Polynomial Representations of GL_n

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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

830 J. A. Green

Polynomial Representations of GL n

Springer-Verlag Berlin Heidelberg New York 1980

Author James A. Green Mathematics Institute University of Warwick Coventry CV4 7AL England

AMS SubjeCt Classifications (1980): 20 C30, 20 G05

ISBN 3-540-10258-2 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-10258-2 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Vertag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

These lectures were given at Yale University semester

1980, while the author was receiving

and hospitality Foundation

of Yale University

versity of Essen,

Germany.

the kind support

and the National

(NSF contract MCS 79-04473).

given at the University of Warwick,

during the spring

Science

Earlier versions were

Engl.and and at the uni-

POLYNOMIAL REPRESENTATIONS OF

GL

n

Table of Contents

Chapter i.

Introduction

Chapter 2.

Polynomial representations of GL (K) : The Schur algebra

18

2.1

Notation, etc.

18

2.2

The categories MK(n),MK(n,r)

19

2.3

The Schur algebra SK(n,r)

21

2.4

The map

23

2.5

Modular theory

25

2.6

The module E mr

27

2.7

Contravariant duality

31

2.8

AK(n,r) as KF-bimodule

34

Weights and characters

36

3.1

Weights

36

3.2

Weight spaces

37

Chapter 3.

e:KF+SK(n,r)

3.3

Some properties of weight spaces

38

3.4

Characters

40

3.5

Irreducible modules in ~_(n,r) K-

44

The modules DX, K

5O

4.1

Preamble

50

4.2

~-tableaux

50

4.3

Bideterminants

4.4

Definition of

Chapter 4.

51

DX, K

53

Vt

4.5

The basis theorem for DE, K

55

4.6

The Carter-Lusztig lemma

57

4.7

Some consequences of the basis theorem

59

4.8

James's construction of Di, K

61

The Carter-Lusztig modules VI, K

65

5.1

Definition of V%, K

65

5.2

VI, K is Carter-Lusztig's "Weyl module"

65

5.3

The Carter-Lusztig basis for V~, K

68

Chapter 5.

5.4

Some consequences of the basis theorem

70

5.5

Contravariant forms on VI, K

73

5.6

Z-forms of V~,Q

77

Representation theory of the symmetric group

80

6.1

The functor

80

6.2

General theory of the functor f:mod S ÷ mod eSe

83

6.3

Application I. Specht modules and their duals

88

6.4

Application II. Irreducible KG(r)-medules, char K = p

93

6.5

Application III. The functor MK(N,r) + MK(n,r) (N L n )

102

Application IV. Some theorems on decomposition numbers

107

Chapter 6.

6.6

f:MK(n,r) ÷ mod KG(r)(r ~ n)

Bibliography

113

Index

117

§I.

Introduction Issai Schur determined the polynomial replesentations of the complex

general linear group in 1901.

GLn(¢)

in his doctoral dissertation [S], published

This remarkable work contained many very original ideas, developed

with superb algebraic skill.

Schur showed that these representations are

completely reducible, that each irreducible one is "homogeneous" of some degree

r ~ 0

(see 2.2), and that the equivalence types of irreducible

polynomial representations of

GLn(~) ,

of fixed homogeneous degree

in one-one correspondence with the partitions not more than

n

parts.

see 3.5).

~I

in

n

kn)

of

are

r

into

Moreover Schur showed that the character of an

irreducible representation of type function

k = (~i'''"

r,

k

is given by a certain symmetric

variables (since described as a "Schur function";

An essential part of Schur's technique was to set up a correspond-

ence between representations of

GL (~) n

of fixed homogeneous degree

and representations of the finite symmetric group

G(r)

on

r

r,

symbols,

and through this correspondence to apply C. Frobenius's discovery of the characters of

G(r)

[F, 1900].

This pioneering achievement of Schur was one of the main inspirations for Hermann Weyl's monumental researches on the representation theory of semi-simple Lie groups [We, 1925, 1926].

Of course Weyl's methods,

based on the representation theory of the Lie algebra of the Lie group and the possibility of integrating over a compact form of

F,

F,

were very

different from the purely algebraic methods of Schur~s dissertation; in particular Weyl's general theory contained nothing to correspond to the symmetric group GLn(~),

G(r).

In 1927 Schur published another paper IS'] on

which has deservedly become a classic.

"dual" actions of

GLn(¢)

(see 2.6) to rederive

and

G(r)

on the

In this he exploited the

~ th

tensor-space

E~r

all the results of his 1901 dissertation in a new

and very economical way.

Weyl publicized the method of Schur's 1927 paper,

with its attractive use of the "double centralizer property",in his influential book "The Classical Groups" [We',1939]. in Chapters 3(B) and

4

of that book has become the

polynomial representations of

GL (~) n

standard treatment of

(and, incidentally, of

representation theory of the symmetric groups this explains the comparative neglect

In fact the exposition

G(r)),

Alfred Young's

and perhaps

of Schur's work of 1901.

I think

this neglect is a pity, because the methods of this earlier work are in some ways very much in keeping with the present-day ideas on representations of algebraic groups. in part

It is the purpose of these lectures to give some accounts,

based on the ideas of Schur's 1901 dissertation, of the polynomial

~epresentatlons of the general linear groups infinite field of arbitrary characteristic.

GLn(K),

where

K

is an

Our treatment will be "elementary" use algebraic interesting

in the sense that we shall not

group theory in our main discussion.

But it might be

to indicate here some general ideas from the representation

theory of algebraic

groups

inverse is not important

(or of algebraic

in this context),

semigroups,

since the group

which are relevant

to

our

work. Let

F

be any semigroup

associative multiplication) field.

A representation

space over

K)

~(IF) = ~V'

•

is a map

for all

identity map on

V.)

: KF-+ EndK(V);

(i.e.

is a set, equipped with an

with identity element of

F

on a

K-space

• : F ~EndK(V )

g, g' E F.

KF

T

~,

and let

V

(i.e.

which satisfies

(For any set

We can extend

here

F

V,

K

a vector T(gg')

= ~(g)~(g'),

we denote by

~V

linearly to give a map of

is the semlgroup-algebra

of

be any

F

the

K-algebras

over

K,

whose

We can make

KF

act

elements are all formal linear combinations

=

whose support on

V

by

Z g~F

supp K = {g ~ F : K

Kv = ~(~)(v) (V,~),

(V',~')

is, by definition,

bijective

representation KF-modules;

or simply

V. a

~'(g)f = f~(g)

is called a

a right

# 0}

g

A

K-map

and thereby get a left KF-module,

f : V ~ V'

for all

g E r.

Kr-modules

(i.e, A

f

(V,~),

is a linear map)

KF-map which is

or an equivalence

One has analogous

KF-module

is finite.

KF-map between such

KF-isomorphism,

~, ~'.

~ K, g

(~ ~ K~, v ~ V),

denoted

which satisfies

K Kgg,

definitions

between the

for right

can be regarded as a pair

(V,~)

where

T

: F ~ EndK(V)

i.e.

~(gg')

is an a n t i - r e p r e s e n t a t i o n

= ~(g')~(g)'

The set

KF

g ~ f(g)f'(g),

element If

~

s ~ F

of

KF

and

are defined

defined

takes each

f ~ KF

K-algebra

Lsf ,

Ls, R s

maps

homomorphism)

a representation

F

rifht

Thus

KF

KF-module

that if

s ~ F

on

and

(using

L).

f ~ KF

takes

that these actions

f ~ K F. f ® f'

: F ~K

KF

while

F.

for all

The identity 1K

of

f

of

K.

by

s

g ~ F.

In particular,

L : s ~ L

K-algebra L s, R s

both

R : s ~ Rs

gives

gives an anti-represent-

s

KF- module

(using

R)

actions by

and a o,

so

we write

and

f o s = L f. s (sof) o t = s o(fot) K F ® K F ~ K FxF

(®

to the function m a p p i n g s, t ~ F.

structure

on

and

F

for all means

F x F ~ K

s, t E F

~)

which

by

This linear map is injective

, and we use it to identify

8 : K F -* K F×F

to

given by

We denote both module

com~ute:

(f, f' ~ K F)

The semigroup

of

into itself and is a

K F ~ K F.

The~e is a linear map

(s,t) ~ f(s)f'(t),

g

is defined

It is easy to check that

K F,

s o f = R f s Notice

ff'

("point")

can be made into a left

and

e.g.

K-algebra,

to the identity element

Rsf

V,

g, g' E F.

R f : g ~ f(gs), s

EndK(KF ) .

ation.

K-space

is s commutative

"pointwise",

g ~ F

belong to the space of

f : F ~K

on the

then the left and r i g h t translates

to be the maps

Each of the operators

F

for all

for every element

L f : g ~ f(sg), s

map (i.e.

~ ~V

of all maps

with algebra operations take

~(IF)

of

KF @ KF

with a subspace of

gives rise to two maps

~ : K F -~ K~

K FxF.

as follows: Both

A,e

if are

f £ K F,

then

&f : (s,t) -~ f(st),

K-algebra maps.

and

~(f) = f(IF)-

We shall say that an element

f ~ K F is

finitary, or is a representative function, if it satisfies any one of the conditions

FI, F2, F3

<see e.g.

[H, Ch. 2]).

FI.

The left

below:

these three conditions are in fact equivalent

KF-submodule

KFof

generated by

f

is finite-

dimensional. F2.

The right

KF-submodule

foK/"

generated by

f

is finite-

dimensional. F3.

Af ~ K F ® K F.

(h

This means that there exist elements

fh' fh E K F

runs over some finite index set) such that

(Is)

Af =

[ih fh ® fh' "

This equation is equivalent to the system of equations

(ib)

f(st) =

~ h

fh(s)f{(t),

all

s, t 6 F.

It is also equivalent to each of the following systems

(ic)

tof =

Z f~(t)fh, h

all

t ( F,

fos =

~ fh(s)f~, h

all

s 6 F.

or (id)

The set

F = F(K F)

of all finitary functions

f: F ~ K

is a

K-bialgebra (see [Sw] for the definitions of coalgebras and bialgebras). It is a

K-subalgebra of

£F ~ F ~ F

K ~,

(this means that if

and is also closed to f

A

in the sense that

is finitary, then the functions

fh' fh F ,

in (la) can be chosen to be themselves finitary).

equipped ~ith the maps

these two structures on fact that

A

and

c

A : F ÷ F ~ F , ~ : F + K ,

F ,

of algebra and coalgebra,

are both

K - a l g e b r a maps

F i n i t a r y functions on

F

d i m e n s i o n a l r e p r e s e n t a t i o n s of on a f i n i t e - d i m e n s i o n a l V ,

z(g)v b = gv b =

Suppose

V °

rab(g)

~ K .

E rab(g)v a , a eB

The functions

c o e f f i c i e n t functions of

If

K-coalgebra;

are linked by the ~,

T

p. i ~ ) .

is a r e p r e s e n t a t i o n of

{vb : b e B)

T ,

or of the

KF-module

for

rab

is a

F

K-basis of

V .

g c F , b e B ;

: F ~ K (a,b g B)

or of the

of these functions is a subspace of T ,

is a

we have equations

(le)

here

K-space

appear as coefficient functions of finite-

F .

K-space

(see

The

KF

KF-module

are called

V = (V,T)

.

The

K-span

called the coefficient space (1)

We denote this space by

cf(V) =

of

E K.rab a,b

;

it is e l e m e n t a r y to verify that it is independent of the choice of the basis {vb} . i.e.

The m a t r i x

R = (tab)

R(gg') = R(g)R(g')

Kronecker delta).

gives a m a t r i x r e p r e s e n t a t i o n of

, R(IF) = (~ab)

for all

tab

(if)

~ r ac ~ rcb , S~rab) ' = 6ab ' eEB

The m a t r i x

,

R = (rab)

C = cf(V)

all

a, b c B .

is sometimes called an "invariant m a t r i x "

are finitary~ hence that

~,

p.140].

it follows that all the coefficient functions cf(V)

is a subspace of

is a s_ubcoalgebra of

As a m a t t e r of fact, every finitary f u n c t i o n space of some f i n i t e - d i m e n s i o n a l

(I)

is the

viz°

From the first equations

shows that

(~ab

These conditions translate into conditions on the

coefficients

Arab =

g, g' c F

F ,

KF-module

F ,

F = F(K F) . i.e.

f : F ÷ K V ;

that

But AC~

rab

(If)

also

C ~ C

lies in the coefficient

for this purpose

In [HI, this is called the ~'space of r e p r e s e n t a t i v e functions" of T, or

V.

we could take

V = KFof

(see

FI).

It is for this reason that finitary

functions are sometimes called "representative functions". If K-space), left

S

is any

mod (S)

S-modules.

shall denote the category of all finite-dimensional Similarly

dimensional right over

K

F(KF), the

K-algebra (possibly of infinite dimension as

mod'(S)

S-modules.

An algebraic representation theory of

could be defined as follows: i.e.

A

is a

K-subspace of

"A-representation theory" of

full subcategory

mOdA(KF )

dimensional left

KF-modules

f : V ~ V' just the

of

F,

first choose a subcoalgebra F(~)

V

such that V, V'

satisfying

~A c A ® A.

cf(V) c A,

of finite-dimensional right way.

The assumption

if

KF-module

cf(V) ~ A ;

A-rational left

V

fh' fhT from

appearing in (ic) (id)

(la)

implies that if

KF-moduleso

A

It is A-rational

mod~(KF)

A-rational in the same f ~ A,

then the functions

can themselves be chosen to belong to

follows that

is

then mOdA(KF)

We can define the category

KF-modules which are

gA c A ~ A

Then

(The morphisms

clear that submodules, quotient-modules and finite direct sums of A-rational.

of

of this category are, by definition,

"rational", or more precisely "A-rational",

modules, are themselves

A

whose objects are all finite-

In some contexts we say that a

is the category of finite-dimensional

F

is defined to be the study of the

mod (KF),

between two objects Ki'-maps.)

is the category of all finite-

is a left and right

A.

KF-submodule of

Then KF ;

also by quite elementary calculations that any finite-dimensional left (or right) (or

KF-submodule

mod~(KF))o

V

of

A

belongs to the category

mOdA(KF)

Examples i.

