf(t) = exp(tA) is a differentiable arc in GL,(K) whose tangent vector at the origin is the element A . Furthermore, one has by (4.43) that f ( t + t’) = exp(tA

+ t ’ A ) = exp(tA) exp(t’A) =f ( t ) f ( t ’ ) .

Thus the set {exp(tA) : - co < t < a} is a one-dimensional subgroup of GL,(K) with tangent vector at the origin equal to A . It can be shown that all of the one-dimensional subgroups of GL,(K) have this form. More generally, if A is an element of the Lie algebra of G, any of the subgroups of GL,(K) under consideration, then one obtains a one-dimensional subgroup H of G by the process indicated. All one-dimensional subgroups of G arise in this manner. Consider the three tangent vectors at the origin, A , , A , , (4.44) EXAMPLE. and A , , of the special orthogonal group SO(3). Then one finds that H i ( t ) = exp(tA,).

To be specific,

1:

H , ( t ) = 0 cos t sip,

1IiI 3.

0 -sin t c ost

1.

We turn to a brief explanation of our interest in these matters. The complex general linear group GL,(K) and its subgroups under discussion are all examples of Lie groups. We will not give a formal definition, but remark that Lie groups are similar to the standard groups. However, the technical problems in topology and differential geometry become substantially deeper for Lie groups in general. Nevertheless, many basic concepts for general Lie groups are analogous to those for matrix groups. In particular. each Lie group G has associated with it a Lie algebra g which is the tangent space t o G at the origin. The algebraic properties of the Lie algebra g strongly influence those of the associated Lie group G. Any two simply connected Lie groups G and G‘ with isomorphic Lie algebra g and g’, respectively, are themselves isomorphic. The relationship remains strong when the Lie group G is connected, but not simply connected.

296

4. Representation Theory of Special Groups

If G is a connected Lie group with Lie algebra g, then there exists a simply connected Lie group G' whose Lie algebra g' is isomorphic to 9. Furthermore, G is a homomorphic image of G'. Actually, G' contains a discrete central subgroup N such that G is isomorphic to G I N .

(4.45) EXAMPLE. The real numbers R under addition and the unit circle C = { z E K : lzl = I} under multiplication are familiar examples of Lie groups. each of which has the real numbers as its Lie algebra. The group R is simply connected while the group G is only connected. The normal subgroup N such that C is isomorphic to R / N can be taken t o be N = {x E R : x = 2nn, IZ EZ}. The groups SU(2) and SO(3) share an analogous relationship. The discrete central subgroup N of S U ( 2 ) is {l, -1} where 1 denotes the 2 x 2 identity matrix. There exists a homomorphismf: SU(2) + SO(3) with kernel N which is a homeomorphism on a sufficiently small neighborhood W of 1. More generally, let G be a connected Lie group with Lie algebra g and let G' be a simply connected Lie group with Lie algebra g' isomorphic to g. Then there exists a neighborhood W of the identity 1' in G' and homomorphismfof G' onto G such that the kernel N offis a discrete central subgroup of G' meeting W only in 1'. Furthermore, the homomorphismfis a homeomorphism on the neighborhood W . A map such a s j i s called a local isomorphism, and a group such as G' is called a universalcoaeringgroup of G. If H any other connected Lie group whose Lie algebra lj is isomorphic to g, then G' is also a universal covering group of H and there exists a homomorphism y : G' H such that for some neighborhood V of I' g is a local isomorphism. A full discussion of this relationship between Sb'(2) and SO(3) is not difficult. However, we must refer the reader t o Gel'fand and Sapiro (1952, p. 213) for it. These observations support the rather vague statement that the nature of a Lie group G in the neighborhood of its identity is largely determined by its Lie algebra. The close relationship is shown for GL,(K) and its standard subgroups by means of the exponential map exp : gl,,,(K) + GL,(K) defined for A E gI,!,(K) by --f

exp(A) = 1

+ A + . . . + A"/n! + . . . ,

and for Lie groups in general by a similar but technically more complicated function. These exponential maps are homeomorphisms of some neighborhood of the 0 element of the Lie algebra 9 onto some neighborhood of the identity I of G. The structure theory of many classes of Lie algebras is known in detail. This is true especially for the Lie algebras of the standard matrix groups. Moreover. the finite-dimensional representation theory of the Lie algebras of these groups is well understood.

4. The Representation Theory of Certain Matrix Groups

297

(4.46) DEFINITION. A finite-dimensional representation t of a real Lie algebra g with Lie product [ , ] is a mapping with domain g and range Hom(V, V) for some r-dimensional real vector space V such that

(4.47)

t(au

+ pv) = at(u) + Pt(u)

and

(4.48)

f([u, v1) = [t(uh t(v)17

where [t(u), t(v)] is t(u)t(v) - t(v)t(u), the usual additive commutator, a, are real numbers, and u, v are elements of g.

fl

Rather than a map t from g into the associative algebra of linear transformations Hom(V, V) one can define a map into the isomorphic algebra R, of r x r real matrices. In this case, one has a matrix representation of the Lie algebra g. There is also the usual technique of rep!acing a representation by a module and the converse.

(4.49) DEFINITION. Let M be a finite-dimensional real vector space. The space M is said to be a module for the real Lie algebra g if (i) there is a left multiplication xm defined for elements m of M by elements x of g; (ii) x(crm, + pm2) = cr(xm,) + p(xm2) for x E g , m , , m2 E M, and 2, fi real numbers; (iii) [x, y]m = x(ym) - y(xm) for x, y E g, m E M . We should be familiar by now with the fact that the concepts of representation and module are mostly different ways o f looking at the same thing. Let G be a standard matrix group with Lie algebra g . Denote by env(G) the real enveloping algebra of C consisting of all real linear combinations

+ ... + a n g n ,

"191

a iE R , g i E G, i _< i _< n. Then env(G) is a real associative algebra which is a closed subspace of the algebra R, of all real M x m matrices. There exists a neighborhood W of the 0 matrix in g such that exp maps W homeomorphically onto a neighborhood of the identity 1 of G. Given A E g, there exists 6 > 0, such that exp(xA) E W when 1x1 < 6. This means that

(exp(xA)

-

l)/x

E

env(C)

when 1x1 < 6. Since env(G) is closed, one has

A is an element of env(G).

=

lim (exp(x.4) - l ) / x

x-0

298

4. Representation Theory of Special Groups

Any finite-dimensional representation T : G -+ GL(V’)extends in a natural manner to a representation of the associative algebra env(G). We denote the extended representation also by T. Define t : g Hom(V’, V’) by --f

t(A)

for any A

E

=T[

lim (exp(xA) - I)/x] = T(A)

x-+o

g. Since T is a representation of env(G), one has

t ( [ A , B ] ) = [ ( A B - B A ) = T(AB - BA) = T(A)T(B)-

=

[W), W ) I = [Q),t(N1.

T(B)T(A)

The linearity of t follows from that of T on env(G). Consequently, t is a representation of the Lie algebra g of G. Thus, every representation T of G leads to a representation t of the Lie algebra g of G. Unfortunately, the converse is false. A representation t of the Lie algebra g need not supply a representation T of G. Nevertheless, the introduction of an associative algebra U(g) such that g can be identified with a subspace of U(g) with [x,y] i n g corresponding to xy - yx i n U ( g ) proves to be fundamental. Warning! This algebra is not the algebra env(G) introduced above.

(4.50) DEFINITION. Let g be a Lie algebra over the real (complex) numbers. A pair { U(g), i} where U(g)is an associative algebra over the real (complex) numbers and i is an injection of g into U ( g ) is called a universalenveloping nl~qcbraof g if: Given any associative algebra A and a map f:g + A which is linear and such that f ( [ x ,y]) = [f(x),f(y)] for x, y E g, there exists a unique homomorphism /7 : U ( g ) -+ A such t h a t f = hi. The map /z is a homomorphism of the associative algebra U(g) into the associative algebra A. The universal enveloping algebra U(g) proves to be unique up to isomorphism. The construction of a satisfactory model and the establishment of all the required properties is a sophisticated piece of mathematics which we do not attempt. See Jacobson (1962, Chap. V). The associative algebra U ( g ) has two fundamental properties from the standpoint of representation theory. The first of these is the following theorem. (4.51) THEOREM. Let g be a real (complex) Lie algebra and V be an rndimensional vector space over the real (complex) numbers. There is a natural one-to-one correspondence between the set of all representations of g on V and the set of all representations of U ( g ) on V where { U ( g ) ,i> is the universal enveloping algebra of g. If t : g+gI,(V) is the representation of g and T : U(g) Hom(V, V) is the corresponding representation of U ( g ) , then --f

t(x) = T(i(x)),

x

E

g.

The second fundamental property is that the representation theory of U ( g ) can be uwrked out in detail for important cases.

299

4. The Representation Theory of Certain Matrix Groups

We must introduce additional terminology in order to discuss further results. Our remarks are restricted to a Lie algebra g over the field F of either the real or complex numbers. Most of the statements are true for Lie algebras over fields of characteristic zero. Let X and Y be subsets of g. The symbol [X, Y] denotes the linear span of all elements of the form [x,y], x E X, y E Y. (4.52) DEFINITION. A subspace f of the Lie algebra g is a subalgebra of g if and only if [f, €1 c f . This asserts that the subspace f is closed under the Lie product.