Let

closed field

K

F

be an affine algebraic group over an algebraically

(see for example ! H, p.21]), and

regular functions on mOdA(KF)

F

(A

A = K[F]

is often called the affin~ ring of

is the category of rational (finite-dimensional)

the usual sense of algebraic group theory~ sub~oalgebra of

F(KF),

remarks apply when

F

In this case,

A

is an affine algebraic semigroupo F(~)

= K F.

2.

If we take

F).

Then

KF-modu!es in is not orlly a

but a sub-blalgebra (see [Se, p. 46]).

finite semigroup, then of course mOdA(Kr) = mod (KF).

the ring of

The same

Let

F

A = K F,

be a then

The (left and right)

K~-module structures on

A,

are dual to tile (right and left) "regular"

KF-module structures on

K

given by multiplication:

we may identify

(KF)

Let

= HomK(KF, K)f

integer, and

3.

F = CL (K)

K

K.

polynomial functions

f : F ~ K

be an infinite field,

We could take

A

(see 2.1).

R = (tab)

F

2.2) by taking

mention that semigroup

g E F

~(n)

Mn(K )

r

M (K), n

F.

matrices

V = (V,T)

in

see 2.2) are

K-bases

(including {Vb}

of

The study of such represent-

the space of polynomial functions in the

n

2

coefficients of a

(see 2.1 for a precise formulation.)

Finally we might

can also be regarded as the affine ring of the algebraic of a l l

n x n

matrices (singular or not) over

we may regard polynomial representations of ations of

n x n

We get another category (denoted

A = ~(n,r),

which are homogeneous of degree

general element

~(n),

obtained by using

ations is the subject of these lectures. in

a positive

the ring of all

The objects

are called polynomial representations of

~(n,r)

n

KF-modules, and the associated representations

the matrix representations

on

~(n),

(we shall later denote this category by

called polynomial

V)

with the dual space

the group of all non-singulsr

with coefficients in

mOdA(K~)

KF

and conversely.

CLn(K),

K,

so that

as rational represent-

Now suppose once more that identity

IF ,

and that

A

F

is an aribtrary

is a subcoalgebra

all flnitary

functions on

to the maps

A : A ~ A ~ A, s : A ~ K.

com(A)

of all right

dimensional which is

F.

A-eomodules;

K-space,

= RV

A

of the space

A-module;

[Se, p. 38] or [Sw, p. 30]). as follows:

if

an object

V

of

the identities

Our category

(le).

relative

com(A)

is a finite-

T : V ~ V ® A

(T ~ ~A)T = (~V ~ A)~ , A-comodule

better references

V E mOdA(KF),

and write down the equations

of

So we may consider the category

(see [G, p. 138], where a right

referred to as a left

F(K F)

is itself a coalgebra,

together with a "structure map"

K-linear and satisfies

(IV ® s)T

com(A),

Then

semigroup with

are [H, p. ].6],

mOdA(KF)

take any

is equivalent

K-basis

Now define

is perversely

{Vb}

of

~ : V -+ V ~ A

to V

to be the

K-linear map given by equations

(Ig)

Y(Vb)

--

% a~B

It is easy to check that using (if) we see that Conversely

given an

the elements

rab

hold, so we may use It is evident that

v a ® rab

T T

satisfies

A ; (le)

such that these morphisms

of the basis

the comodule

(V, T),

{vh}.

Moreover

identities just given.

use equations

to define the left

of(V) c A .

f : V ~ V~

b ~ B.

(Ig)

to define

the comodule identities now show that

can be regarded as a right of morphism

for

is independent

A-comodule of

,

So every

A-comodule,

in

com(A)

KF-module

A-rational,

KF-maps in

KF-module

The definition

(see the references

are the same as

V = (V, ~).

left

and conversely.

(if)

cited)

mOdA(KF).

is

10

This formal transition trivial,

from

but it is nevertheless

20 begin with,

KF-modu]es

worth making,

the basic representation

(~e should here work in the category are possibly infinite-dimensional) discovered (see

[G]).

A-comodules

is rather

from several points of view.

theory of arbitrary

Mod(A),

A-comodules

whose objects

V = (V, T)

follows to a surprising extent the pattern

by R. Brauer and C. Nesbitt

[B],

to

for finite-dimensional

Included here is the possibility

algebra

of a modular

theory,

which we shall discuss below. Next,

the

an important

A-comodule

for the

K-algebra

is the dual of the coalgebra tile product (1)

~D

= Hon~(A,

structure

to be the map of

(~)(f)

(la).

b~longs ~ ~, = ([ o~

A

on A

A,

K).

K

can be regarded as

The algebra structure on

i.e.

into

profit by

if

~,D ~ A ,

we define

which takes the element

to

(lh)

see

also permits us to

fact, namely that every right A-comodule

a left module

f ~ A

interpretation

V,

to v

=

Z ~(fh)~(f~), h

The identity element of com(A),

® ~) (v),

we make for

~v b

=

(see

s : A-* K.

K -module

(1) This product

V = (V, T)

Working in terms of a basis

(ig))

% ~(rab)V a , a~B

for

b E B.

Therefore we have three kinds of matrix representation our original

If

A -module by the rule

into an

~ ~ A , v 6 V.

this rule becomes

(]i)

V

is

A

V = (V, ~),

relstlve

is often called "convolution"

associated

to the basis

{Vb}:

with

{vb}

A

(i)

the representation

g ÷ (rab(g))

whose elements are functions on

F,

of

F ;

(il)

the matrix

satisfying equations

(If), and which

can be thought of as a kind of representation of the coalgebra (iii)

the representation

equetions g ~ F all

(li).

let

We can recover

eg : A ~ K

f 6 A.

Then

e

E A ,

g

K-algebra map

(iii)

with If

e , A

(i)

from

and the map

for

So

modA(KF )

dimensional algebra

r),

for

satisfies may be extended linearly

(i).

is finite-dimenslonal,

A = ~(n,

for each

and if we compoee the representation

and

then it is quite elementary to show mod(A )

mOdA(KF) by composition with the map

Jn the case

e

finally

given by

eg(f) = f(g),

i.e.

e :F + A

are equivalent; this amounts

to showing that every finite-dimensional left in

A ,

A ;

(ill7 very easily: g" ,

g, g' ~ I'.

e : KF ~ A ,

we recover

that the categories

of the algebra

be "evaluation at

e eg, = egg, , elF = s , to a

£ ~ (~(rab))

R = (tab)

e.

A -module

V

yields a module

Schur exploited this fact

end could thereby work with the finite-

AK(n~ r)

= SK(n, r)

(which I have called the

"Sehur algebra" in these lectures; see 2.3, 2.4), instead of with the i1~flnite-dimensicnal and irrelevantly complicated group-algebra If

A

is infinite-dimensional,

V ~ mOdA(KF) is dense in O~ group

F

it is often useful to regard modules

~s modules over some "dense" subalgebra A

~en

if, for every A = K[F]

0 # a 6 A,

by(F)

of

there is some

F

(see

K,

of

A*

~ ~ S

[CFS, §6]).

In ease

F

such that

hy(F)

S

the

is simply-

mOdA(KF)

sets up an equivalence of eategorles (J. Sullivan; see

Moreover in that case

(S

one may take for

connected and semisimple, the correspondence between mod(S)

S

is the affine ring of a connected algebraic

over an algebraically closed field

"hyperalgebra"

KF.

can be identified with an algebra

and [CPS, 6~8]). UK

12

constructed out of the complex semisimple Lie algebra associated with the root system of algebra

UK

F

(W. Haboush:

[CPS, 6.5, 6.6] o r [ H a ,

1.3]).

This

(which is sometimes defined to be the hyperalgebra of

F)

has an explicit basis with sufficiently good multiplicative properties to make it immensely valuable in studying the rational representations of In an important paper

[CL]

F.

R, Carter and G. Lusztig have used the

hyperalgebra -- rather than the Schur algebra -- to investigate the polynomial representations of

F = GLn(K).

Carte1-Lusztig use the idea, which is derived from C. Chevalley's fundamental paper

[Ch]

family of all groups

on split semisimple algebraic groups, that the

GLn(K )

(n

fixed,

of commutative rings) is "defined over

K Z".

varying over some class

K

This makes possible a '~odular

theory" for tile polynomial representations of these groups, which in its essentials corresponds to R. Brauer's modular representation theory for finite groups. as follows. the class

We can give a sufficiently general setting for such a theory

Suppose we have a family K

of all infinite fields,

K~-subcoalgebra of are satisfied. ZI. i.e.

The (a)

O~basis.

F(KFK).

(Q

AQ = (AA, ~Q, ~Q)

AQ , and

Z2~

For each

K ~ K

(here

~

®v~ ,

means

is a semigroup and

K

A~K

in is a

Suppose also that the following two conditions

is a laLtice in

{av.l of

FK

where for each

denotes the rational field.)

Q-coalgebra AZ

{FK, ~K} ,

AQ ,

(b)

there is a and

"e~ten~ion of scalars").

AZ ® K

contains a

which means

z-form

AZ = ~v Zav

&Q(A Z) _c A Z @ A Z ,

for some

~:Q(~Z) --c Z.

K-coalgebra isomorphism is made into a

AZ ,

aK : A Z ® K + A Z

K-coa]gebra by

13

In this case we say that the f~mily means of

Let

n : ~ g -+ EndcE

semisimple Lie algebra dimension n, and let or [St, p. 17 ]). defined by SLn(K ).

~ EZ

be an "admissible lattice" in

its

K ~ K

let

FK

For each pair

(~,v)

showed in over

Z.

[Ch]

of

(see also

The relevant

generated by the

cQ

isomorphisms, .tnd take

(see [Bo, p. A-5]

be the Chevalley group over

K cpKk @' c ~

gc K = L fly X cK

is a

we take this to be

[Bo,

]#thatthe

§4 AZ

the maps

of eK

A0

g = (g~v)

,

we deduce that

family

~..

Chevalley

{FK, AK}

is defined

is just the subring of

are

c~v~t ® 1K ~ cK

K-algebra (as well as

K-coalgebra)

K,

and the family

defined by the

(FK, ~ )

{FK, AK}

Z-bialgebra

AZ .

[Se, p. 46].)

Example 5. K E K. r ~ 0

Z",

A0

for all

is an affine algebraic group defined over is an "affine group scheme over

K

K-subcoalgebra (hence

F(KFK);

Z-form ;

E

of finite

(From the standpoint of algebraic group theory, each pair

See

by

define the coefficient function

From the equations

K-subbialgebra)

E

elements can be regarded as matrices

K-subalgebra generated by all the

even a

Z

be a faithful representation of a complex

over a complex vector space

For each

IY, E Z ;

cK : g ~ gl~v " ~v the

is defined over

AZ .

Example 4.

in

{FK, ~ }

Fix the positive integer

For

AK

(see 2.1).

the fa~aily Az(n, r) {F K, ~ ( n ,

we may take either

and let

~(n),

or

FK = GLn(K)

AK(n,r)

for each

for some fixed

It is completely elementnry to verify that in each case

{FK' AK}

is defined over

are described in 2.5. r)}.

n,

Z;

the relevant

Z-forms

Az(n) ,

in these lectures, we study the family

/4

The first essential of the modular representation family

{FK, AK}

reduction. K E K. which

which is defined over

We shall ~ i t e

Then an object FQ

acts.

equations like

(lj)

If

VQ

Z,

~

for the category

in

MQ

mod~(KFK) ,

is a finlte-dimensional

{Vb, Q : b E B}

N Z r]b(g)v a = " aEB

Here the functions

is a

Q-basis of

rQ ab

,

for

for any

Q-space on

VQ ,

we have

belong to

AQ

and satisfy equations like

'

a subset

Z-form (or admissible lattice) of

VQ

V Z = ~ ZVb, Q

for some

if

VZ (a)

Q-basis

All the coefficient functions

b E B,

g ~ -0 'r

We make the following definition:

(b)

is the process of modular

(le)

gvb, n~ =

~.~hich means

theory of any

rQ ab '

of VZ

VQ

(if)

is called a

is a lattice in

r iVb,Q}

of

VQ ,

VQ ,

and

relative to this basis, lie in

AZ . Another way of expressing condition ~n

AQ-COmodule by means of a map

like

(ig).

Then

(b)

VQ

into

using equations

to

YQ(v z) i V z e A z •

Now suppose that (~

is to convert

~Q : VQ ~ VQ ® AQ ,

is equivalent

(b')

(b)

means

rabK

=

®Z )

(if).

We can make the

into an object of

~K (rQab ® 1 K)

~K : AZ ® K ~ ~

K E K.

of

AK '

~

,

as follows.

usimg the

postulated in Z2.

So we may define an action of

K-coalgebra K rab

These FK

K-space

on

VK

VK = V Z @ K

Define elements isomorphism

satisfy equations like by equations

15

(ik)

Z a6B

gvb, K =

Here

r[~(g)Va,K=u '

Vb, K = Va, Q ® 1 K ,

converted,

via the

A general z-form

Z-form

into

V Z , V~ ,...

VK = VZ ® K ,

V K , V K' ,...

Different

may give non-isomorphic

MK ,

but another general theorem

form to Brauer and Nesbitt)

have the same composition

the notion of decomposition

is

possesses at least one

[G, (2.2d), p. 159]).

VQ

in

VQ

is called modular reduction.

VQ 6 MQ

or

of the same

V~ = V~ @ K ....

(due i~ its original

VK

that every

([Se, Lemma 2, p. 43]

g E F K , b ~ B,

The p~oeess by which

b 6 B.

VZ ,

theor~n guarantees

VZ

Z-forms

for

for

says that all these modules

factor multiplicities;

numbers can be defined

from this

([Se, p. 44] or

[G, (2.5a), p. 162]).

In these lectures we take and study

KF-modules

a fixed homogeneity algebra

SK(n,r )

of

SK(n,r )

V = (V,T)

degree r

where

K

I, above).

and it is shown how

when the latter is given its natural structure

group

G(r)

then every module

V

in

~(n,r)

Schur's multiplication

V

elements

.

in

By definition, homogeneous

SK(n,r)

of degree

r

theorem

SK(n,r)

~(n,r)

An alternative

can

description

as module for the synmletric (2.6e):

if

char K = O ,

in

(see (2.3b)) provides

MK(n,r ) .

are easily expressed Weights

the character of

in

is completely reducible.

method for calculating with modules

~

Schur's

rule for

spaces" of such a module

, for

algebra of the r-th tensor space

E ~r ,

This has as corollary

MK(n,r)

In chapter 2 the Schur

KF-modules

SK(n,r)-modules , and conversely.

is that it is the endomorphism

.

is an infinite field,

which belong to the category

(see Example

is defined,

be regarded as left

F = GL (K) n

V

and characters

For example, in

n

variables

the "weight-

terms of certain idempotent

are discussed

is a symmetric polynomial

in a set of

an effective

over

XI,...,X n .

in chapter 3. Z ,

which is

16

In 3.5 is reproduced

the argument by which Schur showed that the isomorphism

classes of irreducible modules with the partitions

in

% = (%1,...,%n)

Of course Schur considered only minor modification

K ;

yet been determined

of

for an arbitrary

we write this

are in one-one correspondence r

only the case

of an irreducible module of type of

~(n,r)

~%,p

% For

except in special

into not more than

K = ¢ ,

n

parts.

but his argument requires

infinite field

K .

The character

depends only on the characteristic p # O

these characters

cases.

For

~%,p

p = 0 ,

p

have not

Schur showed in

IS] that they are the symmetric functions now known as "Schur functions".

A

proof of this is given at the end of 3.5 - our proof uses some identities involving symmetric functions which can be found, Macdonald's

recent book

for example,

in I.G.

[M].

In chapters 4 and 5, I have departed widely from Schur's dissertation. These chapters K ,

are concerned with the construction,

of two modules

"explicit"

has a unique irreducible

{F%,K}, parts,

V~, K

in our category

in the sense of the "contravariant"

has a unique minimal

then

and

as

X

submodule,

factor module;

F%, K ~ VX, K ~ D%, K .

in

They are

They are dual

this is denoted

% of

gives a full set of irreducible modules

.

duality described

which is isomorphic

ranges over all partitions

% and for each

MK(n,r)

in the sense that a basis can be given for each.

to each other, V%, K

D~, K

for each

to r

F%, K .

F ,K .

D%, K

The set

into not more than

P~(n,r)

But for char K = p # 0 ,

in 2.7.

.

If

knowledge

n

char K = O , of the

F%, K

is still very incomplete.

The history of the modules indebted

to J. Towber

D%, K

and

V%, K

is interesting,

(see IT]) for much of the following

is generated by certain determinantal

expressions

and I am

information.

(here denoted

D%, K

(T%:Ti)),

17

whose significance in 1892.

as "primary covariants" was noted by J. Deruyts

Although Schur refers to two later papers of Deruyts,

no sign in [S] that he appreciated set of irreducible modules in "standard"

(T~:T i) ,

observation that the equivalently that the by G. Higman

M~(n,r)

.

The discovery of the basis of the [Y, 1902].

The

generate a Z-form V~, K

D%, Z

(and the Z-form

called these "Weyl modules",

M. Clausen

V%, K .

G.D. James describes,

are isomorphic to the D%, K .

They

Towber

~]

showed that

([DKR]) to construct both modules in his book

~a],

some

Chapter 6 r e t u r ~ t o

theory of D%, K

KF-modules which

His construction is quite different from those D%, K

in the sense of algebraic group theory.

Schur's disseration.

procedure by which he constructed G(r) .

D%, K

and more

we show in 4.8 that it yields the important and deep fact that

is an "induced" module,

group

1974].

[CI] has used recent combinatorial

G.-C. Rota and his collaborators

above;

were constructed,

are dual to each other - his framework is "functorial"

general than ours.

and

~L,

D%,Q - was made

and their construction was based on methods used

in the theory of semisimple algebraic groups. V%, K

in

V%,Z)

independently of all this, by R. Carter and G. Lusztig

and

The

can be constructed over an arbitrary field - or

(T%:Ti)

~Hi, 1965].

there is

that Deruyts had really given a complete

seems to go back to A. Young D%, K

[D!,

KF-modules

I have "reversed" the elegant from modules for the symmetric

This provides an interesting illumination of some recent work

of James on the modular representation

theory

of the symmetric groups.

§2.

Polynomial

2.1

Notation, Let

representations

of

GLn(K):

The Schur algebra

etc.

n

be a positive

integer,

K

an infinite field, and

F = GL (K) n

the group of all non-singular of elements of ates to each the of

n = {l,...,n}, g ~ F

K-subalgebra A

its

of

the

c

matrices over

let

c

KF

generated

the polynomial

For each pair

D,v

be the function which associg~v"

Denote by

independent over

over K

A

c v~,v

functions on

regarded as the algebra of all polynomials

c v(~,~ ~

K.

by the functions

are algebraically

~v

~ ~

~,v)-coefficient

are, by definition,

infinite,

n × n

in

Since

K,

so that

n

the elements

~ n);

F.

2

~K(n)

or

is

K A

can be

"indeterminates"

n_).

For each

r > 0

we denote by

of the elements expressible

~(n,r)

the subspace of

A

as polynomials which are homogeneous

consisting of degree

r

(n2+r-l) in the

c v.

particular

Then

AK(n,r)

~(n,0)

= K-IA,

1 A : g -+ iK(g ~ F).

The

(2.1b)

has finite dimension where

1A

K-algebra

A = AK(n ) =

If integers

n,r

set of all functions

i : r ~ n.

vector or "multi-index" group on the set on the right on

(both

A

has the standard

Z" r~0

by

i~

K-space;

in

function

grading

AK(n,r).

are given, we write

Such a function

i = (il,.°.,i r)

r = {l,...,r} I(n,r)

~ i)

denotes

as

r the constant

G(r)

for the

is usually written as a

with values

is denoted

I(n,r)

or

= (i (i) .... ,i (r)) ,

i~ ~ _n" G.

The symmetric

It acts naturally

so that

~ ~ G(r)

~9

acts as a "place-permutation" on the set

I(n,r)

× l(n,r)

that the elements j = i~ and

for some

g = j~

i,J

as

= (i~,j~).

Similarly

(i,j) ~

AK(n,r)

in bijeetive

The categories The maps

& : KF~

with

K r×F c

) =

the

IF

A(Cp,q)

= 1

P,q

or

k = i~

that

AK(n,r)

is

is not uniquely

if and only if monomials

G(r)-orbits

of

(i,j) ~

(2.1b), I(n,r)

determined (k,~).

and these are

× I(n,r).

(n2+r-l) . r -

is

~ g : KF ~ K

O,

=

(see introduction)

behave

as

(~,v 6 n_):

k6nZ

c k ~' ckv

of

F.

for any "multi-indices"

5

notice

(i,j)

,

from the rule for multiplying

for the unit element

(2.2b)

that

that

MK(n) , MK(n,r)

&(c

These follow

the pair ci, j = Ck, ~

of these orbits

on the functions

(2.2a)

i.e.

... c i . , r Jr

the set of distinct

correspondence

2.2

deduce,

in fact

K-basis

the number

follows

Of course

(2.1b);

Thus

to indice

by the monomials

i,j f I(n,r),

has as

means

act also

i ~ j

G(r)-orbit,

(k,g)

of the use of this notation,

K-module,

by the monomial

G(r)

We write

are in the same

ci, j = c . . ci2J2 i131

for all

We make

, ~ G(r).

(2.1b)

Here

i ~ I(n,r).

(i,j)~

I(n,r)

~ ~ G(r).

As an example spanned,

by

of

for some

on each

Z s61

Since p,q

g(c v) =

6Gv.

two matrices,

&,c

are both multiplicative

6 I(n,r)

of length

Cp, s ~ c

according

r ~ I,

6 s,q'

as

and from the formula

p = q

g(Cp,q) or

p

=

~ q.

P,q

we

20

These formulae show that bialgebra) of (for

r = 0

MK(n,r)

F(~),

A = ~(n)

is a sub-coalgebra (hence also a sub-

and that each

this is because

~(n,r)

Al A = 1A ~ IA).

for the categories

modAK(n)(kT)

is the category of finite-dimensional representations of

F = GLn(K);

and

(left)

~K(n,r)

is a sub-coalgebra of

AK(n)

We shall write

and

ME(n)

mOdAK(n,r)(F$).

Thus

MK(n)

~Y-modules which afford "polynomial"

the subcategory consisting of those

affording representations in which all the coefficients are polynomials homogeneous of degree IS, p. 5] in case

r

in the

K = ~,

c v.

By an argument first given by Schur

but valid for any infinite field

K

of any character-

istic, we have (2.2c) V =

Theorem

~ re0

V r,

Each

KF-module

where for each

V (~(n)

r e 0

v '

cf(V r) E AK(n,r) ,

i,e.

has a direct sum decomposition is a sub-module of

V

with

r

Vr ¢ MK(n'r)"

In other words each polynomial representation of

F

is equivalent to

a direct sum of homogeneous ones. Remark A

In fact (2.2c) follows from n general theorem on

is any coalgebra which is a direct sum

(p then

ranging over an index set V =

%~

Vp,

where for each

maximum sub-comodule of Vp (com(Ap)

P).

V

A

of sub-coalgebras

p

This theorem says that if p ( P

such that

[G, p. 156, (io6C).

A = Z~ P

A-comodules, where

the space

cf(Vp) E Ap,

Vp i.e.

V = (,~)

~ com(A),

is the unique such that

The proof given there does not depend on

the assumption that the sub-coalgebra summands

R

P

are minimal].

A

21 2.3

The Schur algebra

SK(n,r )

Theorem (2.2c) shows that each indecomposable module homogeneous,

i.e

V ~ MK(n,r)

confine our attention fixed, and define

for some

r > O.

to homogeneous modules.

SK(n,r )

As

K-space,

S K ~ , r ) has basis

{ei, j : i,J E l(n,r)} of

SK(n,r )

of

i

ci, j

if and only if

r

r ~ 0

be

~(n~r).

: i,j 6 I(n,r)}

For

dual to the basis

i,j 6 I(n,r),

~i,j

is the element

given by

~i'j(Cp'q) As with the

From now on, let

= HOmK(~(n,r) , K).

{~i,j

~(n,r).

is

This means that we may as well

to be the dual space of

SK(n,r) = ~ ( n , r )

V E ~(n)

=

if if

(i,j) ~ (p,q) (i,j) ~ (p,q)

all

•

p,q ~ I ( n , r ) .

we have an e~uality rule to take into account:

(i,j) ~ (k,g).

The dimension of

SK(n,r)

is

~i,j = ~k,~

of course

/ = dim AK(n,r ) . Since

algebra.

AK(n,r )

We saw in

is a coalgebra,

its dual

§i that the product

is defined as follows:

if

c ~ ~(n,r)

&(c) =

~

SK(n,r)

of elements

~,~

of

SK(n,r)

and if

Z ct @ ct t

where the sum is finite and the

ct, c t ~ AK(n,r),

(2.3a)

= ~ ~(ct)~(e ~),

(~n)(c)

is an associative

then

t The unit element of for all

c E AK(n,r).

SK(n,r)

will be denoted

a;

it is given by

s(c) = c(l r)

22

Applying (2.3a) to a basis element

of

c = Cp,q

AK(n ,r),

we ge<

(see (2.2b)) ( ~ ) (Cp, q) = s~ I(n,r) Specializing to the case where SK(n,r),

~ (c p, s)~ (c s,q ).

= ~i,j' ~ = ~k,g

we deduce a

Multiplication rule for

(2.3b)

SK(n,r ).

~i,j~k,Z

=

% {Z(i,j,k,g,p,q). IK} ~p,q, P,q

where the sum is over a set of re~r_esentatives of

are basis elements of

l(n,r) × l(n,r),

(p,q)

of the

G (r)-orbits

and

Z(i,J,k,~,p,q_)_ = Card{s 6 I(n,r)

: (i,J) ~ (p,s)

and

(k,~) ~ (s,q)}.

This multiplication rule (rather differently expressed) in due to Schur [S, p. 20]. (2.3c) (i)

Some special cases are worth noticing.

For any

i,j,k,~ 6 l(n,r)

~i,jSk,g = 0

i,i i,j

unless

i,J

j ~ k,

and

s, p,q k ~ s, From

with hence

distinct

~i,i

~i,j~k,g ~ 0,

(i,J) ~ (p,s)

and

then by (2.3b) there must

(k,g) ~ (s,q).

This implies

j ~ s

j ~ k.

(2.3c) follows that

Of course if

and

i,j j,j "

For example, (i) holds because if exist

there hold

i ~ J,

then

~2 :'i,i =

(i,i) ~ (J,j)

and

~i,i~j,j = 0

if

and hence

~i,i = <J,j"

But the

~i,i'

i # J.

form a set of mutually orthogonal idempotents, and their sum

is the unit element

e

of

SK(n,r).

This last equation,

23

sum over a set of representives of the

(2.3d) of

L = Z ~i,i, i l(n,r)

C (r)-orbits

c

is proved by evaluating both sides at all the basis elements Of great importance in the modular theory for for fixed

n,r,

the scheme or family of algebras

in the following sense. K ~i,j

elements Sz(n,r)

of

of

Let us use a superscript

SK(n,r).

SQ(n,r),

K

is "defined ever

is a

Z-order in

K-algebras

Z",

to denote the basis Z-submodule

SQ(n,r).

is

And for any field

Sz(n,r) @ K ~ SK(n,r)

which takes

K

~ ,j ® IK ~ ~i,j

The map

e : KF ~ SK(n,r)

For each

g E F

we define the element

eg(C) = c(g) (c E AK(n,r)). for all

g,g' E F;

the map

g ~ e

of

SK(n,r)

Q ~i,j(i,j E I(n,r)),

which is generated by the

there is an isomorphism of

each

2.4

is the fact that,

n

It is clear from (2.3b) that the

multiplicatively closed - - i t K,

GL

~(n,r) .

of

P,q

g

also

eg E SK(n,r )

by

It is clear from (2.3a) and (i) that

eI =

s by the definition of

linearly we get a map

~.

e : KF ~ SK(n,r)

egeg, = egg,

So if we extend which a morphism

K-algebras. Any function

f : KF ~ K. <=Z~g

(2.4a)

g

f E KF

has a unique extension to a linear map

With this convention,

EKF,

is "evaluation at

e(~)

' c -+ c(K),

the image under <";

all

e

of an element

i.e.

e E ~(n,r).

Propositions (2.4b,c) give the most important facts about

e.