A subspace b of the Lie algebra g is called an ideal of g (4.53) DEFINITION. if and only if [g, b] c b. The 0-subspace is always an ideal of g. One can show by means of the Jacobi identity that if b an ideal of g, then [b, g] = [g, b] is an ideal of g contained in 6. Furthermore, [b, 6'1 is an ideal whenever b and b' are ideals. (4.54) DEFINITION. The sequence of ideals, defined recursively by g1 = [g, g] and g'+' = [g', g], 1 i, forms a descending chain (4.55) of ideals which is called the lower central series of g. (4.56) DEFINITION. A Lie algebra g (an ideal or subalgebra b of g) is called nilpotent if and only if the lower central series of g (of 6 ) terminates in the zero ideal after a finite number of steps. (4.57) DEFINITION. Let g be a Lie algebra. The sequence of ideals of g, defined recursively by g' = g(l) = [g, g] and g ( ' + l )= [g"), g")], 1 5 i, forms a descending chain (4.58)

g(l)

...

g(n)

.. .

called the derived series of g .

A Lie algebra g (an ideal or subalgebra b of g) is called (4.59) DEFINITION. solvable if and only if the derived series of g (of b) terminates in the zero ideal after a finite number of steps. Every Lie algebra g contains a maximal solvable ideal n (4.60) DEFINITION. called the radical of g. A Lie algebra is called semisimple if and only if it has radical (0). Let g be a Lie algebra with no proper ideals for which g' = [g, g], the derivedalgebra of g, is not (0). Then g is a simple Lie algebra. The structure theory of finite-dimensional simple and semisimple Lie algebras over the complex field is known in great detail. The term Lie algebra, without additional qualifications, denotes a simple or semisimple Lie algebra over the complex or real field in the sequel. A Lie algebra of linear transformations is a subspace

300

4. Representation Theory of Special Groups

S of Hom(V, V) such that A , B E S implies that A B - BA E S . All of the Lie algebras of the standard groups are Lie algebras of linear transformations.

(4.61) DEFINITION. Let g be a Lie algebra of linear transformations acting on an m-dimensional vector space V over F. A linear mapping LY: g --t F is called a weight of g with respect to V if there exists a non zero vector v E V such that ( A - M(A)I)”‘A’V = 0 for some integer m(A), depending on A , for every A ~ gThe . set of all such vectors (including zero) for which this condition is satisfied form a subspace V, of V called the weight space of g corresponding to the weight a. (4.62) THEOREM. Let g be a nilpotent Lie algebra of linear transformations acting on the m-dimensional complex space V. Then g has only a finite number of weights with respect to V. Each weight space W of V is invariant under the action of g. Furthermore, V is the direct sum of the weight spaces of g. In addition, let

v =v,@ . . . @ V ,

be a decomposition of V into subspaces V i , 1 i i 5 k , such that each V i is invariant under the action of g. Suppose also that (i) the restriction of any A E g to Vi is a linear transformation with a single characteristic root a,(A) (not necessarily of multiplicity one) ; (ii) for i different from j , there exists B E g such that ai(B) # aj(B). Then the mappings a i : g + K are the weights of g with respect to V and the spaces Vi are the corresponding weight spaces. (4.63) REMARK. The methods of study of Lie algebras somewhat parallel those of the study of associative algebras. One represents a Lie algebra g on itself, so to speak. For any x E g, let ad x denote the element of Hom,(g, g) defined by ad x(m)

=

[x, m],

m Eg

It follows immediately from the definition that ad x is a linear transformation on g which is read “add ex.” The map ad: g + Horn&, g) is a representation, called the adjoint representation, of the Lie algebra g whose representation space is g. Note that the equations ad[x. yI@)

=

[[x, PI, ml = [x, [Y,m11 + [Y, [x,mll x(ad y(m)) - ad y(ad x(m)) = (ad x ad y - ad y ad x)(m)

= ad

follow from anticommutativity and the Jacobi identity. The result shows that the map ad preserves the Lie product. The remainder of the argument that

4. The Representation Theory of Certain Matrix Groups

301

ad is a Lie algebra homomorphism is easy to supply. The adjoint representation plays the same crucial role in the study of semisimple Lie algebras that the left regular representation plays in the study of semisimple associative algebras. A Lie algebra Ij is said to be abelian if and only if [lj, Ij] = (0). Every semisimple Lie algebra g over the complex numbers has associated with it a family of very special abelian subalgebras which satisfy the followingdefnition. (4.64) DEFINITION. A subalgebra Ij of the semisimple Lie algebra 9 is a Cartan subalgebra of g if and only if: (i) The subalgebra Ij is abelian, but is not properly contained in any abelian subalgebra of g, that is, Ij is a maximal, abelian subalgebra of g. (ii) For each element x E Ij, the linear transformation ad x, regarded as a linear transformation on g, is semisimple. This means that the Lie algebra g decomposes into invariant subspaces which are eigenspaces of ad x. This semisimplicity of the transformation ad x is at the basis of the analysis of semisimple Lie algebras over the complex numbers. The classification of the finite-dimensional semisimple Lie algebras over K is based on an analysis made possible by Theorem (4.62). Select a Cartan subalgebra Ij of g. Then g decomposes as the direct sum of weight spaces of ad I). (4.65)

g = Ij 0 w,, 0.. .0 wak.

The Cartan subalgebra Ij itself is the weight space corresponding to the zero weight of ad Ij. If one takes {Ij,, . . . , Ij,) to be a complex basis of Ij, then the dimension r is normally called the rank of the Lie algebra g. There may exist nonisomorphic semisimple Lie algebras of the same rank. Each weight space Wmicorresponding to a nonzero weight aiproves to be one-dimensional over K. Furthermore, if tli is a weight, then - a i is a weight. However, tli, - x i , and 0 = Oa, are the only multiples of ai which are weights. Thus one can select a family of vectors {e,,) corresponding to the nonzero weights such that each Wuiis spanned by e a i . There is a great deal of arbitrariness in all of these choices. Consequently, to find a standard description of the Lie algebra g, all of the choices must be made in a very special way. The full treatment of the classification problem for a semisimple Lie algebra g over the complex field is an elegant piece of linear algebra beyond the scope of this book. A change of terminology is made when discussing the decomposition of g under the action of the niepotent Lie algebra ad Ij. The weights arising are called roots of g and their weight spaces called rootspaces. A special class of roots (or weights), known as simple roots, is needed in the analysis. Their definition requires the introduction of an ordering in a real subspace Ij,* of the dual space Ij* of Ij. One denotes by IjR* the set of all r e d linear

302

4. Representation Theory of Special Groups

combinations of the roots of g. The real linear space selecting any basis {I.,, . . . , 2s} for it. Then an element =

bR* is

ordered by

r, J . ~+ . . . + <,$a,

is said to be pasitirc or grcwtc’r //inn 0 if and only if its first nonzero coefficient ti is positive. The vector x is greater tlzan the vector y if and only if the e x - y is positive. A yositice roof a is one such that 0 < a. (4.66) DEFINIT~ON. A simple root is a positive root which is not the sum of two positive roots. Clearly one obtains different sets of simple roots for different choices of the basis ti.,, . . . . AT}. It turns out that any set {al,. . . , a,.} of simple roots is a real basis of IjR* and a complex basis of Ij*. Thus the number of vectors in any set of simple roots is equal to the rank of the semiGniple Lie algebra g. Every root a can be written TX

=nlctl

+ ... + n,a,

;IS an infcyrnl h c u i . c o ~ h i n a t i o nof the simple roots where the set (n,} of integers are either all nonnegative or nonpositive. When 0 < a, they are all nonnegative. When s( < 0, they are all nonpositive.

(4.67) DEFINITION. There exists a symmetric bilinear form ( , ) on every finite-dimensional Lie algebra g. This form is called the Killing form and is defined by

(x,y)

= tr(ad

x a d y),

x, y

E g.

It is a famous theorem of Cartan’s that a Lie algebra over a field of characteristic 0 is semisiinple if and only if its Killing form is nondegenerate.

The Killing form remains nondegenerate when restricted t o a Cartan subalgebra 11 of a semisimple Lie algebra $1. This means that given any root a in i)*, there exists a unique element h, in Ij such that .(X)

= (h,.

x

x),

E

1).

where ( , ) denotes the Killing form restricted t o 5. The correspondence r/ + h, enables one to define an inner product { . 1 on I?,*. Given roots a and [I i n detine { a , /I> = ( h a

>

ha).

where { ,I denotes the inner product on bR* and ( , ) denotes the Killing form on Ij. The space I),* is an r-dimensional real Euclidean space under the norm I/ / / induced by { , >. Given a set {ri} of simple roots of g. the Lie algebra is completely characterized by nieans of a canonical selection of elements {hi,e i ,fi} for each

303

4. The Representation Theory of Certain Matrix Groups

simple root, 1 I i 5 r, together with a certain r x r integral matrix called a Cartan matrix of g relative to the Cartan subalgebra 5. These matters are efficiently described by means of Dynkin diagrams. We will present some examples of Dynkin diagrams in the sequel. Given a root a , from the set {g1, . . . , ar} of simple roots, there exists a unique element hai E 5 such that .i(x)

x 9.