24

(2.4b)

Proposition

(ii)

Let

Then

f ( AK(n,r )

Proof

(i)

exist

some

(i)

Y = Ker e,

e

is surjective.

and let

f

be any element

if and only if

If lme were a proper 0 # c E ~(n,r)

of

F K

f(Y) = 0. subspace

such that

of

SK(n,r)

eg(C)

= ~(n,r)

= c(g) = 0

,

there would

for all

g ( F,

a

contradiction. (ii)

If

f (AK(n,r)

by (2o4a).

So

K I' such that

and

f(Y) = 0. f(Y) = 0.

K E Y,

we have

Now suppose By

(i)

e(K) = 0

conversely

there

and hence

that

f

f(~) = 0,

is any element

of

is an exact sequence

0 ~ Y ~ KF ~ SK(n,r)

~ 0, *

from which it is clear y(e(~))

= f(<)

for all

there exists = e(<),

that there

c (1~(n,r)

then, we have

K E KF.

(2.4c)

Proposition

y (SK(n,r)

y(<) = <(c)

f(K) = e(K)(c)

and the proof of (2.4b)

some

By the natural

such that

f = c,

Let

exists

= c(<),

such that

isomorphism

SK(n,r)

~ ~(n,r),

for all

< (SK(n,r).

Put

all

K ~ KF.

Therefore

is complete.

V E mod(KF).

Then

V (~(n,r)

if and only if

YV = 0. Proof

Let

{v b}

by the action of if

rab(Y ) = 0

saying

be a basis KF

But of course the proof

of

V,

on this basis

for all

that all the

of

rab

a,b.

(rab)

(see (i)).

the invariant Clearly

By the last proposition,

lie in

AK(n,r) ,

this is the condition (2.4c)

and

is complete.

for

V

YV = 0

afforded

if and only

this equivalent

that is, that to belong

matrix

to

cf(V)

to

~ AK(n,r).

MK(n,r),

and so

25

There propositions are equivalent,

and in a very elementary

can be transformed

Kv

= e(K)v,

to relate the action on actions determine

V

of submodule,

all

[S,

on a module

V ~ ~(n,r),

F

action of

SK(n,r)

on a basis

We might mention

and

SK(n,r).

it holds for all

etc. coincide

on

Since both

V,

the

in the two categories.

that the action of

which is given by (2.4d),

{Vb}

of

% ~(rab)V a a

For it's clear that

outlined V

SK(n,r)

= AK(n,r)

is the same as that which

in the Introduction.

is given by equations

all

~ ~ SK(n,r) ,

(2.4d) holds, whenever

K f K

If the

(I~),

then the

and

K = g

Vb ~ B

and

v = v b.

By linearity

v ~ V.

theory

R. Brauer's

theory of modular representations

by Brauer himself

[B] and by Nakayama

More recently Serre [Se]

affine algebraic

groups,

GL

K

F K = GLn(K).)

n

(see also

of finite groups was

[N], to finite dimensional

[G]) extended

or rather to affine algebraic

group scheme

"reduction"

KF

is given by

~v b =

the group

in either category

was one of the main techniques used by Schur in his

p. 21].

action of

algebras.

V

mod(SK(n,r))

v E V

of the two algebras

is obtained by the general procedure

extended,

~ ~ KF,

module homomorphism,

dissertation

M0dular

way; an object

and

the same algebra of linear transformations

This category equivalence

2.5

MK(n,r )

into an object of the other, using the rule

(2.4d)

concepts

show that the categories

is a functor, which associates We can describe

or "decomposition"

the theory to

group schemes.

to each commutative

the characteristic

process as follows.

(The ring

modular

26 Let

Az
be the subsets of

consisting of those polynomials in the Z.

These are

"Z-forms" of Q-basis

{c~,j}

a(Az(n,r) 5 Z

(see (2.2b)).

of

i,j ~ l(n,r).

The

of all elements

Z-order

Now let

VQ

taking

such that

be any object of

which (i)

is the

Z-span of some

closed to the action of defined by rab

lie in

{vb}

Az(n,r).

That every

If

K,

there is a

c ,j ~ iK ~ ci, j

K-coalgebra

for all

~(Az(n,r) ~ Z.

M(n,r); we shall regard By a

Z-form of

Q-basls

{vb}

RQ = (rab)

where

(VQ,~) VQ

in

is the

of

VQ

as

VQ

is meant a subset

V,

and (ii) i~

is the invariant matrix

Az(n,r)-comodule determined hy

MQ(n,r)

contains at least one

Z-form,

, p. 256, §6] or [Se, p. 43, lemme 2] or [G , p. 158, (2.2c)].

If we now take any infinite field

hence as a

A~(n,r) ~ AZ(~,r) ~ ~(n,r),

Still another formulation of (ii) is that

QFQ-mOdule

follows from [B

VK = VZ ® K

is the

(see (I e )), then condition (li) just says that all the

T(V Z) ! V Z ® Az(n,r), VQ.

Sz(n,r ),

Az(n,r)

which we defined in (2.3), is the set

SQ(n,r)-module when this is convenient. VZ

and

For any infinite field

Sz(n,r)

~ ~ So(n,r)

for example

AQ(n,r),

Az(n,r) ® K ~ AK(n,r)

respectively,

whose coefficients all lie in

AQ(n), AQ(n,r);

Z-span of the

isomorphism

c

AQ(n), AQ(n,r)

K,

it is clear that the

can be regarded as a left module for ~K-mOdule in

M~(n,r).

K-space

SK(n,r ) ~ Sz(n,r) ~ K,

The transition from

VQ

to

VK

is

particularly easy to express in terms of invariant matrices; the invariant matrix

RK

defined by the

K-basis

{vb ® 1K}

of

VK,

(tab) = RQ

is the Invariant matrix defined by the basis

case where

K

has finite characteristic

the coefficient of

RQ.

is

(tab ~ IK) , where

{vb}

of

VQ.

In the

p, this amount to "reducing mod p"

27

Our notation in the preceding discussion conceals a disadvantage: there are in general many different VQ E MQ(n,r),

Z-forms

and the corresponding

may be not all isomorphic.

Vz,V~,...

KFK-mOdules

of a given

~Q-mOdule

V K = V Z ® K, V~ = V~ ~ K,...

However one of the classical results of modular

theory, deducible from [B , p. 257, (8)] or [Se, p. 44, theorhme 2] or [G, p. 162, (2.5a)], says that, for any type of simple E MK(n,r),

the multiplicity

depends only on case that d ~k'

VQ, i.e.

VQ = V

~ ( V K)

of

Lk

as composition factor in

is the same for all

is a simple

KFK-mOdule

Z-forms

VZ

of

VQ.

VK

In the

QFQ-mOdule, this multiplicity is often written

and referred to as a decomposition number for the modular reduction

MQ(n,r) ~ MK(n,r).

2.6

The module

E~r

Fix our infinite field E = E K = K.e I @ ... • K-e n {e

v

: v E n}

on which

=

gev

K,

and write

be an

n-dimensional

E

Z e ~E~ g~v ~

is an object of

Now let E<%r = E ~ . . ®

r ~ 0 E

Let

K-space with a basis

F acts "naturally":

=

% ~En

c v(g)e v ,

Since the corresponding invariant matrix is KF-module

F = F K = GLn(K ).

all

g E F, v E ~.

C = (c),

we see that the

AK(n,I).

be given, and then

in the usual way

(®

F

acts on the

here means

~).

{ei = eil ® "'" ~ eir : i E l(n,r)},

and relative to this the action of

F

is given by

r-fold tensor power Eer

has

K-basis

28

gej = ge

®.. ~

ge. ]r

Jl =

Z i(I(n,r)

The invariant matrix is E~r ( M K ( n , r ) . as an

(2.6a)

Sej =

Z i( I (n,r)

ci, j(g)ei,

all

. gilJl

g ~ F,

(ei,j) = C x ... × C,

According

SK(n,r)-module,

=

to

what

ei "" girJr

j ( l(n,r)

and this shows that

was said in (2~4),

E®r

can be regarded

by the rule

Z <(ci,j)ei, i(I(n,r)

all

< (SK(n,r) ,

j (l(n,r).

In a very famous paper [S'] which appeared in 1927, Schur rederived all the results of his 1901 dissertation ~r

[S] by an analysis of this module

Although his method gives a complete answer only when

char K = O,

it

is still valuable for fields of finite characteristic.

We make the s}~mmetric

group

act on the right of

Ef~r

G(r),

and hence also its group-algebra

KG(r),

by

(2.6b)

ei~

= ei~ ,

all

6 G(r).

i (l(n,r),

It is clear that this action commutes with that of with that of all

SK(n,r);

~ (SK(n,r),

KF,

we can verify from (2.6a) that

x ( Eo~r

and

~ (G(r).

or (what is the same) (~x)~

= ~(x~r)

We have however a stronger

statement. (2.6c)

Theorem (Schur)

representation afforded by the (i)

Im @

(ii)

Ker @ = 0.

Hence

= EndKG(r )(E ~ ) ,

SK(n,r ) ~ EndKG(r ) (E@r)

Let

~ : SK(n,r) ~ EndK(E~)

SK(n,r)-module and

E ~r.

Then

be the

for

2g

Proof

Each element

basis

{ei}

of

E®r.

of course, and the EndKG(r)(~ 'r)

0 E EndK(E Here

i,j

Ti, j ~ K.

Tin,j ~

the set

)

has matrix, say

(Ti,j)

for all

has a

lies in

Ti, j = i

and all

~p,q

of

SK(n,r )

I × I,

or

0

0

namely if

co

according as

(p,q).

~0 is such an orbit,

to be that

e E EndK(E~r)

whose

(i,j) ~ ~0 or not. (p,q) E I × I,

~r

is represented on

G(r)-orbit containing

~ ~ G(r).

K-basis in one-to-one correspondence with

G(r)-orbits On

has

i,j ~ I

it follows very readily from (2.6a) that, for any

~0 is the

e

From (2.6b) follows at once that

Ti,j,

EndKG(r)(E~r )

~ of all

element

I = I(n,r) ,

run independently over the set

define the corresponding basis element matrix

relative to the

(Ti,j) ,

if and only if

(2.6d)

Consequently

~r

by

Therefore

the basis

~(~p,q) = ~0 '

~

Now

where

induces an isomorphism

SK(n,r) -~ EndKG(r ) (E~r) , and this proves the theorem. Remark

The proof of (2.6c) shows that

sentation by the algebra of all condition having ~i,i

(2.6d).

Ti, j = 1

nr × nr

The basis element or

0

SK(n,r)

matrices

~p,q

according as

has a faithful matrix repre(Ti, j)

which satisfy

is represented by the matrix

(i,j) ~ (p,q)

or not.

The idempotents

are represented by diagonal matrices, and the "orthogonal" decomposition

(2.3d) is easy to deduce from this. (2.6e) then Proof

Corollary to (2.6c) (Sehur [S, S']). SK(n,r )

is semisimple.

Hence every

Under the given conditions on

is semisimple K-G(r)-module,

(since

char K

If

V ~ MK(n,r)

char K,

does mot divide

and in particular

E~r,

char K = 0,

or if char K = p > r,

is completely reducible.

the group algebra IG(r) I = r!).

K-G(r)

Therefore every

is completely reducible.

But the

endomomorphism algebra of a completely reducible module is semisimple,

so by

30

(2.6c), and

SK(n,r)

mod SK(n,r)

is semisimple.

is clearly "defined over is simply a version of

{~r}

Z"

,

r

fixed but with

Z.

GL -module, n

We say that the family

a

of

in the category

VQ,

{V K}

and for each

~(n,r).

varying,

GL

being regarded

n

See [Se, p. 46]).

Suppose that for each infinite field

VZ

K

in the sense of the following definition (which

V K E MK(n,r).

and

with

the definition of

as affine group scheme over

Z-form

~(n,r)

now completes the proof of (2.6e).

The family of modules

Definition

The equivalence of the categories

K

K

we have a

is defined over

an isomorphism

More exactly we say

{VK}

KFK-mOdule Z

if there is

6 K : VZ @ K ~ VK

is

Z-defined by

VZ

{6K}.

Example I.

Take

of

VQ

(we write

of

EK, E®r K ),

MK(n,r), ively.

for all

is defined over

Definition

The module

Suppose that

K

the

K

Z i~l(n,r)

Z.e i

is a

Z-form

for the basis elements

6K : V Z ~ K ~ V K

is an isomorphism in

taking ~(n,r).

So

Z. {VK} , {WK} Z,

Suppose we have for each

We say that the family

K-map

i E l(n,r),

both defined over

and for each

VZ =

e~,K, ei,K = eil,K ® "'" ® eir,K

and for each

e i ® 1 K ~ ei,K, {~K r}

V K = E~ r.

{eK}

By K

are both families of modules in VZ

and

{6K} ,

a morphism

is defined over

Z

GQ~ K

VK

>

OK - - >

and

{~K},

eK : V K ~ W K

the diagram shown commutes VZ ® K

WZ

WZ ® K

WK

if

eQ

maps

in

respectM~(n,r).

VZ

into

Wz,

31

Example 2. the

r th

Define the

r th

monomial

Dr, K

of a given

Z-form

of degree

in the variables

r

Dr, Z

The isomorphism

for all

i E I(n,r).

defined over

Z

in

by

becomes a

{Dr,K}

is defined over

belong

e I = el,Q,..., en = en,Q, takes

which have coefficients

e(i ) ,Q ~ 1K ~+ e(i ) ,K

V

In this sectlo~ we keep = HomK(V,K)

of a

(v E V).

K

one defines

a left

action

However the

to our category

which is still in

with this

{O K } is

KF-module if

fixed and write V E MK(n,r )

f E V ,

To make

V

g E F

(denoted

KF-module

NK(n,r).

V

by a d o t )

F-action

~L.(n,r). K-

KF-module

F

to "reverse" on

V

so defined will not, in

B u t i f we r e p l a c e

We denote by

can be

g ~ g-i of

F = F K.

define

into a left

(transposed matrix) in the above definition, we get a left V

Z ;

is the set of all homogeneous polynomials

K -module in a natural way:

by (g'f)(v) = f(g-lv). general,

Dr, K

which takes

It is clear now that the family of morphisms

(fg)(v) = f(gv)

multiplication;

KF-map:

is the restriction to

the traditional practice in group-theory is to use the map

on

There is

in the sense of the last definition.

The dual space made into right

Dr, Q

@K

K[el,.°.,en]

~]K : Dr,Z ~ K -~ Dr, K

Contravariant duality

fg E V

g E F,

We can show that the family

the relevant

2.7

K[el,...,en];

KF-module, such that

K-algebra automorphism of

-~ ge (~ E n).

Z.

to be

are regarded as commuting indeterminate..

has a unique structure as

of the unique

in

EK

K-map

in fact tha action on

e

of

e K : Eer -~ Dr(EK) taking e i = ell @ ... ~ e i to the K r e(i) = ell "'" eir ' for all i E I(n,r). It is well-known that

a surjectivc

r,K

Dr, K = Dr(EK)

homogeneous subspace of the polynomial

e I = el,K,..., e n = en, K

D

symmetric power

V°

g

-i

by

g

tr

KF-module structure

the space

V ,

equipped

32

(2.7a)

V°

(g-f)(v) = f(gtrv)

all

is called the "contravariant dual" to

g E F, f E V , v E V.

V;

an analogous dual applies to

rational modules over all semisimple algebraic groups a great

deal in recent years by Wong

F,

and has been used

[W], Verma [V] and Jantzen

[J].

It is convenient to express (2.7a) in terms of the action of It is easy to see that the J(
all

K-linear map

i,j E I(n,r)

In fact one has clearly, for any

J(~)(ci,j) = ~(cj,i),

and by taking

~

= e

we find that

is regarded as

the contravariant dual

(2.7c)

MK(n,r) ,

V°( = V )

J(eg) = e t r ' g

(,) : V × W ~ K

V, W

g E F.

gives an exact contravariant function on V ~ (V)

of

K-spaces,

V ~ (V°) °.

be modules in

MK(n,r).

Then a

K-bilinear form

is called contravariant if it has the property

(~v,w) = (v,J(~)w),

So if

~ E SK(n,r) , f E V , v E V.

and that the usual natural isomorphism

Let

all

reads

all

V ~ V°

gives a natural isomorphism Definitio N

i,j E I(n,r),

SK(n,r)-module the action (2.7a) which defines

(~.f)(v) = f(J($)v),

It is clear that

~ E SK(n,r),

all

g V E MK(n,r)

given by

is an involutory anti-automorphism of

(2.7b)

(2.7d)

J : SK(n,r ) -~ SK(n,r)

SK(n,r).

all

~ E SK(n,r) , v E V, w E W.

The proof of the next proposition is standard.

33 (2.7e)

Proposition

correspondence A

: V~

W°

If

V, W 6 ~ ( n , r )

between contravariant

in

~(n,r),

(,)

forms

(,) : V × W + K,

and morphisms

given by

A(v)(w)

The form

are given, there is a bijective

= (v,w),

is non-singular

all

v ~ V, w 6 W.

(= non-degenerate)

if and only if

F

is an

isomorphism. Example 1

We can use the last proposition to show that the module

self-dual,

that is that

~r

~ (~r)o

: E~r × ~ r

<,>

i,j ~ I(n,r).

But a simple calculation,

on

defined by

condition

<ei,ej>

=

6ij,

using (2.6a), shows that

(2.7d). Call <,>

all <,>

the canonical form

E~r .

Example 2

Let

Z-defined by

{VK} VZ

and

be a family of modules {6K}

V Z.

For each

a basis of

K,

write

(V K E ~ ( n , r ) )

as in the last section.

running over some finite index set Va, K =

B) be a basis of 6K(Va, Q ® 1 K)

For each Then a,Q

K,

write

{fa,K}

for the basis of

VzO = {f ~ VQ : f(Vz) --c Z} }.

which is

Let VQ

so that

{Va,Q} which

(~ Z-generates

{Va, K : a 6 B}

is

V K.

We can now show easily that the family

{f

is

For there is clearly a non-singular

bilinear form

satisfies the contravariant

~ K

E~r

is a

It is not hard to see that

6K : VZo ® K ~ VKo

which take

° {V K}

V K* = V oK

Z-form of {VK} o

is

fa,Q ~ IK °+ fa,K'

is defined over

o VZ,

dual to having

Z-defined by all

a 6 B.

Z.

{Va,K}°

Z-basis V oZ

and the maps

34

Example 3

Let

which are

Z-defined by

K

and

{WK}

is defined over

and for each

be families

VZ, {6 K}

we have a bilinear form

{(,)K } K,

{VK}

and

(VK

WZ, {~K }

(')K : VK x WK ~ K,

Z

if

v Z E VZ,

(,)Q

wZ E WZ

maps

and

WK

both in

respectively.

M~(n,r)~

If for every

we say that the family

Vz x Wz

into

z,

and for each

there holds

(SK(Vz ® IK), nK(wZ ® IK)) K = (Vz,Wz).l K. If all the (')K

are contravariant, W Ko

then we can

(given, for each

show

K,

by Proposition

family of morphisms

AK : VK

is defined over

For example, the family of canonical forms

Z.

EK~)r is defined over EK~r ~ (E~r) ° K

2.8

AK(n,r)

Z,

is a

these two

KF

on

derived from these forms.

as

KF-bimodule

KF-bimodule,

i.e.

actions commute.

there holds an equation (see 2.3).

<'>K

(2.7e))

and so therefore is the family of isomorphisms

We saw in the introduction (p.4) that the space f:F ÷ K

that the

KF

it is a left and right If we take an element

&(c) = Z c t ~ c't ' t

of all functions KF-module, and

c E AK(n,r)

for suitable elements

,

then

c t ,c't c AK(n,r)

By formulae (ic), (Id),

(2.8a)

g o c = ~ c~(g)c t , t

for any

g g F .

KF-actions, hence

c o g = Z ct(g)c' t' t

This shows that ~(n,r)

AK(n,r) is closed to both the left and right

is itself a

K -bimodule.

linearity, to give the action of an arbitrary element

Now extend (2°8a) by < c K :

35

c o < = Z ct(K)_- c tv

< o c = Ec'(K)ctt'

By (2.4b), we find (2.4c) shows that MK(n,r)

.

~(n,r)

AK(n,r)

Similarly of right

bimodule,

(2.8b)

K o c = c o ~ = O , ,

regarded

AK(n,r)

belongs

KF-modules

as left

K ~ Y = Ker e . K£-module,

to an analogously

(see 4.4).

AK(n,r)

belongs

defined

becomes

Then

an

to

category

SK(n,r)-

with actions

~ o c = Z$(c~)e t ,

c o % = Z~(ct)c ~ ,

Now let us define a bilinear rule:

for any

if

~ E SK(n,r)

form

, c c AK(n,r)

,

(

,

for

~ E SK(n,r)

•

by the

):SK(n,r)~K(n,r ) ~ K

then

($,c) = J ( O ( c )

This is non-singular, to

(

,

)

(see (2.7b)).

~,q E SK(n,r)

(2.8c)

in fact

,

{cj, i}

are dual bases with respect

The reader may check that for any

c e AK(n,r)

there holds

(~n,c) = (~,J(~) o c) = (~,c o J(q))

In fact, using

(2.3a),

all three expressions shows that left

{$i,j},

(

,

)

(2.7e) we deduce

(2.8b) and the d e f i n i t i o n just given are equal

is eontravariant,

SK(n,r)-modules that

.

to

SK(n,r)

of

( , ) ,

E(~,c~)(q,c t) • and

AK(n,r)

(and even when they are regarded AK(n,r)

~ (SK(n,r)) ° ,

we see that But

(2.8c)

being regarded

as right modules).

an isomorphism of

as

By

KF-bimodules.

§3

Weights and Characters

3.1

Weishts In this chapter we describe

of modules

in

MK(n,r),

in terms of the Schur algebra

results go back to Schur researches

of Weyl and Chevalley.

ideal domain,

Let

n,r ~ 1

G(r)-orbits

in

weights

(more precisely,

weight

6

of reductive

see [Se,

number of

The elements

as unordered

such that

partitions

of

any

It acts on

w E W,

w

~ = (~w(1)'''''

dominant weight,

W

i.e

into

of

denote the set of all of

A(n,r)

GLn,

i.e.

n

acts on

for each

a weight

to (ordered)

parts.

A+(n,r)

~v

is the

can also be regarded

can be identified with the Weyl group wi = (~(il),... , w(i r)

if

W-orbit of X

A

which gives the

~ f ~ ' ~

r).

parts (zero parts being allowed).

7~(n,r):

Each

will be called

of dimension

These vectors

on the left,

weights correspond Denote by

to the category

This action commutes with that of

~w(n) )"

in classical

split over a

6 = (61,... , 6 n)

~ 6,

W = G(n)

l(n,r)

i ~ I(n,r).

right, and therefore -i

e,~ ....

i p = v. r

The symmetric group GL . n

A(n,r)

they are weights

i = (i I .... ,ir)

p ~ -r-

group schemes

All our

§3].

is specified by the vector

content of any

of

For the generalization

be given~ and let

I(n,r).

SK(n,r) o

[S], and all have been generalized

of rational representations principal

the theory of weights and characters

such that

partitions

of

w ~ W, A(n,r)

~ ~ A(n,r)

on the then

contains exactly one

X1 ~ "'" ~ Xn. r

G(r)

for

Thus dominant

into not more than

the set of all dominant weights.

n

37

3.2

Weight spaces Take a fixed infinite field

weight < .

e E A(n,r)

K.

If

i E l(n,r)

we shall denote the idempotent

This is reasonable, since

<j,j

belongs to the

{.

.

(see

if and only if

2.3) by i ~ j.

The

orthogonal decomposition (2.3d) now reads

s

=

If we apply this to any module

(3.2a)

V =

a decomposition of

V

In fact space

Va

of

[[ <~ ~(A(n,r) V ~ MK(n,r)

we get

~]~> < V, aEi(n,r)

as direct sum of subspaces ([S, pp. 6,7])

V,

< V

Tn(K )

a-weight

which is defined as

all

x(t) E Tn(K)}.

is the diagonal subgroup (a maximal split torus) of

consisting of all diagonal matrices tl,... , tn E K character

~ V.

coincides with the

V a = {v E V : x(t)v = tI ,..tnnv ,

Here

•

X

= K\{0}. : Tn(K ) ~ K

To show that

(3.2b)

ex(t) =

For each ,

by

x(t) = diag(tl,... , tn) c~ E A = A(n,r),

F K = GLn(K)

with

define the multiplicative

51 an X (x(t)) = t I ...t n

~i~V = V~,

first verify the formula

~i en aEAZ tI ... tn ~

,

all

x(t) E Tn(K) ,

by evaluating both sides at each deduce

c. . (i,j E I(n,r)). If v E ~ V , we 1,3 51 ~ V s" x(t)v = ex(t)v = t I ...tnnv; hence ~ V ~ But for distinct

~,~ E A(n,r)

the multiplicative characters

k , X~

are unequal (since

38

K

is infinite),

spaces that

V

and by a familiar argument

, ~ E A(n,r),

< V = V~

is direct.

for each

it

follows

Comparing

that the sum of the

this with

(3.2a), we see

e:

(3.2c)

V =

%~ V ~. ~Efi

Remark.

It follows from (3.2b), and the fact that

image of K'Tn(K) of

SK(n,r )

which has the

commutative, Example V =

under the map

split,

For each

ArE

is any

~

(see 2.4)

(5 E A(n,r))

is infinite,

is the subalgebra as

K-basis.

that the

DK(n,r)

This is a

semisimple algebra. r

satisfying

(= Altr(E)) r-element

e

K

is a

0 ! r ~ n,

KF-module

subset of

~

in

the

r

th

~(n,r).

exterior power

If

(i I < i 2 < ... < ir)

s = {il, .... ir}

write

e s = e i A ... A e i r

Then

V

has a basis consisting

x(t) E Tn(K) = ~(s)

then

x(t)e s = til~.,

is the weight containing

r-element

subsets

s,s'

the weight spaces r-element

3.3

subset

V~

Let

V

n

and

of weight

Let

($)

elements

e s.

Moreover

~I an t.1 es = t I ... tn es' r i = (il,...,ir).

V~ = 0

1

or zero:

for all other

distinct

~(s), ~(s').

So

v~(S)

for any

= K.e

s

~ E A(n,r).

spaces ~(n,r),

w E W = G(n).

and

~

an element of

Then the

if

where

Clearly,

give distinct weights

all have dimension

be a module in

Proposition

are isomorphic.

of

s ~,

Some properties

(3.3a)

of these

K-spaces

A(n,r).

V ~, V w(~)

39

Proof

Let

el,...,

en

of

n~ 1

verify

hence that

(3.3b)

n

be the element of

w E

to

ew(1),... , ew(n)

v ~+ n v w

gives a

Let

0 ~ VI ~

V ~ -~ V ~2

0

Since

Now let to

be an exact sequence in

sequence of

is obtained

is idempotent,

K-spaces

by applying

KF-module

~(n,s)

to every term of

the result follows.

be any non-negative

MK(n,r) ,

regarded as

(3.3c)

~a

r,s

* t I ..... t n E K ,

for all

is exact.

The second sequence

the first.

It is simple to

V ~ ~ V w(~) .

0 ~ V 1 ~ V -~ V 2 ~ 0

Then the naturally induced

elementary

respectively.

K-isomorphism

MK(n,r).

belonging

which maps the basis elements

x(t I .... , tn)n w = X(tw(1),... , tw(n)) ,

Proposition

Proof

F = GL (K) n

integers and

respectively.

V, W

Clearly

in the usual way, belongs

to

be

~-modules

V ~ W = V ®KW ,

~(n,

r+s).

It is

to verify the

Proposition.

Let

where the sum is over all

y E A(n, r+s). e E A(n,r),

Then

(V ® W)~ =

8 (A(n,s)

Z~

V~ ~

such that

(~i ..... an) + (~i ..... Pn ) = (~I ..... ~n )" Next suppose with a subset of i,j E l(n,r).

L

is a field containing

SL(n,r)

Then

~K ~ =

K ~i,j

~L ,

~ E A(n,r).

into an

identify

with the subset

(3.3d) V~=

Pr@position.

$~K V.

for all

SL(n,r)-module

with

~° ,j'

V ® 1L

of

VL,

VL = ~

SK(n,r) for all

So if we make

by "extension of scalars",

The weight-space

In particular,

We identify

by identifying

VL = V ®K L V

K.

and

we have the L

dimK V~ = dim L V L .