= (x, hai),

Select any root vector ea,corresponding to for x

E

5. Since

E

such that

[x,euil= ai(x)eai= (x, hui)ea,, - x i is also a root, there exists a root vector e P a isuch that [x, e - J = -ai(x)e-ai

for x

CI,

=

-(x, hui)e-u,

b. It turns out that [eai,e-,J

=

e-,,)haZ

and that e-ui can be selected such that (eai,e-ui)= 1. The elements eal and e-,i are unique only to scale factor, but this last condition is a partial normalization. One defines the set {hi, e , , fi} by (4.68)

hi = 2hui/{ai,a,},

e, = eai,

f,

= 2e-,,/{ai, .,}.

These vectors satisfy the following multiplication table. (4.68)’

[hi,ej] = Aijej, [hi, fj]

where the matrix { A i j } , A i j matrix.

=

= 2{r,,

- A . 1J. f .J ’

[e,, fj] = 6 IJ. . hJ .’

a j } / { a i ,a , } , 1 I i, j

< r,

is the Cartan

The elements of the set {el, . . . , e,} are called simple or elenirntary raising operators while those o f the set {fl, . , . , f,} are called simple or elementary lowering operators. (4.69) EXAMPLE. Up to isomorphism, there is only one simple Lie algebra g of rank one over the complex numbers. A Cartan subalgebra of g is spanned by all complex multiples of a vector h E 9. There are two root spaces W, and W-, corresponding to the nonzero roots a and -3. The set {h, e, f } corresponding to the simple root CI is a basis of g for which the multiplication is determined by (4.71‘). Jacobson (1962) refers to this algebra as the split threedimension simple Lie algebra. The adjective split is used to indicate that the characteristic values of all the elements of ad 1, are included in whatever field of characteristic 0 is being considered. The real Lie algebras so(3) and Sll(2) are real forms. see below, of the simple Lie algebra g of rank one over the complex numbers. To obtain the

304

4. Representation Theory of Special Groups

complex Lie algebra g from so(3). one merely takes all complex linear combinations of the set { A l , A , . A 3 } of matrices spanning 5-43). The physicists are accustomed to introducing a different set { I f j } , 1 < j _< 3, of generators as an intermediate step. These are matrices defined by the equations

1<j<3.

Hj=iAj,

The commutators of these elements assume the form (4.70)

[HI. H,]

= iH,,

[ H , , H,]

= iH,,

[ H , , HI] = iH,

for this new basis. Raising and lowering operators are defined by H,

=HI

+ iH,.

H-

= HI

- iH,,

so that H , , H- , and H , give still a different basis for g in which the commutators assume the form [ H 3 , H + ]= H +

.

[Hj, H - ]

=

[ H +, H - ] = 2 H 3 .

-H-,

Finally. one obtains a cawnical basis for g by defining (4.71)

e=H+.

f=H-.

h=2H,

The commutators for the set {h, e, f } are given by (4.71')

[h, e] = 2e,

[h, f ] = -2f,

[e, f ] = h.

The three-dimensional simple Lie algebra g over the complex numbers and its representation theory play a central role in both the structure and the representation theory of semisimple Lie algebras. The principal theorem on the irreducible representations of g , see Jacobson (1962, p. SS), is the following. (4.72) THEOREM. Let g be a three-dimensional simple Lie algebra over the complex numbers. Then for each nonnegative integer rn there is, u p t o isomorphism, exactly one irreducible g-module V of dimension m + 1. The space V has a basis {v,. , . . . v,,,} such that the actions of the members of the set {h, e, f } are given by

(4.73)

hvj = (t72 - 2 j ) v j , fVj = v j r l , fv, = 0, ev, = 0,

Osjsm.

evj = j [ m - j + I]vj-,,

1 2 j < m.

O _ < j < m - 1,

(4.74) REMARK. Each such (m + I)-dimensional representation is characterized by the integer n7 which is the highest eigenvalue of h which occurs. The weight CI which assigns to h the eigenvalue m(h) = rn is called the highest weight

4. The Representation Theory of Certain Matrix Groups

305

of the representation. All complex finite-dimensional, irreducible representations of any semisimple Lie algebra are characterized by such a highest weight. It is perhaps worth noting that the family of representations described above is frequently seen in the physics literature, but in a slightly different normalization. According to Eq. (4.71), h is taken to be 2H,, where H , is the generator of the Cartan subalgebra generally used by physicists. Consequently, H 3 has for its highest eigenvalue some integral multiple of in the irreducible representation described by physicists. The interested reader can find the details discussed in Gel’fand and Sapiro (1952, pp. 223-232). We have observed that the Lie algebra su(2) of the Lie group SU(2),the universal covering group of SO(3), is isomorphic to so(3). The finite-dimensional, irreducible representations of eu(2) and SU(2) are in one-to-one correspondence. Thus Theorem (4.73) completely determines the finite-dimensional irreducible representations of SU(2). Furthermore, all irreducible representations of SU(2) are finite-dimensional. The odd-dimensional, irreducible representations of m ( 2 ) % 5 0 ( 3 ) give rise to ordinary, irreducible representations of SO(3). However, the even-dimensional, irreducible representations of 50(3) correspond t o what are called spinor represenfations of SO(3). These come from those irreducible representations of SU(2) with kernels K not containing the kernel N of the natural map of SU(2) onto its factor group SO(3), that is, t o the irreducible representations of SU(2) which do not map

+

l -: -:I1

onto the identity. The spinor representations of SO(3) play a significant role in physical applications. They are discussed at some length in Gel’fand and Sapiro (1952).

(4.75) REMARK. The relationship between the uses of the real and the complex numbers in Lie groups and Lie algebras is quite bewildering on the first encounter. The Lie algebras which appear in a natural way are real Lie algebras, but the ones which get analyzed are complex Lie algebras. Since one begins with Lie groups whose underlying space is either a subset of real Euclidean space or at least is locally like real Euclidean space, the Lie algebra of tangent vectors at the origin forms a real vector space. The situation arises since differential geometers frequently deal with real geometries. Unfortunately, the algebraic problems connected with the solution of polynomial equations make it easier to deal with complex Lie algebras and to classify the complex simple and semisimple ones rather than the real. Fortunately, one can go backwards and forwards among the complex and real Lie algebras. Given any real Lie algebra g of real dimension k with basis {x~,. . . , x,}, there exists a complex Lie algebra gc, called the complexijcation of g, of

306

4. Representation Theory of Special Groups

complex dimension k whose elements are all formal linear combinations x = c i x i ,ci E K . The operations are defined by

2

%(Ici xi) 1 ixcixi, c cixi + In i x i = 1 + =

[Ici xi 1 d j *

(Ci

Xi]

=

CI

E

K,

di)Xi

11 c,d,[x,, Xi].

This is, of course, 9 0K in an informal dress. Conversely, given a complex Lie algebra g of dimension k with basis {x,, . . . , xk), there exists a real Lie algebra gR of dimension 2k with basis {xi, _ . . .x , , i x , , ..., ix,) whose product is inherited from g itself. It sometimes proves possible to determine a real Lie subalgebra r of gR. such that g R assumes the form r -t ir. Such a real subalgebra r of gR is called a real form of g itself. Generally, a complex Lie algebra g may have several different forms. In the case of a complex, simple Lie algebra, among the real forms there is always precisely one called the cor?ipact r.ml,fi)rriz. The term is used because the compact real form is ;i real Lie algebra which is the Lie algebra of a compact Lie group. The classification problem for real Lie algebras can be thrown back into the classification problem for complex, but the task is a nontrivial one. Each simple real Lie algebra can be one of two mutually exclusive types: (i) simple complex Lie algebras regarded as real algebras, or (ii) real forms of simple complex Lie algebras. A good discussion of these matters can be found in Helgason (1962, see i n particular pp. 152-156), and a less formal one i n the work of Belinfante and Kolman (1972). Any finite-dimensional g-module V of a seniisimple Lie algebra g is also an 6-module for any Cartan subalgebra b of g. Therefore V decomposes as the direct sum of weight spaces of 5. Each weight I of b is an element of the space bR*, that is, each weight is a real linear combination of the roots of l). Consequently, there is an ordering of the finite set of weights of l) belonging to V. (4.76) D E I ' I N I T I O N . The largest among the weights of called the /zig/zrst w i g h t of on V.

l) belonging

to V is

Hence [I,* contains a set of highest weights of all of the irreducible gmodules. Given 0-modules U and V, their tensor product U 0V is also a 9-module. If u @ v E U @ V and x E g, then x(u Ov) = X U O V

(4.77)

+UOXV,

a rule which is a bit surprising. Let lL and A be the highest weights of the irreducible g-modules U and V with weight vectors u and v, respectively. Then one has X(U

8 V)

= I.(x)(u 0V)

+ A(x)(u @ V)= (L(X)+ A(x))(u 0 V)

4. The Representation Theory of Certain Matrix Groups

307

so that U @ V has A + A for a weight which proves to be the highest weight of U @ V. There is a standard process for finding an irreducible g-module X of U @ V with I + A as its highest weight. This irreducible submodule X is called the Cartan composition of U and V. Thus, if 1 and A are highest weights, then I + A is also a highest weight so that the set of bR* consisting of the highest weights of g is closed under addition. A highest weight I is said to be a basic highest weight of g if it is not the sum of two other highest weights. The number of basic highest weights of g is equal to the number of simple roots of g. If A = {A1, . . . , A,} is the set of basic highest weights, then the set {al, . . . , a,} of simple roots may be enumerated so that (4.78)

2{4, c r j } / { c r j , a j } = d i j .

This equation says that the Bravais lattice i?* generated by the basic highest weights is (up to scale factors) the reciprocal lattice of the Bravais lattice i? generated by the simple roots. The Bravais lattice 2*plays a fundamental role in the representation theory of g. Every weight A of any finite dimensional representation of g corresponds to a point of the lattice 2*. The highest weight A of an irreducible g-module V is an integral linear combination

A =nlAl

+ ... + n,I,

of the basic highest weights with nonnegative coefficients. Jacobson uses a slightly different terminology. We recall that each mi of the set (q,. . . , a,} of simple roots is associated with a triple {hi, e , , fi}. All of these triples together determine a set B = {hl, . . . ,h,} which is a basis of 9. A linear functional I of b* is said to be integral if &hi) is an integer for every hi E B. One has Ai(hj) = (hn, ,hj) = (hn,, 2h,

/{aj,~ j } = )

2{&, a j } / { u j ,aj) = 6 i j .

Thus the basic highest weights and all their integral linear combinations are integral. An integral function A is said t o be dominant if A(hi) 2 0, hi E B. The nonnegative integral linear combinations of the basic weights, and only they, are dominant. The fundamental theorem on $nite-dimensional irreducible modules of a semisimple Lie algebra over the complex numbers is given below. See Jacobson (1962, Chapt. VII).

(4.79) THEOREM. Let g be a semisimple Lie algebra over K . Let b be a Cartan subalgebra of g. There is a one-to-one correspondence between the dominant integral functionals on E, and the finite-dimensional irreducible modules V of g. Let A be the highest weight of ij which occurs in V. Then 1 is a dominant integral linear functional on 8. Conversely, given a dominant integral linear functional A on b, there exists a finite-dimensional irreducible g-module V, unique up t o isomorphism, with A as its highest weight.

308

4. Representation Theory of Special Groups

The highest weight A of an irreducible g-module V has a one-dimensional weight space. Let v be any weight vector of A. Then eiv = 0 for every simple raising operator of g. Such a vector is called an extreme vector. Given a nonzero extreme vector v in any finite-dimensional g-module V, there is a systematic method of using v to generate an irreducible g-submodule of V by means of the elementary lowering operators. The set 2l3 of all weights of lj occurring in any irreducible g-module V constitute a symmetrical set in the Bravais lattice i?* spanned by all integral linear combinations of the set A = {Al, . . . ,A,}. The set 2B is called the weight diagram of V. The weight diagram determines V up to isomorphism since the highest weight in \II! is sufficient for this purpose. There are a number of rules satisfied by the weight diagram 9 . 3 of V of which we mention two : (i) For any root CI of g, there exist nonnegative integers p and q such that if w E %3, then w + na E 1' 13 for any integer n with - p I n 5 q. This sequence of weights is sometimes called the cc-ladder through w. (ii) For any root CI of g and any weight w of %J3, the functional w' = w - 2{w, a}/(cc,C Y belongs }~ to !ID. Geometrically the weight w' is the reflection of w in the plane through the origin in f&* perpendicularto the direction of CY. The dimension of the weight space of w' is the same as that of the weight space of w. The group generated by all such reflections is called the Weylgroup. In summary, the weight diagram is invariant under the Weyl group. We turn to the consideration of some examples. The simple Lie algebras over K of rank two have been investigated vigorously from 1960-1970 by physicists. The papers of de Swart (1963) and of Behrends et al. (1 962), among others, contain rather detailed applications of the ideas of this section t o these particular algebras. There are three nonisomorphic simple Lie algebras over K of rank two. These are SL[, , 'B2,and G 2 in the notation introduced by Cartan. The simple Lie algebra 21, of rank m, m 2 1 , over K is characterized intuitively by the following facts: (i) the real Lie algebra 91mRis isomorphic t o the real Lie algebra $,,+l(K) of the Lie group SLm+l(K). (ii) the Lie algebra 91, itself is the complexification of its compact real form eii(m+ l), the real Lie algebra of the Lie group SU(m+ 1). The dimension of PI, over K is m(m + 2). The simple Lie algebra 212 of rank two is the complexification of 4 3 ) and has dimension eight over the complex numbers.

The simple Lie algebra 23, of rank m, m 2 2, over K is characterized intuitively by the following facts:

4. The Representation Theory of Certain Matrix Groups

309

(i) the real Lie algebra SrnRis isomorphic to the real Lie algebra so(2m + 1, K ) which is the Lie algebra of the Lie group SO,,+,(K) (complex orthogonal group). (ii) the Lie algebra Bmitself is the complexification of its compact real form so(2rn + 1, R ) which is the Lie algebra of the Lie group SO(2m + 1, R) (real orthogonal group). Thus 8,is the complexification of the real Lie algebra of the Lie group SO(5, R) which is the group of rotations of a fivedimensional, real Euclidean space. The complex dimension of Srnis m(2m 1) so that of 23, is ten.

-+

The Lie algebra 8,is called an exceptional Lie algebra. It does not arise as the Lie algebra of one of the classical groups. Its complex dimension is fourteen. There is a wealth of information on these kinds of things in Helgason (1962, Chapt. IX). The Dynkin diagrams of the three simple Lie algebras a,, 23, and 6, of rank two are (i) 212 (ii) 23,

- o---o -

D

(iii) 8,

The two circles in each of the three Dynkin diagrams show that a set

{a1, a,} of simple roots for each of these algebras contains two simple roots.

The single line in the diagram for 21u2implies that the angle between the simple roots a1 and c(, is 120". The double lines joining the circles of the Dynkin diagram for 23, imply that the angle between the simple roots a, and a, is 135". The triple lines in the case of (5, indicate that this angle is 150". Since each of the circles of (i) are of the same shade, the two simple roots of '? are I,of the same length. The fact that one is darker than the other in (ii) and (iii) means that one simple root is shorter than the other. For the case (ii) of 2 lines, the ratio of the lengths is J 2 : 1. For the case (iii) of 3 lines, the ratio of the lengths is J 3 : 1. There is a result of Brown (1964) stating that the sum of the squares of all the root vectors is equal to the rank of a semisimple Lie algebra. Since the dimension of %, is eight and its rank is two, the number of nonzero roots is six. They are all of the same length so that it follows from the result of Brown that the common length is J3/3. Similarly, one finds the length of the short root a1 of 8, is J6/6 and that of the long root a2 is ,/3/3. For 8,,one finds the short length is J3/6 and the long lj2. The three root diagrams are shown in Figs. (4.80)-(4.82).

310

4. Representation Theory of’Special Groups

(4.80)

(4.81)

(4.82)

- Y2

i

-r 2

t

3x1 + Y 2

L

21, i T 2 =

31, i

i,

2r,

=

i,

311

4. The Representation Theory of Certain Matrix Groups

The simple roots in each of these diagrams are labeled g1 and u 2 , and the basic weights are labeled 1, and 1,. Figure (4.80) is the root diagram of U , . The lengths and coordinates of the simple roots and basic weights are la1 I = J3/3,

121 I =

1/39

I a, I = $/3, \ A 2 1 = 1/3,

a1 = (J3/6,

lP),

4= (J%,

W),

a2 = (J5/6,

- 1/2).

= (J3/6,

- 116).

1,

Figure (4.81) is the root diagram of 23,. The lengths and coordinates of the simple roots and basic weights are

I a1I = J6/6, I 4 I = J3/6, I a2 I = J 3 / 3 , IA2 I = J6/6,

a1 = (,/3/6,

J3/6),

1, = (J3/6,0), x , = (0, - J3/3),

A2

= (J3/6,

- J3/6).

Figure (4.82) is the root diagram of 8,.The lengths and coordinates of the simple roots and basis weights are

( u l I = $/6,

al = (fiI12, 1/41,

1 1, 1

= &16,

Ic(2I

=

a2

I 1, 1

112,

=

112,

L2 = (J3/4,

4 = (J3/6,0), =(O,

- 1/21?

- 1/4).