VL

is the

L-span of

W ~,

40

Contravariant spaces.

If

duality (see 2.7) behaves well with respect to weight-

V,W E MK(n,r )

and

(,) : V × W ~ K

is a contravariant

form, then

(V~,W ~) = 0

for any distinct weights

[W, p. 42],

[J, p. 6]).

For

(~aV,~W)

= (V, ~ W )

= 0.

only if the restrictions E A(n,r).

(3.3e)

Taking

J(~)

,

(,)~ : V

× W

(see

so by the contravariant

It follows that

W = V°

Proposition.

= ~

~,~ E A(n,r)

(,)

~ K

is non-singular

are non-singular

bilinear

property if and

for all

there follows the

dimK V~ = dimK(V°)~ ,

for all

V E MK(n,r )

and

E A(n,r). Finally let as in 2.6.

{V K}

Because the idempotent$

we have a direct sum These

(3.3f)

in

and

Z-defined by

SQ(n,r)

actually lie in

~ V Z = V Z ~ VQ~ Z-module

VZ, {5 K}

VZ,

for all

Sz(n,r) ~ E A(n,r).

are themselves free

We have the

Proposition

6K : V Z ® K ~ V K

3.4

V Z = Z @ ~ Vz,Q

%aYQVz, being surmnands of the free

Z-modules.

free

be a family of modules,

Z-module

For each ~Q V Z ® 1K

of ~Q~ VZ,

K, •

VK

is the

Hence

K-span of the image under equals the rank of the

dim K V aK

and so is independent of

K.

Characters Let

V

be a

KFK-mOdule in

we assign the monomial

~i ~n X 1 ... Xn

MK(n,r).

To each weight

of degree r

in

n

~ E A(n,r)

indeterminates

XI,...,X n

over the rational field

Q.

character,

cf.

is defined to be the polynomial

[J', p• 274]) of

V

Then the character

(or formal

4]

@v(XI,...,Xn)

~i • X1

dimK v~

Z

=

an ... X n

~EA(n,r) Thus

~V

is an element

homogeneous

of degree

(3.4a)

of the polynomial r.

By (3.3a)

ring

Z[XI,.,.~Xn] ,

it is symmetric,

and is

Jn fact

Z dimK V% • mt(X 1 ..... Xn) ,

+v(XI ..... Xn) =

k the sum being over all dominant monomial

symmetric

of the distinct Xl,... , X n. character

function

monomials

V =

ArE

k E A+(n,r).

(see for example

obtained

For example,

of

weights

let

r

(see 3.2)

from

[M, p. ii]),

kI A n X1 ... X n

be in the range is the

r th

Here

~k

is the

i.e.

the sum

by permuting

0 ! r ! n,

elementar7

then the

symmetric

function

e = ~(i,i,...,I,0 ..... 0) = XIX 2 ... X r + ... --r The propositions If

0 ~ V1 ~ V + V 2 + 0

in 3.3 give rise to propositions is an exact

sequence

in

~(n,r),

about characters. then

~V

=

~V 1

+

2 ,

by (3.3b).

It follows

series for

V,

by induction

on the length

g

say

V : V ° D V1 D %/2 D .. . D Vii = 0,

that

(3.4b)

~V = o=ig %o-I/Vo From

(3.3c) we have

of a composition

42 when

V E MK(n,r),

W E ~(n,s);

this extends of course to a similar formula

for tensor products of any finite number of factors. = (~i''''' ~r ) ![i + "'" + ~ r

is any partition of

= r)

r

....> I~r ~ 0

~I

and

then the s~wmetric function

~ ( X I ..... Xn)

is the character of the module ~V

(i.e.

For example if

lies in the ring

~(n,r)

~i

.o°

e

~Ir

£ ~i E ®... ® A Zr E.

Since every character

of all symmetric functions in

Z[XI,... , Xn],

and since by the fundamental theorem on symmetric functions ([M, (2.4), p. 13]) S(n,r) is Z-spanned by the e (3.4d)

Theorem.

by all characters

above, we have the well-known

The additive subgroup of ~V' V E MK(n,r),

group is independent of the field

is

Z[XI,,.., Xn]

S(n,r).

In particular, this additive

K.

We must next connect our "formal" character character

~V

of

V = (V,p) E ~ ( n , r ) ,

~v(g) - Trace p(g),

~V

is an element of

~(n,r),

(rab)

(3.4e)

Theorem (see [S, p, 17]).

g E rK = GLn(K)

afforded by any basis

there holds

are the eigenvalues of

g.

Let

~V

with the natural

defined hy

all

in fact

matrix

which is generated

g E r

~V

[Va }

= GLn(K ).

is the trace of the invariant of

V.

V E MK(n,r ).

~T(g) = %T(
Then for any where

43

Proof

By (3.3d), the character

module L.

VL = V ® K

L (~(n,r)

@V

obtained by extending

Since this process replaces

coincides with that

K

on

FK'

~V

K.

C

be the

by a function on

by the

to a larger field I = GLn(L)

which

we may legitimately assume, in proving (3.4e),

n x n

Define elements

(I)

matrix

(c v) ,

and let

fl' .... fn ( ~ ( n , r )

u

be an indeterminate

by

det(ul-C) = u n - fl un-I + ... + (-l)nfn

It is clear that

f (g) = ~r(~ , ~ ), r 1 ~''" n %

of

~(n,r)

by

~

% (K1 . . . . . ~n )

-I

g

r,

=

as above.

Then we have, for any

having

dim V ~ diagonal terms

traces

we have

~i ~i

g ( FK

~"(g) "

Relative to a basis of

V = Z~ Vg~ diag(~l,...,~ n)

Now we may write

Define the element

is diagovalizable i.e. that there is some

= diag(~l,...,~n).

decomposition

of

I < r < n. -- -

(bg ( Z ) .

~ = Z (b .iK) f~l ...f~rr "

(2) Now suppose

for

~i ~r = Z b~ -~I "'" ~r

the sum being over the partitions

zgz

K

V

is algebraically closed. Let

over

@V

is unchanged if we replace

V

z ~ FK

such that

which is adapted to the

is represented by a diagonal matrix

an "'" ~n '

for each

~ (A(n,r).

Taking

44

2

We have now two polynomials in

n

and

~(g) = ~v(g)

~V'

and by (2) and (3,),

diagonalizable elements of are distinct belongs to satisfying polynomial

d(g) # O, (i).

Corollary.

F K = f~n(K).

D,

we have

where

~ = ~V'

~i''''' ~t

namely

for all

g

~(g) = ~v(g )

for all

~i''''' ~t

Proof

the natural characters ~i''''' ~t elements of If

~(n,r).

z .

are the characteristics of a set of VI,... , V t ~ ~ ( n , r ) . S(n,r).

(see [CR, p. 184, (27.13)] shows that FI,... , F t

are linearly independent

is a non-trivial relation

p = char K

Then by (3.4e)

g ~ FK

(It is here that we need absolute irreducibility.)

Zl~ 1 + ... + zt~ t = 0

assume, in case

of

of

and this completes the proof of (3.4e).

are linearly independent elements of

A theorem of Frobenius-Schur

D

is the discriminant of the

mutually non-isomorphic, absolutely irreducible modules Then

in the set

Since every matrix whose eigenvalues

d ~ ~(n,r(r-l))

It follows

Suppose that

variables c v ,

is finite, that

p

(z~ E Z),

we may

does not divide all the

(Zl.iK)~l(g) + ... + (zt'iK)~t(g) = 0

for all

g ~ FK-

But this contradicts the Frobenius-Schur theorem.

3.5

Irreducible modules in

~(n,r)

The next theorem, forerunner of far-reaching generalizations by Weyl [We]

and Chevalley(

=ee[S~])

, is due in the case

K = ~

to Schur

Is, p. 37]. The leading term of a polynomial in to the usual lexicographical of monomials

X1

... X

an n

Z[XI,... , Xn]

(Gaussian) ordering of weights

(see [S, p. 17]).

is taken relative a ~ A(n,r),

or

45 (3.5a)

Theorem

Let

any infinite field. (i)

For each

FX, K E ~ ( n , r )

n,r

be given integers,

whose character

These

(iii)

Every irreducible module

Proof to

(i)

k.

(k E A+(n,r))

[S, p. 37]

Let

U

X kl 1 ... X kn n .

a E K

such that

U' = {u 6 U : 0(u) = a.u} equal to the scalar map (ii) for

Z-basis of

V E ~(n,r)

~ = (~l,...,~r) V =

~(n,r). Fk, K

for

We may take

be the partition conjugate A~IE

~V U

of

• k,K(k E A+(n,r))

V

~

and since e(u) = a.u

e

to be

FX, K.

Uk

u E U k.

is a submodule of

kn ... X n .

By

To prove that

U

KFK-endomorphism

Since our assumption on

must map

for

kl

has character

whose character has

it is enough to show that every

U,

into itself, there

But the set

hence

U = U'

and

@

is

a-~. mk

If we express the functions

equations of the form

~... ® A~rE

is therefore

U

The monomial symmetric functions S(n,r).

~I Xn X 1 ... X n

is isomorphic to

is scalar (see [CR, p. 202, (29.13)].

shows that dim U k = i,

is some

form a

The leading term of

is absolutely irreducible,

~U

has leading term

there is some composition factor

leading term

of

be

~ E A+(n,r).

~V = hi~I " " ~ ~r

0

~,K

We saw in 3.4 that the module

(3.4b)

K

these exists an absolutely irreducible module

(ii)

exactly one

Let

Then

k E A+(n,r)

~,K

n ~ I, r t 0.

~X,K = ex +

Z

~

also form a basis for

(k E A+(n,r))

form a basis

@X,K

in terms of these, we get

~

and it follows at

,

S(n,r).

46 (iii) by

Suppose

Fk, K

L

L

is an algebraically closed field containing

the module

Fk, K L .

irreducible, so is ¢~ = ~k,K

as

~ , K ® K L 6 ~(n,r).

6 A+(n,r))

Fk,KL ,

Fk, K

Denote

is absolutely

Fk, K L has the same character

On the other hand

FX, K , by (3.2d).

isomorphic to one of the

Since

K.

Any irreducible

X 6 ~(n,r)

must be

since otherwise the characters

~X' ~k

would be linearly independent by (3.4e) Corollary, and that

would contradict (ii) above. Now let

V

be any irreducible module in

a minimal submodule of that

Fk,K L ~ X "

V L = V ~K L.

hence the space

HomSK(n,r)(Fk,K,. , V) # O.

Therefore

MK(n,r) , and let

There is some

k 6 A+(n,r),

IIomsL(n,r)(Fk,KL , VL) # 0,

X

be

then, such it follows

V ~ F>~,K, and the proof of Theorem

(3.5a) is complete.

Remarks. (i)

The proof above shows that

isomorphism in

~(n,r),

FX, K

by the properties

is defined, uniquely up to

(a)

Fk, K ~ ~(n,r)

is

irreducible, and (b) FX, K has character CX,K whose leading term is kI kn X 1 ... X Moreover if L is any field containing K, then n

Fk,K ® K L 6 ~(n,r)

has the corresponding properties

fact its character is the same as that of ~k,K = ~k,L ~k,K Write

if

K, L

for

~k,K'

{~h,p : k6 A+(n,r)} dk~

are infinite fields with

is the same, for all infinite fields ~k,p

Fk, K.

is a

if

p = char. K. Z-basis of

which appear in the equations

K

In other words, K _c L.

It follows that

of given characteristic.

For each

~(n,r).

(a), (b), since in

If

p,

the set

p > 0,

the coefficients

47

=

Z ~EA+(n,r)

@k,O

k E A+(n,r),

dk~'k, p,

are the decomposition numbers relative to the modular reduction from to

~(n,r),

(ii)

where

K

is any field of characteristic

Suppose that

K

is fixed.

ring for the category corresponding

~(n),

to a module

was said in 3.3, that R(~(n))

R(MK(n ))

and let

[V]

V E ~(n).

[V] -~ ~V

onto the ring

Let

p

MQ(n,r)

(see 2.5).

be the Grothendiock

be the element of

R(MK(n))

Then it as clear from (3.4a) and what

defines an isomorphism of rings from

S(n) =

% _S_(n,r) of all symmetric functions in r>0

Z[X I,..., X n]. (iii)

The symmetric functions

~k,0

were determined by Schur [S,

and are now known as Schur functions or finite characteristic

p,

S-functions

for

and write

X1

... X ~n n ; s(w)

~k

,P

For

have not yet

We sketch here a proof, rather different from

Sehur's, of his theorem for characteristic X~

(see [M, p.24]).

the irreducible characters

been calculated explicitly.

23],

recall that

for the "sign"

(+i)

zero.

W = G(n)

If

e E A(n,r)

acts on

A(n,r)

of a permutation

w E W.

we write (see 3.1), Define

a = ~ ( X l , .... Xn) = Z s(w) X w(~), an alternating function in =~ wEW which in fact is expressible as the n × n determinant Xl,..., 1 Let 6 = (n-l, n-2,...,l,O) E A(n, ~ n(n-1)). Define the S-function iX

= [k(Xl ..... Xn)

for any

(see [M, (3.2). (4.3)]) (in fact

the

k E A+(n,r)

~k (k E A+(n,r))

has leading term X%), n

~l = a~+6/-9~ " form a

-

Then

Z-basis of

S(n,r)

and there holds the formal identity

1

l _ X vV ~ ~,v=l

by

~. "'IXi]I.

Z + kEA (n)

S_.(X)S=x(Y),

48

where and

Y = (YI""' A+(n) =

Yn )

is a set of variables independent of

X = (XI,..., Xn),

~ A+(n,r). r~0

Schur's theorem [S, §23]

is that

¢'k,0 = ~

'

for all

k 6 A+(n,r).

To prove this, it is enough to show that (I) remains true when replaced by

#~0's.

given degree

r

(2)

For then, considering only the part of (I) which is of

in (both)

%

X

and

Y,

S%(X)Sx(Y) =

k~A+(n,r) = If we write

~% =

Z vk~ ~

we shall have

Z +

=

~k o(X)@x,O (Y)"

k(A (n,r) (k 6 A+(n,r))

be a signed permutation matrix, i.e.

the

to order and sign (cf. [M, p. 35]). X k,

we must have

So we must prove (I), with

'

we get an integral matrix

which, on account of (2), must be orthogonal.

leading term

S=>'s are

This matrix must therefore

S=k's coincide with

But since both

S) = ....

~)

k,O

Sk's

g=X

for all

replaced by

and

~k,0's ~'k

coincide on the two sides of the proposed equation.

Z (r)

I] X ~v y ~v ~,v=l

=

~+

X~A (n,r)

r Let

module

have

~k,0's.

It is clearly r ~ O,

So we must prove

~X,o(X)~x,O(7),

where the sum on the left is over all non-negative that

up

X 6 A+(n,r)

sufficient to prove that the sums of terms, of each given degree

(3)

(v~v)

integrsl

matrices such

= r. K

@r EK

irreducible faithfully on

be an infinite field of characteristic is completely reducible Fk, K (k E A+(n,r)) E 6Y,

zero.

Because the

(2.6e), and contains each (absolutely)

with positive multiplicity

see (2.6c)) we have

(SK(n,r)

acts

49

•~i(n,r) =

Now

~(n,r)

is a

~(n,r).

y = diag(y I ..... yn )

cf(~l,K ) .

K~-bii'~odule, using right and left translation

operators g o c,c o g (see sub-bimodule of

~ ~ + h~A (n,r)

~ i. Each coefficient space Take diagonal matrices

(~ ,yy E K ),

ef(F~.,K)

is a

x = diag(xl,...,Xn),

and calculate the trace

f(x,y)

of

the linear transformation

(c ~ y o c o x : c E ~ ( n , r ) ) .

Using the basis of monomials is obtained by substituting

ci, j x

for

for X,

But from (4) we get another basis of A+(n,r)

(r~b)

for

the coefficient functions

F~, K,

relative

~(n,r), y

~(n,r), rab

to some b a s i s of

theorem (see 3.4) shows t h a t t h e s e f u n c t i o n s If we choose a basis for Fk, K

for

we find that Y

f(x,y)

in the left side of (3).

by taking for each

appearing in an invari~nt matrix

F~, K. (The F r o b e n i u s - S c h u r k rab are l i n e a r l y i n d e p e n d e n t . )

which is adapted to the weight-space

decomposition, so that each basis element belongs to some weight-space it is easy to show that the trace of the map is

~(x)~k(y);

hence by (4),

of the variables

(c -~ y o c o x : c E c f ( ~ , K ) )

f(x,y) = Z~ ~ ( x ) ~ ( y ) .

since it holds for arbitrary substitutions XI''''' Xn' YI''''' Yn"

F~, K,

But this proves (3), * Xl''''' Xn' Y I ' ' ' " Yn ~ K

§4.

The modules

4.1

Preamble

Dk.~K In this section and the next we shall define, for each

k = (kl,... , kn) Dk, K

and

in

Vk, K

A+(n,r)

(see Introduction).

We shall define in

~(n,r)

case

(see 4.4).

K = ~,

and for each infinite field

~,K

Both have character

K,

the modules

~k,0 = S~\(XI''"'Xn)"

in terms of certain determinantal functions

These modules have been known, in the "classical"

for a very long time -- they were discovered (under the guise

of "primary convariants") by J. Deruyts in 1892 [D, p. 71] (1) .

More recently

they have been described, for fields of arbitrary characteristic, by M. Clausen [CI, p , i 8 0 ] , and by G. D. James [Ja, p. 129]. refers to the module

DX, K

in fact James

as a "Weyl module", but we prefer to reserve this term for

Vh, K

which was defined by Carter-Lusztig

as we show in 5.2, is the contravariant dual of However James's construction in [Ja ] ours, since it can be used to identify

Dh, K

[CL, p. 211], and which,

DX, K. gives more information than

with the "induced module" of

a one-dimensional character on the lower triangular Borel subgroup

Bn(K) 4.2.

of

F K = GLn(K).

k-tableaux

We mention this identification

For the rest of

The diagram (or "shape") of

§4, k

[k] = {(s,t) : 1 e s,

in 4.8.

k = (k I ..... kn) ~ A+(n,r)

is fixed.

is defined to be the subset i ~ t ! k } S

of

Z x Z

(cf. [DKR, p. 66]).

bijective, of

(I)

[k]

into a set.

A

k-tableau is a map, not necessarily Since

[k]

has

r

elements, there exists

I am indebted to J. Towber for this reference. In his article IT], Towber

gives an account of the history of Dh,K; see particularly IT, po 448].

51

at least one bijection

T : IX] ~ r.

We shall arbitrarily

biJection,

and call it the basic k -tabLeau ~

of

is

(s,t)

(4.2a)

x(s,t),

we may depict

. . . . . .

x(2,1)

x(2,2)

...

x(3,1)

x(3,2)

...

x(l,k I)

x(2,k2)

°i,

p 6 r

appears

we say that

p

R(T)

is the subgroup

preserve

T

as

x(l,2)

Thus every element

T

If the image under

x(l,l)

• .,

of

T

T = Tk .

choose one such

is in row

s

once in (4.2a).

and in column of

the rows of (4.2a),

exactly

G = G(~)

t of

T.

p = x(s,t),

The row stabilizer

consisting

and the column

If

of all

stabilizer

n E G

C(T)

which

is defined

similarly. If k-tableau

i = (il,... , Jr)

is an element

iT : [k] ~ ~

T i.

= x(s,t), (s,t)

we often

place,

of

by

refer

to

of

In general, i

I(n,r), Ti

as the entry

P

we denote

the

is not bijective. in place

P ,

If

or in the

T.. I

Example basic

Suppose k-tableau

i i 3 i5)

r = 5, T = I1 \2

Ti =

i2

4.3°

Bideterminants

of l(n,r). formula

i4

n = 3 3 4

51 /

The entries

We define

Let

and

K

k = (3,2,0).

Then,

of

T.l

for any

belong

be an infinite

an element

(T i : Tj)=

We might

take for our

i 6 I(3,5),

to the set: _3 = {1,2,3}.

field,

and let

(T i : Tj) K

of

i,J

be elements

AK(n,r)

by the

52

(4.3a)

(T i : Tj) =

Here

s(o)

ci,jo

=

c

~ s(o)ci,jo = o~C(T)

is the sign of -i

io

"

o.

for any

Z o~C(T) s(°)ci°'J

The second equality comes from the fact that

~ ~ G(r).

,j

Apart from the interchange of rows and columns, bideterminant

in the sense of Desarmenien,

= (~I''''' ~r )

be the partition of

might not be an element of Then it is easy to see that t

of

[k],

of the

Gt x ~t

x(s,t),

(If

~t = O,

we take this determinant

to be

s(o)(T i : Tj),

Let

for all

(T i : Tj)

22

'

Ti =

Cla

Clb

Cl d

Clelclf|

C2a

e2b

C2d

C2e I

be

n

parts.)

Ti,

= 1,...,~ t

i.)

From this, or directly

is zero if there are equal entries at or of

Tj.

Also

(T i : Tjo) = (Tio : Tj) =

a E C(T).

k = (3,2,0),

T~ =

(notice that

determinants

s,s'

any two places in a column of

k

Let

is the product, over all columns

x(s',t)

from (4.3a), we see that

Example I.

conjugate to

since it might have more than

(T i : Tj)

is a

Kung and Rota [DKR, p. 67].

~

A+(n,r),

(T i : Tj)

T

be as in the example in 4.2. .

Then

(Tg : Ti)

is equal to

Let

53 ,Example 2.

Let

(4.3h)

be the element of

Tg

i

1

2

2

3

3

whose

l(n,r)

•. ,2 , .

k-tableau is

I1 • J

In other words,

gx(s,t) = s,

for all

(s,t) ~ [hi. Then

(Tg : T~) = C(~l)C(~ 2) ... where, for any integer of the

4.4.

n × n

matrix

Definition of The space

m(0 ~ m ~ n),

denotes the

DX, K

AK(n,r )

h,j ~ I(n,r) = I

is a blmodule for

KF,

Z ~(ci,j)Ch, i i~l

(4.4a')

Ch, j o ~ =

Z ~(Ch,i)ci, j, iEI

~(n,r), ~(n,r)

As left module

acting SK(n,r).

~(n,r)

and

belongs to the category

and as right module it belongs to the analogously defined category of all right, finite-dimensional

coefficient space lies in ~(n,r),

F = FK

we have (see

~ o Oh, j =

~ ~ SK(n,r ).

with

Equivalently, it is a bimodule for

(4.4a)

for all

mth leading minor

C = (c v).

by right and left translations. If

c(m)

~(n,r).

KF

(or

SK(n,r))-mo~ules whose

In fact the coefficient space of

whether regarded as left or right

KF-module, is precisely

AK(n,r).

54

Now let that

~

~

be the element of

belongs to the weight

Definition

Dk, K

is the

l(n,r)

defined in (4.3b).

It is clear

k = (kl,... , in).

K-span of the bideterminants

(Tz : T!),

for all

i E l(n,r).

Dk, K

is therefore a subspace of

(4.4a), multiply by

L ~.~.

s(o),

~ o(Tg,j) =

It follows that

Dk, K

~(n,r).

and sum over all

a

% ~(ci,j)(T g : Ti) , i61 is a left

Replace in

all

SK(n,r)-submodule

h

C(T).

by

Zo

in

We get

< E SK(n,r ).

of

AK(n,r).

If we

compare (4.4b) with (2.6a), we see also that the surJective linear map = ~K : EK ®r + Dk,K' an

SK(n,r)-module

Remark basic

k-tableau

i 6 l(n,r),

T

and of

(see 4.2).

T' = ~T

and if

g

for some

Z, = Z -i

In case

~ 6 G(r).

we have

k = (r,0,...,O)

T i = (i I .... , Jr)

map this to the monomial Dr, K

j E l(n,r),

is

is isomorphic to the

and

ell "'" eir rth

depends our choice of

However any other bijective

diagram is a single row of length

i E I(n,r),

for all

(Tg : T i)

Then

T i' = TiT

k-tableau

(% of

and

Therefore

T.

we shall ~ i t e r,

T'

for any

(Ti, : Ti) = (Tz : T i ).

is in fact independent of the choice of

Example i. k

e3 -" (T~:j)

homomorphlsm.

Our definition of

can be written

Dk, K

which takes

Dr, K

for

Tg = (I,I,...,i).

: T i) = Cl,il ... Cl,ir,

DX,K.

For any If we

2.6, Example 2, we see that

symmetric power of

E.

The

55 Example 2.

Assume

we shall write and

r ~ n.

In case

k = (i,i,...,I,0,...,0)

with

r

l's,

for Dk~ K. The k diagram is a single column, (Ir),K has entries 1,2 .... ,r. For any i E I(n,r), ( % : Ti) is the

T

D

r-rowed determinant we see that

D

det(c~, i ).

Mapping this onto

is isomorphic to the

rth

ell A...A e ir

(see 3.2), r

exterior power

A EKO

(ir),K 4.5.

The basis theorem for As usual,

K

is an infinite field.

(4.5a) ...............Basis ...... theorem for such that

Ti

Dk, K

Dk~ K.

DX, K

Our aim is to prove the

has

K-basis consisting of all

is "standard", i.e. the entries in each row of

Ti

(Tg : T i)

are weakly

increasing

(!)

from left to right, and the entries in each column are strictly

increasing

~)

from top to bottom.

This theorem generalizes the theorem of Specht-Garnir for Specht modules (see [P] and [Ja, §§8, 13]).

It may be deduced from a more general basis

theorem of D@sarm~nien, Kung and Rota [DKR, p. 78]. also to prepare for the transition to the module

For completeness, and

Vk, K

in

§5,

we give here

a proof of (4.5a) based on a combinatorial lemma of Carter-Lusztig (see 4.6). We begin by showing that (4.5b) All the

The set

{(T~ : Ti) : T i

(Tg : T i) (i E I(n,r))

k~(n,r) = ~(n,r) o ~k"

in = j.

lie in the "right"

eg, i (i E I(n,r))

if and only if there is some

The condition

stabilizer of the basic

is linearly independent. k-weight-space

For it is clear from (2.3c) that

is spanned by the elements o~, i = cg,j

standard}

~ E G(r)

g~ = g is equivalent to X-tableau (4.2a).

(remember

$o

kAK(n,r) ~k= ~,~).

such that

~ E R(T), c~, i = cg,j

~

Now = g and

the row if and only if

56

T i, Tj

can be obtained from one another by a row permutation (thus the

correspond to James's For each ~s(i) if

(4.5c)

"k-tabloids" [Ja. pp. I0, 127]).

i 6 I(n,r), define

If

i,j 6 I(n,r)

and

~(i)

usual lexicographic way.

If

Ti

s

(~l(i) ..... ~n(i)), of

T i.

Clearly

where

~(i) = ~(j)

by what we have just said; hence

We order these vectors

(4.5d)

~(i) =

is the sum of the entries in row

c4, i = cg,j,,

c~,i

~(i) # ~(j),

then

cg, i # c

(they are in fact elements of

j.

A(n,r))

in the

The reader may verify

is standard and if

1 # ~ ~ C(T),

then

~(i~) > ~(i).

Now suppose we have a non-trivial linear relation

Z fi(Tg : T i) = O, i6H

fi 6 K,

where the sum is over a non-empty subset that

Ti

is standard.

(4.5e)

H

of those

fi # 0

i ~ I(n,r)

such

By (4.3a), this gives

Z i~H

Z fi s(~) cg = 0. ~fC(T) ,in

By (4.5c) we can equate to zero the partial sum of all terms (i,~)

in (4.5e)

such that

the

~(i~)

least of the for

is equal to any given

~(i~)

(i ~ H,

i ~ H, ~ ~ C(T),

~ ~ A(n,r).

~ C(T)). only if

By (4.5d),

~ = i.

Take for

6,

we can have

~ = ~(i~),

So the partial sum has the

form Z f =0, i~H' i c~,i for some non-empty subset

H'

of

H.

But since

Ti

is standard for all

57

i E H',

the

c~, i

appearing in (4.5f) are linearly independent, and this

gives a contradiction.

4.6.

This proves (4.5b).

The Carter-Lusztig lemma To complete the proof of the basis theorem (4.5a) we must show that

every bideterminsnt combination of

(Tg : Ti)

(Tg : T i)

(i E I(n,r))

for which

Ti

is expressible as a linear

is standard.