We include a few weight diagrams of 41, to illustrate some of the general ideas. The reader is referred to the paper of de Swart (1963) for many others. (4.83) EXAMPLE. The weight diagrams for the basic (highest) weights L1 = (J3/6, 1/6) and ,I2 = (J3/6, - 1/6) are shown in Fig. (4.84),where 1, = wt =

312

4. Representation Theory of Special Groups

(\/3/6, 1/6). wZ = ( - J 3 / 6 . 1/6), w3 = (0, - 1/3), w,' = (0, 1/3), ,I2 = w2' = (,,h/6, - 1 /6), w3' = (-\/3/6, - 1/6). In Diagram (I). there are three examples of x-ladders. {wl. w3}, {wl,w2}, and {wz, w3}. The first of these is either an a1 or a - El-ladder. Considering it as an &,-ladder, one has p = 1 and q = 0 while the ladder consists of w, + n r , for - p = - 1 5 n 5 0 = q. The sides of the triangle are named with the positive root a of which the vertices constitute an a-ladder. All of the weights have a one-dimensional weight space so that each of the irreducible representations is three-dimensional. These two representations are famous in physics as the " quark " representations. We sketch two other weight diagrams to give the reader a feeling for the symmetry of them (Fig. 4.85). All weight diagrams for 91, tend toward triangular or i

(4.85)

vv

hexagonal shapes. In both cases, the horizontal and vertical axes pass through the center of the weight diagram. We wish to make a few remarks about the simple Lie algebra 913 of rank three over K which is the complexification of the real Lie algebra su(3). Since the Dynkin diagram of 213 is

a set { x i . x 2 , aj) of simple roots of the algebra contains three elements with a common length. The complex dimension of '$!I3 is 3(3 2) = 15 by our basic formula. I t follows that 213 has twelve nonzero roots which prove to be of equal length 4.The angle between a, and a2 is 120", that between ct2 and a3 is 120'. and that between a1 and a3 is 90". The nonzero root vectors are the twelve vectors which join the origin to the midpoints of the sides of a cube with center at the origin and orientation that of the coordinate axes. Geometrically, the picture is as shown in Fig. (4.86). There are many different sets of these twelve vectors which will serve as a set {x1,a 2 , a 3 } of simple roots. We have selected xl = ( p , 0, p ) , 2, = ( - p , p, 0), and a3 = (p, 0, - p ) to be specific. All of the positive roots are labeled in Fig. (4.86). The negative ones can easily be read from these. It is well known that the Bravais lattice spanned by these vectors is the face-centered cubic lattice. Therefore, the lattice

+

313

4. The Representation Theory of Certain Matrix Groups

(4.86)

spanned by the basic highest weights is the body-centered cubic lattice. We are speaking of the integral linear combinations of the simple roots and basic highest weights, of course. A summary of the data is:

I @I1 I 11 I I @21 I1, I

=

= $/4, =

1/29

= J2/4, =

113 I

112,

1/2,

= 4614,

@1 = (J5/4,0,

4 = (J5/8,

J2/4),

J%3,

42/81?

% = ( - J2/4, J2/4,0),

1,

= (0, J2/4,

@ 3 = (J2/4,0,

O), -J2/4),

d3 = (J2/8, J2/8, -J2/8).

The Lie algebras 81,. m 2 1, constitute one of the great families of classical Lie algebras. As such, there is a world of information available on them. We make a few brief remarks about an excellent summary of ltzykson and Nauenberg (1966). Here one can find detailed descriptions of certain Cartan subalgebras of \urnand the related elementary raising and lowering operators. In particular, there is a useful discussion of the relationships between the Lie algebra approach to the representations of BI, and SU(m+ 1) and the analysis of those of S U ( m + I ) determined by means of Young’s tableaux. We make two basic points: (i) Let I denote the family of irreducible representations of S U ( m + l ) which correspond to the irreducible representations of 91, arising from the basic highest weights of am. Then the set I of irreducible representations of

314

4. Representation Theory of Special Groups

SUfm+ 1) is the set of irreducible representations obtained from frames F with but a single column :

These tableaux determine the antisymmetric tensors of ranks 1, 2, . . . , m which are the representations of SU(m + 1) arising from the basic highest weights of %, . (ii) The representation and character theory of the symmetric groups can be used with considerable effectiveness in the analysis of the representations of SU(m+ 1). We recommend this paper to readers who have practical computations to make concerning the representations of SU(m + 1). PROBLEMS

1. List the ordered partitions of 2, 3, 4, and 6. 2. List the frames corresponding to the partitions of 2, 3, 4, and 6.

3. Write down all the tableaux which correspond to the frame ' . . 4.

(a) Write down all the tableaux which correspond to the frame ' . (b) Write out the action of (123) E S3 on each member of this list.

5.

Determine e(T) for each tableau T of Problem 4.

6 . Given

123 T, =45 6

and

125 T2 = 3 6 , 4

find a permutation s E S6 such that e(T,)s = se(T2).

7. Given the tableaux

show directly that e(T2)e(7',)= 0.

315

Problems

8. (a) Write down all the frames belonging to 6 . (b) Determine the number of elements in the conjugacy class of S6 belonging to each of the frames. (c) Write down a typical permutation from each class. 9. Carry out the details of Lemma (1.19) for the group S, and some tableau T corresponding to (2, l}. 10. Find the dimension of the irreducible representation of S8 corresponding

to the frame

.... : . * ’

11. (a) Find the dimension of the irreducible representation of S7 corre-

....

sponding to :* . (b) Write out the set of standard tableaux corresponding to this frame. 12. (a) Find the dimension of the irreducible representation of Slo

.....

corresponding to the frame : ‘ . (b) Determine the lengths of the principal hooks in the hook diagram. (c) Determine the standard box with entries the principal hook lengths. (d) Find the value of the character of the corresponding irreducible representation for the class K with cycle structure 2’ 3’.

13. Find the value of the character of the irreducible representation of S , corresponding to the partition (4, 3, 1, l} for the class K with cycle structure 1 2 3,. 14. Find the matrix of the transposition (23) corresponding to Young’s rational seminormal form for the irreducible representation of S5 determined by the partition (3, 2). 15.

Let

=)It 4

=I1

112 1/2 and e2 1/2 1,211 denote idempotent generators of the minimal left ideals J, and J, of the

algebra K , of 2 x 2 complex matrices. (a) Show that K2 = J, @ J, even though the product e1e2 is not zero. (b) Find a decomposition of the identity 1 of K2 as the sum f, fi of orthogonal idempotents which are generators of J1 and J, , respectively.

+

16. Consider the irreducible representation T of S , determined by the frame (a) (b) (c) (d)

...

.

Find the dimension of T. Write down the list of standard tableaux of the frame. Determine the set of permutations pi,li in this case. Work out the matrix of the transposition (23).

316

4. Representation Theory of Speciul Groups

17. (a) Write out the Cayley table of the group S, . Take the elements of A , as the first twelve elements. (b) Compute the irreducible representations of S, by the methods of Chapter 3. (c) Check your answers by the methods of Chapter 4. 18. Determine a set of orthogonal idempotents for the decomposition of the group algebra of S, from first principals without using the techniques of Young outlined in the text.

19. Use the ideas of this chapter to give a general description of the irreducible representations of S, without going into details. Let M be a left A-module of the algebra A. Prove that the dual space M* is a right A-module according t o Definition (2.12). 20.

21. Let A be a symmetric algebra and M be an A-module. Show that D = Hom,(M, M) is a complex algebra. 22. Complete the argument thatf [ x [ y is an element of D = Hom,(M, M).

23. Consider the action of S, on M = V 0 V 0 V, where {vl, .. . ,vs} is a basis B of V. Let s denote the permutation (132). (a) Find the action of s on the vector m = (vl 0 v2 8 v,) + 2(v2 0 v3 0 v4) + 3(v, 0v4 0VS). (b) Write out the specific relations between the coefficients of m and sm in this particular case. Let M = V 0V 0V, where V has a basis {vl, vJ. Let s = (123) belong to S, and let g E GL(V) have the matrix

24.

1; 4

with respect to the given basis. (a) Verify directly that s(mg) = (sm)g, where m is the basis tensor v1 Ov, 0 v 1 . (b) Carry out the same direct verification for other extended basis elements for M. Let M = V @ V 0V 0V, where {vl, v2 , v3} is a basis of V. (a) Determine the number of equivalence classes of the relation R associated with the space of symmetric tensors W of M. (b) Select a standard representative element from each set.

25.

Problems

317

26. Let M = U @ U @ U @ U, where {ul, uz} is a basis of U. Let T =

123 4

and let W be the space of tensors symmetric under the rows of T. (a) Sort out the index diagrams into equivalence classes. (b) Select a row-ordered index diagram J R from each class.

27. Assuming the space U of Problem 26 has a basis {q,. . .,u,}, determine the set of standard index diagrams of T in the case where the set {jl,. . . ,j4} consists of the integers 1 through 4.

28. Let 1234 T = 567 89

be the canonical tableau of the frame diagram

.... :: . Denote by D the standard index *

1234 567 89 (a) Find the images qD where q runs through the set (15), (26), (37), (26)(37) and (269), each contained in the column group Q(T). (b) Row-order the resulting index diagrams of (a) and observe that D < p f q D whenever the new index diagram is standard.

29. Use row transformations to transform the column-ordered diagram E presented below into a standard index diagram. 6 3 12 8 14 11 19 13 22 18 24 20

1 2 523 4 921 7 16 10 17 15

30. Let M denote the GL(V)-module V @ V @ V @ V where V has dimension three. (a) Determine the dimension d of the canonical GL(V)-module N(F, V), where F is the frame :. ' . (b) Write down the standard index diagrams for F and compare their number with the dimension determined in part (a).

318

4. Representation Theory of Special Groups

31. Let M be the GL(V)-module of Problem 30. (a) Find the dimensions of the irreducible representations of S, determined by the frames:

(b) Find the dimensions of the standard canonical GL(V)-modules N(F,. V), W,, V). N ( F , , V), NF4,V) and N ( F 5 , V). (c) Write down the symbolic decomposition of M into irreducible S,-modules. (d) Write down the symbolic decomposition of M into irreducible GL(V)-modules. (e) Show that the dimensions check for parts (c) and (d).

References

BEHRENDS, R. E., DREITLEIN, J., FRONSDAL, C., and LEE,W. (1962). Simple groups and strong interaction symmetries, Rev. Modern Phys. 34, 1-40, BELINFANTE, J. G. F., and KOLMAN, B. (1972). “ A Survey of Lie Groups and Lie Algebras.” SIAM, Philadelphia, Pennsylvania. BELINFANTE, J. G., KOLMAN, B., and SMITH,H. A. (1966). An introduction to Lie groups and Lie algebras with applications, SIAM Rev. 8, 11-46. BOERNER, .H. (1963). “Representations of Groups.” Wiley, New York. BROWN, G. (1964). A remark on semisimple Lie algebras, Proc. Amer. Math. Soc. 15, 518. BWRCKHARDT, J. J. (1947). “The Motion Groups of Crystallography.” Birkhauser, Basle, Switzerland. CHEVALLEY, C. (1946). “ Theory of Lie Groups,” Vol. 1 . Princeton Univ. Press, Princeton New Jersey. COXETER, H. S. M., and MOSER,W. 0.J. (1965). “Generators and Relations for Discrete Groups.” Springer-Verlag, Berlin and New York. CURTIS,C. W. (1956). Commuting rings of endomorphisms, Cunud. J. Math. 8, 271-292. CURTIS,C. W. (1958). Modules whose annihilators are direct summands, Pacific J. Math. 8, 685-691. CURTIS,C. W., and REINER,I. (1962). “Representation Theory of Finite Groups and Associative Algebras.” Wiley (Interscience), New York. DE SWART,J. J. (1963). The octet model and its Clebsch-Gordon coefficients, Rev. Mod. Phys. 35,916-939. DORING,W. (1959). The Strahldarstellungen der kristallographischen Gruppen, 2. Narurforsch. 14, 343-350. GEL’FAND, I. M., and SAPIRO,Z. YA. (1952). Representations of the group of rotations in three-dimensional space and their applications, Usp. Mat. Nuuk. ( N . S . ) 7, 3-1 17. 319

320

References

HAMERMESH, M. (1962). “Group Theory and Its Application t o Physical Problems.” Addison-Wesley, Reading, Massachusetts. HARTER, W. G. (1969). Algebraic theory of ray representations of finite groups, J. Math. Phys. 10, 739-752. HAUSNER, M., and SCHWARTZ, J. T. (1965). “Lie Groups; Lie Algebras.” Gordon and Breach, New York. S. (1962). “ Differential Geometry and Symmetric Spaces.” Academic Press, HELGASON, New York. HOCHSCHILD, G. (1965). “The Structure of Lie Groups.” Holden-Day, San Francisco. ITZYKSON, C., and NAUENBERG, M. (1966). Unitary groups : representations and decompositions, Rev. Mod. Phys. 38, 95-120. JACOBSON, N. (1962). “Lie Algebras.” Wiley (Interscience), New York. JANSSEN, T. (1973). “ Crystallographic Groups.” Amer. Elsevier, New York. KAPLANSKY, I. (1963). “Lectures on Modern Mathematics,” Vol. 1. Wiley, New York. G. F. (1957). Space groups and their representations, Solid SfnrePhys.5, 173-256. KOSTER, LoMoNr, J. S. (1959). “Applications of Finite Groups.” Academic Press, New York. LYUBARSKII, G. YA. (1960). “The Application of Group Theory in Physics.” Pergamon, Oxford. PONTRJAGIN, L. (1946). “Topological Groups.” Princeton Univ. Press, Princeton, New Jersey. ROTMAN, J. J. (1965). “The theory of groups. An Introduction.” Allyn and Bacon, Boston, Massachusetts. ROBINSON, G. DE B. (1961). “Representation Theory of the Symmetric Group.” Univ. Toronto Press, Toronto, Ontario. J. C. (1965). “Quantum Theory of Molecules and Solids,” Vol. 2. McCraw-Hill, SLATER, New York. J. R. (1971). “Computational Methods for Problems in Finite Solvable Group TALBURT, Theory ” Dissertation, Univ. of Arkansas, Fayetteville. Unpublished. E. P. (1959). “Group Theory and Its Application to the Quantum Mechanics of WIGNER. Atomic Spectra.” Academic Press, New York. WOLF,J. A. (1967). “Spaces of Constant Curvature.” McGraw-Hill, New York.

Index A A-basis, 20 A-homomorphism, 21,99 A-module, 19 A-submodule, 20 A-torsion free, 20 Abelian group, 3 internal direct sum, 18 irreducible representations of, 80-83 torsion free, 17 torsion subgroup of, 17 Abelian Lie algebra, 301 Action of permutations on tensors, 246-247 Adjoint, 49 linear transformation, 49 representation, 300 with respect to a nondegenerate form, 52 Affine motions, group of, 171 Afford a matrix representation, 68-69 Algebras axioms of, 84 Lie, 292-293 linear representations, 92 multiplication or vector product, 84 over K, 84 symmetric, 236 a-ladder definition of, 308 examples of, 311-312 Annihilators, left and right, 96 Antisymmetric tensor, 254

Arc, 283 Ascending chain condition (A.C.C.), 20, 31 on modules, 20 on vector spaces, 31 Associative bilinear form, 236 Automorphisms, 5 group, 5 inner, 5 ring, 13 Axioms of an algebra, 84 of a group, 2 of a Lie algebra, 292-293 of a module, 19 of a ring, 10 of a vector space, 23

B Balanced map, 158 Basic tensor, 145 Basic unit cell, 182 Basis of an abelian group, 17 dual, 48 existence of for vector spaces, 25 extended basis of a tensor product, 144 of a module, 20 natural, 112 standard or natural, 88 standard, space of r tuples, 27 symmetry, 112 symmetry adapted, 112, 154

321

322

Index

of a topology, 282 of a vector space, 25 Bijection, 55 Bilinear forms associative, 236 mapping, 47, 145 nondegenerate, 47 symmetric, 236 Bimodule, 159 Body centered cubic lattice, 194 Box, standard, 226 Bravais class, 175 Bravais lattice. 175

C c z 0c,

Cayley table of, 125 character table of, 125 Canonical bilinear mapping, 145 Canonical form, 4 6 4 7 Jordan, 46-47 of a matrix, 39 Canonical GL(V)-module, 264 Canonical linear transformation, 35 Canonical v-dimensional vector space, 27 Canonical representation, 264 Canonical tableau, GL(V)-submodule corresponding to, 267 Cartan composition, 306-307 Cartan matrix, 303 Cayley-Hamilton theorem, 42 Cayley’s theorem, 62 Cayley tables, 3 of Cz (3 Cz , 125 of C, :I) C, , 83 of Ca iv) 5’3,208 of C 7 , 206 of C p , 82 of the dihedral group D 6 , 210 of the dihedral group D7, 209 of group 7 order 16 (M. Hall), 150 of the octahedral group 0, 181 of the quarternions Q, 125 of S3, 87, 124 of the symmetries of the tetrahedron, 181

of the tetrahedral group, 186 of (2, 2, 3) (2s metacycfic), 135

Cells, primitive, 174 Center, 113 of an algebra, 1 I5 of a group, 113 Central idempotents, expressions for, 134 Chain, 20-21 factors of,20-21 of submodules, 20-21 of subspaces, 31-32 Character tables, 117, 121 of Cz 0Cz , 125 of Cz 0C, 156 of group 7 order 16 (M. Hall), 151 of the quarternions Q, 126 of SB , 124 Characteristic of a field, 16 of a ring, 16 Characteristic polynomial, 42 Characteristic subspace, 44 Characters, 115-1 16 afforded by a representation, 116 of a direct sum of a representation, 116-117 orthogonality relations, 117, 120 principal, 123 Characters, standard notation, 117, 121 Class function, 116 Clebsch-Gordon coefficients, 154 Clifford’s theorem, 163 Column group, 217 Commutative group, 3 Commutative ring, I I Commutators, 134 Compact, 284 Compact real form, 306 Complementary subspace, 29 Complex general linear group, 278 Complex of a group, 2 Complex special linear group, 279 Cornplexification of a Lie algebra, 305-306 Components homogeneous, 164 ring tensor, 260 simple, of an algebra, 106 Composition of functions, 55 Composition factors, 21 Composition series, 21 factors of, 21 length of, 21

323

Index of a module, 21 of vector spaces, 31-33 Compound lattice, 174-175 Congruent, 218 Conjugacy classes, 6 of a direct product, 155-156 of an element, 6 number of elements in for S, ,223 Conjugate element, 6, 59 Conjugate subgroup, 6 Conjugates complete set of conjugate representations, 163 module, 162 Connected, 284 Continuous map, 282-283 Coset, 6 left, 6 right, 6 ring, 13 Crystallographic point groups, 176 Crystallographic space groups, 176 Crystals diamond, 183-185 zinc blende, 183-185

D Decomposable, 65 Degree of a matrix representation, 66 of a representation, 66 Derived group, 134 Derived series of a Lie algebra, 299 Descending chain condition (D.C.C.), 21, 31 on modules, 21 on vector spaces, 31 Diagrams index, 253 node, 224 Diamond crystal, 183-185 Dictionary order, 149 Dihedral group, 4,209,210 Dimensions, 27 of a component of KG, 107 of an integral representation, 224 of a vector space, 27

Direct sum of abelian groups, 18 vector spaces, 28 Disjoint KG-modules, 166 Division ring, 11 of quarternions, 12 Divisors of zero, 11 Domain, integral, 11 Dominant integral functional, 307 Dual basis, 48 Dual space, 34 Duality between KS,-modules and GL(V)-modules, 267-268

E Eigenspace, 40 Eigenvalue, 40 Element, representative, 6 Endomorphisms, 5 group, 5 ring, 13 Energy levels, 148 Energy operator H, 147 Enveloping algebra, 297 Epimorphisnis, 5 group, 5 ring, 13 Equivalent left ideals, 92, 99 Equivalent matrices, 39 Equivalent, representation spaces, 77 Equivalent representations, 66 Essentially idempotent, 217 Euclidean motions, group of, 171 Exceptional Lie algebra, 309 Exponentiation abelian group, 17 nonabelian group, 3 Extended basis, 144 External direct product, 8 External direct sum, vector spaces, 28 External semidirect product, 172 Extreme vector, 308 F

Face-centered cubic lattice, 182, 183, 194 Factor group, 6 Factor module, 20

324

Index

Factor representation, 76 Factor ring, 13 Factor set, 202 Factor space, 30 Factors of a chain of modules, 20-21 Field, 1 1 finite, 16 Finite generated abelian group, 17 basis for, 17 Finitely generated group, 17 First brillouin zone, 195-197 First isomorphism theorem of groups, 9 of rings, 14 Form, 55 bilinear, 47, 145 positive definite, hermitian symmetric, 49 Frames, 215 Frattini subgroup, 139 Free abelian group, 17 Basis of, 17 Free vector spaces, 24 Frobenius’ theorem, 161 Full linear group, 65 Full matrix group, 65 Functional, linear, 34, 47 Functions, class, 116 Fundamental theorem, 7 of abelian groups, 18 of group homomorphisms, 7 of ring homomorphisms, 14 vector spaces, 27

G Galois fields, 16 Generators, 17 idempotent, 96 of the ideal L, 96 Glide reflection, I73 Gram-Schmidt theorem, 51 Group algebras, 85 scalar multiplication, 85 structure constants of, 127 vector product, 86 vector sum, 85 Groups, 2 abelian, 3 of affine motions, 171 automorphism, 5

axioms, 2 column, 217 commutative, 3 complex general linear, 278 complex of, 2 complex special linear, 279 complex unimodular, 279 crystallographic point, 176 cyclic, 3 derived subgroup, 134 dihedral, 4 endomorphism, 5 epimorphism, 5 of Euclidean motions, 171 factor group, 6 finitely generated, 17 Frattini subgroup, 139 full linear, 65 full matrix, 65 homomorphism, 5 inertia, 200 isomorphism, 5 Lorentz, 52 monomorphism, 5 multiplication, 2 octahedral, 4, 179 order, 2 real general linear, 279 of real orthogonal matrices, 279 real special linear, 279 real unimodular, 279 of rigid motions, 167-173 of rotations, 279 row, 217 standard matrix, 278-279 subgroup, 2 of symmetries of the cube, 177 of symmetries of the tetrahedron, 177 table, 3 tetrahedral, 179 torsion subgroup, 17 transformation, 69 translation, 168, 175 unitary, 52, 279 unitary unimodular, 279

H Hamiltonian operator H, 147 Hamilton’s quaternions, 12

Index

325

Highest weight, 308 High symmetry vectors, 196-198 Holohedry, 175 Homogeneous components, I64 Homomorphic image, 5 group, 5 ring, 13 Homomorphisms, 5 A-homomorphism, 157 group, 5 natural, 7 ring, 13 Hook, 224 Hook graph, 224 h-surface, 285 1

Ideal, 12 left, 12, 90 of a Lie algebra, 299 minimal, 91 nilpotent, 105 nontrivial, 90 prime, 13. right, 12,90 solvable, 299 trivial, 90 two-sided, 90, 12 Idempotent, essentially, 21 7 Idempotent generator of the ideal L, 96 Idempotents, 96 central, expressions for, 134 in the ring of n x n matrices, 137 orthogonal, 97 primitive, 97 Identity element, 2 of a group, 2 of a ring, 11 Image, 5 of a function f, 38 homomorphic, 5 Improper motions, 171 Inclusion map, 283 Indecomposable, 65 Independent KG-modules, 166 Index diagrams, 253 row-ordered, 255 standard, 256

Index symbols, 245-246 Induced matrix representations, 160 Induced module, 159 irreducible, 166-167 Induced representations, 159-160 Inertia group, 200 Inertia subgroup, I64 Injection, 55 Inner automorphism, 5, 59 Inner product, 49 Inner product space, 49 Integral domain, I 1 Integral linear functional, 307 Integral representations, 25 1-278 dimension of, 224 Interior vector of the Brillouin zone, 202 Internal direct product, 9, 59 Internal direct sum, vector spaces, 28 Internal product, 9 Internal semidirect product, 172, 173 Invariant subgroup, 6 Invariant subspace, 41, 45 Inverse, 55, 56 left, 55 right, 55 Inversion, 173 Irreducible, 65 Irreducible induced modules, 166-167 Irreducible module, 20 Irreducible representation space, 67 Irreducible representations, 57 book notations for matrices of, 57 two-dimensional, group 7 order 16 (M. Hall), 151, 152 of T d , 189-193 of the tetrahedral group, 186-188, 190 Isomorphism group, 5 local, 296 ring, 13 Isotopy subgroup, 170

J Jacobi identity, 292 Jordan canonical form, 4 6 4 7 Jordan-Holder theorem, 33

326

Index K

K-module, 24 K-space, 24 k-vectors, 196 Kernel, 5 of a group homomorphism, 5 Killing form, 302 Kronecker product, 149-150

L Lagrange, theorem of, 3, 58 Lattices body-centered cubic, 194 Bravais class, 175 compound, 174-175 dual or reciprocal, 194 face-centered cubic, 182, 183, 194 simple, 174-175 Left A-module, 157 Left annihilator, 96 Left coset, 6 Left factor niodule, I57 Left ideal, 12, 90 Left inverse, 55 Left multiplication, 86 defined by n, 56 Length of a vector, 50 Lie algebras, 292-293 abelian, 301 ideals of, 299 nilpotent, 299 semisimple, 299 simple, 299 solvable, 299 split, 303 subalgebra of, 299 Lie product, 292 Linear combination, 25 Linear functional, 34, 47 integral, 307 Linear mapping, 34 Linear operator, 34 Linear representation, 66 of an algebra, 92 Linear transformation, 34 adjoint, 49 adjoint with respect to a form, 53 characteristic polynomial of, 42

characteristic subspace, 44 Hilbert space adjoint, 53 minimal polynomial of, 42 nilcyclic, 46 nilpotent, 46 null, 38 self-adjoint with respect to a form, 54 semisimple, 47 Linearly dependent, 25 Linearly independent, 25 Local isomorphism, 296 Locally compact, 284 Lorentz group, homogeneous, 52 Lower central series of a Lie algebra, 299

M Maps balanced, 158 continuous, 282-283 inclusion, 283 linear, 34 Maschke's theorem, 13 Matrix of a linear transformation, 36 Matrix representations, 66 afforded by a linear representation, 68, 69 degree, 66 equivalent, 66 format in this book, 66 induced, 161 of a Lie algebra, 297 Matrix units, 110 of a minimal two-sided ideal, 112 Maximal ideal, 12 Minimal ideal, 12, 91 left, 12, 91 right, 91 two-sided, 91 Minimal module, 20 Minkowski space, 52 Modules, 19 A-torsion free, 20 axioms of, 19 basis, 20 composition factors of, 21 composition series of, 21 conjugate, 162 disjoint, 166

327

Index

factor module of, 20 generators of, 20 induced, 159 irreducible, 20 left A-module, 157 minimal, 20 product in, 19 right A-module, 157 subduced, 161 submodule of, 20 unitary, 197 Monomorphisms, 5 group, 5 ring, 13 Multiplication, 2, 10 in a group, 2 right, 98 in a ring, 10 N

Natural basis, 88, 112 Natural homomorphism, 7 Nearest neighbors, 184 Neighborhood, 283 Nilcyclic linear transformation, 46 Nilpotent ideal, 105, 235 Nilpotent Lie algebra, 299 Nilpotent linear transformation, 46 n-fold basic tensors, 146 n-fold tensor products, 146 Nodes, 216, 224 diagram, 224 principal, 224 Nontrivial ideal, 90 Nontrivial vector space, 25 Norm of a vector, 50 Normal set of vectors, 50 Normal subgroup, 6 Nucleus of a module over a symmetric algebra, 238-239 Null linear transformation, 38 Nullity of a linear transformation, 38

0 Octahedral group, 4, 179 1-function, 55

1-representation, 123 One-parameter subgroup, 289-291 Open set, 282 Operator, linear, 34 Order, 2 of an element, 3 of a group, 2 of an orbit, 164 Order, dictionary, 149 Orbit of a subgroup, 164 Orthogonal set of vectors, 50 Orthogonality relations, 117, 120, 122 Orthonormal set of vectors, 50 Outer tensor products, 155

P Partitions, 215 Permutations, symmetries of tetrahedron as, 180 Physics, 147 Clebsch-Gordon coefficients, 154-155 energy operator, 147 Hamiltonian operator, 147 Pierce decomposition, 108-1 10 Point groups, 176 of first kind, 176 of second kind, 176 Polynomial, 42 characteristic, 42 minimal, 42 monic, 42 Positive definite, hermitian symmetric form,

49

Positive root, 302 Prime ideal, 15 Primitive cells, 174 Principal character, 123 Principal left ideal, 96 Product, 6 of characters, 150-152 of complexes, 6 external, 8 external direct, 8 inner, 49 internal direct, 9 Kronecker, 149-150 Lie, 292

328

Index

module, 19 of subgroups, 59 vector, space, 23 Projective representation, 201 Pure translations, 171

Q Quarternion group Q Cayley table of, 125 character table of, 126 Quarternions, 12 Quotient topology, 284

R Radical of an associative algebra, 105, 235 Rank, 38 column, of a matrix, 41 of a Lie algebra, 301 of a linear transformation, 38 of a matrix, 41 row, of a matrix, 41 Real form of a Lie algebra, 306 Real general linear group, 279 Real special linear group, 279 Reducible, 65 Reflection, 173 Regular A-module over a symmetric algebra, 239 Regular representations, 84-89 of a group algebra, 94-95 Relative topology, 283 Representations, 66 of abelian groups, 80-83 adjoint, 300 alternating, 123 belonging to a normal subgroup, 123 belonging to a simple component, 128 by means of structure constants, 126-1 33 Canonical, 264 of cyclic groups, 79 on coset spaces, 136 definition of, 66 degree, 66

equivalent, 66 factor, 76 induced, 156, 159-160 integral, 267 integral of GL(V), 251-278 irreducible, of abelian groups, 80-83 irreducible, of C, 0C, , 83 irreducible, of the cyclic group C8, 82 Lie algebra, 297 linear, 66 matrix, 66 I-representation, I23 projective, 201 rational, 280-281 regular, 84-89 self-conjugate, 187 semirational, 281 semirational integral, 28 I spinor, 305 star of, 205 subduced, 161 of symmetric group, 214-234 unitary, 120 Representation, space, 66 Representative element, 6 Residue class ring, 13 Restriction of a linear transformation, 45, 74 Right A-module, 157 Right annihilator, 96 Right coset, 6 Right ideal, 12, 90 Right inverse, 55 Right multiplication determined by A, 61 Right translation (multiplication), 98 Rigid motions, groups of, 167-173 Rings, 10 addition in, 10 automorphism, 13 axioms of, 10 characteristic, 16 commutative, 11 division, 11 epimorphism of, 13 factor, 13 homomorphism of, 13 ideal of, 12 isomorphism, 13 maximal ideal of, 12 minimal ideal of, 12

329

Index

monomorphism of, 13 multiplication in, 10 subring of, 12 Ring tensor component, 260 Root spaces, 301 Roots of a Lie algebra, 301 positive, 302 simple, 301 Rotation reflection, 173 Rotations, 168-170 octahedral group as, 180 Row group, 217 Row-ordered index diagrams, 255

S Scalar multiplication, 16 Scalar product, 16 module, 19 vector space, 23 Schur’s lemma, 72 Schwarz’s inequality, 50 Screw axis, 173 Self-adjoint with respect to a nondegenerate form, 53-54 Self-conjugate element, 6 Self-conjugate representations, 187 Self-conjugate subgroup, 6 Semidirect products external, 172 internal, 172, 173 Semisimple associative algebra, 105, 235 Semisimple Lie algebras, 299 Semisimple linear transformation, 47 Similar matrices, 40 Simple associative algebra, 105, 235 Simple closed arc, 284 Simple components, 106 representations belonging to, 128 Simple lattice, 174-175 Simple Lie algebras, 299 Simple lowering operators, 303 Simple raising operators, 303 Simple roots, 301-302 Simple tensor, 145 Simply connected, 284 Simply reducible groups, 154

Space, dual, 34 Space, Minkowski, 52 Space, of an irreducible representation, 67 Space, representation, 66 Space, tangent, 168 Space groups, 176 of diamond crystal, 177 equivalent, 176 nonsymmorphic, 177 point groups of, 176 symmorphic, 177 translation groups of, 176 of zinc blende crystal, 177 Spinor representations, 305 Split Lie algebra, 303 Standard basis, 88 of space of r-tuples, 27 Standard box, 226 Standard matrix groups, 278-279 Standard primitive cell, face-centered cubic lattice, 183 Standard primitive idempotents, 226 Standard subgroups, 279 Standard tableau, 225, 256 Star of a representation, 205 Structure constants, 127 representations by means of, 126-133 Subalgebra of a Lie algebra, 299 Subbasis of a topology, 282 Subduced module, I61 Subduced representation space, 161 Subduced representations, 161 Subgroups, 2 Frattini, 139 generated by K, 2-3 inertia, 164 invariant, 6 isotopy, 170 normal, 6 torsion, 17 translation, 168, 175 Submodules, 20 Subring, 12 Subspace characteristic, 44 complementary, 29 invariant, 41 Sum, 28 of vector spaces, 28 Surjection, 55

330

Index

Symmetric algebras, 236 Symmetric bilinear form, 236 Symmetric group action on a tensor product, 244-247 of degree n, 56 number of elements in a conjugacy class, 223 Symmetric tensors, 248-249, 253-254 Symmetries of the cube, 177-181 Symmetry basis, 112 in the columns, 264 in the rows, 264 Symmetry adapted basis, 112, 154 Symmetry group of the Hamiltonian H , 147 of an operator, 147

T Table, 3 Cayley, 3 character, I 17, 12 I group, 3 Tableau, 215 canonical, 215 standard, 225, 256 Talburt, method of, 210-21 1 Tangent space, 168 Tensor products of A-modules, I58 of characters, 150-1 52 of linear transformations, 146 n-fold, 146 outer, 155 of representations, 146-147 of vector spaces, 144 of vector spaces, n-fold, 146 Tensors antisymmetric in the columns, 253 basic, 145 basic, n-fold, 146 index symbols for, 245-246 symmetric in the rows, 253 Tetrahedral group T, 179 Cayley table, 186 Theorem Cayley, 62

Clifford, 163 Frobenius, 161 Lagrange, 3, 58 Maschke, 73 Wedderburn, 235 Topology, 282 quotient, 284 relative, 283 Translation, right, 98 Torsion free, 17 Torsion subgroup, 17 Transformation, linear, 34 Transformation groups, 69 representation afforded by, 69 Translation group, 168, 175 Translations, 168 nonprimitive, 175 pure, 171 subgroup, 168, 175 Trivial ideal, 90 Two-sided ideal, 12, 90

U Unit cell, basic, 182 Unitary basis, 51 Unitary group, 52, 279 Unitary matrix, 52 Unitary modules, 197 Unitary representations, 120 Unitary unimodular group, 279 Units of a ring, 15 Universal covering group, 296

V Vectors high symmetry, 196-198 interior to the first brillouin zone, 202 wave, 196 Vector spaces, 23, 49 axioms of, 23 bases of, 25 dimension, 27 direct sum of subspaces, 28

331

Index

factor space of, 30 inner product, 49 linear independence in, 25 rules of computation, 24 sum of subspaces, 28 Vector sum, group algebra, 85 Viergruppe, 125 W

Wave functions of the hydrogen atom, 148 Wave vectors, 196 Wedderburn’s theorem, 235 Weight basic highest, 307 of a Lie algebra, 300 Weight diagrams definition of, 308 example of, 310

A 8 5 C 6 D 7

E B

F 9 G O

H

I

1 2 ) 3

Weight space, 300 Weyl group, 308 Wigner-Zeitz cell, 196, 198

Y Young’s algorithm for representations, 230 Young’s integral representations, 223 Young’s rational normal form, 228

Z

Zero, divisor of, 11 Zinc blende crystal, 183-1 85 2-module, 19