This is the harder

part of the proof of (4.5a), and we deduce it from a combinatorial lemma (4.6a) of Carter-Lusztig.

The proof of (4.6a) is given in [CL, pp. 214, 215],

and does not depend essentially on other results in

[CL].

The reader may

note some variations of expmession between Carter-Lusztig's paper and this one: our

their "semi-standard" is our "standard"; their Ti ;

their

oT

to the "weight" of

is our

i.

the "type"

k'

corresponds to

of

T( = T i)

corresponds

Finally, in our (4.6a) we have no > restricted

to tableaux of a given type

(4.6a)

Tio ;

T

f

k v.

Carter-Lusztig lemma Let

f : I(n,r) ~ F

be any map with values in an abelian group

F,

satisfying the following three conditions: (i)

f(i) = O,

if

Ti

has equal entries at two distinct places in the

same column. (ii)

f(io) = s(q)f(i),

(iii) (Garnir relations) for any ~+i and

i E l(n,r)

of the basic G(J)

for any

i E l(n,r)

and

~ E C(T).

Z s(v)f(iv) = O, vEG(J)

and any non-empty subset k-tableau

T.

Here

h

J

of the

(h+l)th column

is any element of

is a transversal of the set of cosets

{vX : v E Y},

{1,2 .... ,r-l}, where

Y

58

is the subgroup of

G(r)

consisting of all

element outside

U J,

and

Then {f(i) : T. i Remarks

~

Im f

~ G(r)

which fix every

X = C(T) @ Y.

lies in the subgroup of

F

generated by the set

standard}.

Condition (ii) shows that the sum in (iii) is independent of the

choice of transversal

G(J).

Condition (iii) is equivalent, by an argument

given in [CL, p. 212], to condition (37) of

[CL, p, 214].

A slightly weaker

form of (4.6a), which uses a larger set of "Garnir relations" (but is still adequate for our purposes) can be proved by an adaptation of the proof of Garnir [Ga] for Specht modules; see [P, pp. 93, 94] or [Ja, pp. 29, 30]. Lemma (4.6a) completes the proof of the basis theorem (4.5a), as soon as we observe (4.6b)

The function

f(i) = (Tg : Ti)

satisfies conditions (i), (ii) and

(iii) of (4.6a). Proof

That

f(i)

satisfies (i) and (ii) has already been said in 4.3.

The proof of (iii) is like that for Specht polynomial~. ~ Every element of the set with (with

v ~ G(J)

and

B = Y • C(T)

~ ~ C(T).

f(i) = (T~ : Ti))

Z v~G(J)

has unique expression

transposition in

becomes

Z ~C(T)

{~,~}, R(T).

, = vo,

So the left side of the Garnir relation (iii)

s(~)s(~)cg,iv °

An argument given in [P, pp. 92, 93] shows that union of subsets

Noos as follows.

in which

K

For such a pair

=

Z ~EB B

s(~)c~ i~' '

can be written as disjoint

(which depends on

~)

is some

5g

s(.)c~,i~

because

c%,i~ K

+ s(~)c~,i~ K

= O,

~K = % for all

= c~K,i ~ = c%,i~ (notice

K ~ R(T)).

Hence the sum above is zero, as required.

4.7.

Some consequences of the basis theore______mm Let

~. ~

~ { A(n,r)

Putting

~(=

be a given weight, and

~a,a )

for

o (Tg : Tj) = (Tg : Tj)

each bideterminant i.e.

~

or zero, according as

(Tg : Tj)

(4.7a) with

j.

(j E I(n,r)) Dk,K,

j E e

or not.

Consequently

is a "weight element" of where

~

Dk, K,

is the weight

Then (4.5a) gives the first statement below:

For each i E ~

an element of

in formula (4.4b) we see that

it belongs to the weight-space

containing

a 6 I(n,r)

and

~ ~ A(n,r), Ti

Dk, Kq

standard.

has

K-basis consisting of all

Hence the character

~D

(T~ : T i)

is equal to X,K

~ ( x I .... , Xn). The second statement in (4.7a) follows fact that the coefficient of number of

h-tableaux

Ti

X~

in

from the first, and from the

S=%(XI,... , X n)

which have "content"

can be proved by a direct combinatorial

(i.e.

is precisely the weight)

~

-- this

argument from the definition of

(see [M, p. 42]). Since when

(4.7b)

~

char K = O,

If

is the character of an irreducible module in we deduce

char K = 0,

then

DX, K

is irreducible.

~(n,r)

~A

60 We show next that the family infinite field

K,

we may write

is defined over

{~,K }

(~

Z.

For each

to denote the element

: Ti) K

(~

: T i)

defined in 4.3.

(4.7c)

The

Z-span

(i E l(n,r)) is a D~, Z to

~,Z

of the elements

Z-form of

~,Q.

6K : ~ , Z

K~

and the maps

(~

: Ti) Q

The family ~,K

{~,K }

is

Z-defined by

which take each

(T~ : Ti) Q ® 1K

D~,K~+ k~(n,r)

is also defined

(Te : Ti) K. Moreover the family of inclusions

over

Z.

Proof

The Carter-Lusztig !emma (4.6a), applied to the function

f(i) = (T~ : Ti)Q,

shows that

(Tg : Ti) Q

{(Tg : Ti) Q : T i But by (4.5a), this set is a basis of to the left action of Dk, Q.

The maps

5K

Sz(n,r ).

is in the

standard}. Dk, Q.

By (4.4b),

It follows that

described above are clearly

are isomorphisms by (4.5a) applied to

Z-span of

DX, K.

DX, Z

D~, Z is a

is invariant Z-form of

SK(n,r)-morphisms , and

The last statement of (4.7c)

is immediate from the definitions. Remark

Dk, Z

is a direct summand, as

Z-module, of

inc Az(n,r ) the exact sequence 0 ~ DX, Z -----> from (4.7c) and the next lemma.

is

Az(n,r);

Z-split

equivalently,

This follows

61 (4.7d)

Lemma.

Z-defined by

Suppose VZ

{VK} , {WK}

and {6K} , W Z

family of morphisms in (i)

If all the 0 + VZ

(ii)

If

}~(n,r),

@Q----> W Z

is

also defined over

eK : V K ~ W K

is a

(see 2.6).

Then

Z

Z-split.

9K

are surJective, then the exact sequence

VZ ~ W Z ~ 0

is

Z-split.

Let of

M = (m~)

be the matrix of

VQ, WQ J which {9K}

and that for each

K,

appropriate bases of has rank

d

Z-generate

are equal to

M' = (m~)

9Q,

VZ, W Z

is defined over ~

= (mB~ • IK)

VK, W K.

= dimQVQ,

relative to bases respectively.

Z

implies

M

9K

The assumption

9K

relative to

are injective, this means

and that it still has rank p.

{v },

is an integral matrix,

is the matrix of

If all the

reduction modulo any rational prime M

Suppose

~(n,r),

are injective, then the exact sequence

that the family

M

{~K }.

all the

Proof {w~}

9K

and

are families of modules in

d,

even after

Hence all elementary divisors of

i, and therefore there exists an integral matrix

such that

M'M = identity.

This proves (i). The proof of (ii)

is similar.

4.8

James's construction of

DX, K

G. D. James has given [Ja, p. 129] a construction of a W X, D~ K"

KFK-mOdule

which he calls a "Weyl module", but which is in fact isomorphic to If

e,~ E A(n,r),

then the elements

ci,j(i ~ ~, J E ~)

may be identified with James's "s-tabloids of type not necessary that either

~ = (~i ..... an)

dominant, i.e. be proper partitions of X~(n,r)

in our language (see 4.5).

r.)

or

of

~(n,r)

~" [Ja, p. 127].

~ = (~i .... , ~n )

James's space

S°'%

(It is

be becomes

He defines, for all pairs of integers

62

s E {i .... ,n} and

v E {0,..., ks+ 1 }

a linear map

k ~s,v :

~(n,r)

~ ~(n,r)

by the rule:

(4.8a)

Ss,v(C~,i) = Zh

sunmed over all ks+ I - v

h 6 l(n,r)

of the entries

Definition such that

%s,v(C) = O

obtained by replacing,

s + I by

[Ja, p. 129]

Ch'i'

Wk

in

e

Tg),

s.

is the set of all elements

for all

(or in

s E {i .... ,n}

c E k~(n,r)

v E {0, .... ks+ I - i}.

By a rather delicate combinatorial argument, James proves a theorem [Ja, 26.3, p. 128] from which it follows readily that the set

{(T$ : T i) : T i

standard}; hence

~

= Dk, K.

definition has a very important group-theoretical

Wk

has as basis

However James's

interpretation,

which we

shall now give. Let

U

= U (K)

be the subgroup of

lower triangular unipotent matrices in generated by the elements

Us(t)

F.

F = GLn(K)

consisting of all

It is well-known that

U-

is

(s 6 {I .... ,n-l}, t E K) where

I 0

"°° i

us(t)

t 1

=

0

(row

S)

(row

s+l)

.

'. i

Now write F

on

~(n)

g = Us(t)

and

(see Introduction)

i E l(n,r).

By definition of the action

63

(4.8b)

cg i o g = '

it is clear that

(4.8c)

For all

such that

h E l(n,r) the entries

then

Cg,h(g ) = tw,

s + i

by

s.

c £ k~(n,r),

ks+l ks+l-V Z t v=0

s E {!,...,n-l}

w

is the number of

Ce,h(g ) = tw,

in

Z

(or in

it is clear that

Te) ,

w

t 6 K.

of

(i E l(n,r)).

%s,v(C) = c, K

Naturally we prove

for all

c E X~(n,r).

is infinite, we see that for any

the conditions

~s,v(C) = O,

all

s E {i, .... n-l},

v E {0,..., ks+l-l}

are equivalent to the conditions c o Us(t) = c,

all

s E {i .... ,n-l},

which in turn are equivalent to the conditions

(4.8e)

for any

~s,v(C),

and

c = ce, i

So by (4.8d), and using the fact that c E kAK(n,r) ,

where

Thus (4.8b) gives a formula

(4.8d), by taking first the case v = ks+l,

(s+l,s)},

In other words,

which can be obtained by replacing,

c o Us(t) =

If

is zero unless

p E r, (gp,h@) E {(i,i) .... ,(n,n),

(gp,hp) = (s+l, s).

(4.8d)

for all

C~,h(g)ch, i •

Cg'h(g) = gglhl "'" gP~rrh

and if (4.8c) is satisfied, p

Z hEI(n,r)

c o u = c,

all

u E Un(K).

t E K,

64 k

~(n,r)

is the right

Extend the character Bn(K) = Tn(K)Un(K) ,

X k of

k-weight-space of Tn(K )

by defining

allows us to reformulate James's

(4.8e)

Theorem

c £ ~(n,r)

The module

Xk

~(n,r)

FK = GL (K)), n of

for all

u ~ Un(K).

This

theorem as follows.

Dk, K = W

k

is the set of all elements

which satisfy the conditions

This shows that

for

(see 4.5).

(see 3.2) to the Borel subgroup

Xk(u) = 1

c 0 b = Xk(b)c,

category

~(n,r)

Bn(K).

Dk, K

all

b 6 B-(K). n

is the induced module

Ind~-(Kk)

in the

(or, in fact, in the category of all rational modules of a

K-B-(K)-module n

Kk

which affords the character

For a discussion of a theorem equivalent to (4.8e), for

semisimple algebraic groups, see [J"~ §i.].

§5

The Carter-Lusztig modules

5.1

Definition of Let

%~,K

~ ~ #~+(n,r)

Denote by

NK

~,K

be given, and let

the kernel of the

defined in 4.4.

Let

VX, K

the canonical form

on

E~rK ' Vk,K

{x (

is also a submodule of

o Vk, K ~ (Dk, K) ,

contravariant

to

DX, K

If

K

is an

NK,

relative to

SK(n,r)-submodule

of

Since

<, > is non-singular,

we

form

(,) : Vk, K × D k K -~ K

by

@r x ~ Vk,K, y ~ E K .

all

dual modules in

V%,K

~ = Sk(X 1 ..... Xn) o ' Vk, K is irreducible, and is isomorphic

has finite characteristic,

V~~,K

and

5.2

Vh~,K is Carter-Lusztig's

then in general

are not isomorphic.

We shall identify

Vk, K

Lusztig in [CL, pp. 211, 222]. more exactly.

"

(by 3.3e))

if char K = 0, then

(see 4.7). Dk,K

,

NK

and because (contravariant)

have the same character

In particular,

and E~.

(x, ~K(y)) = <x,y~

l~(n,r)

--> 0

.~r : <x, Nk;~ = 0}

is contravariant

may define a non-singular,

Hence

,K

(see 2.7, example I):

Vk, K =

<,>

D~

~(n,r)

be the orthogonsl complement to

<,>

(5.1b)

~r ~K : EK ~ ~ , K

SK(n,r)-epJmorphism

®r ~ K --> EK

0 --~ N K

Since

be any infinite field.

We have then an exact sequence in

(5.1a) Delinition.

K

"Weyl module" with the module

~

defined by Carter-

First we must describe

N K = Ker ~K

@6

(5.2a)

NK

(i)

is the

R1

K-span of the subset

consists of all

ei

R = R1 U R2 U R 3

such that

i E I(n,r)

and

of

E~,

where

Ti

has equal

entries in two distinct places in some column. (ii)

R2

cons~ists of all

e i - s(o)elo ,

(iii)

R3

consists of all elements

where

Y

i ~ I(n,r)

s(w)eiv,

where

and

o ~ C(T).

i ~ l(n,r) _

o~G(J) and

J

is a non-empty subset of

Proof the

All the elements of K-span

N'

F = EK®r'-'/m onto function

of

R.

~,K'

R

Ch+I(T)

lie in

for some

N = NK,

h E {1,2 ..... r-l}.

by (4.6b), and so

There is therefore a well-defined given by

~(x+N') = ~K(X),

all

K-map

x ~ E~ r.

N ~

contains from

Now the

f : I(n,r) ~ F given by f(i) = ei+N' clearly satisfies the hypotheses

of (4.6a), hence

F

is K-spanned by the set {e i + N' : T i standard}.

this set onto a basis of implies

NC

N';

hence

~,K'

by (4.5a).

N = N',

Therefore

and (5.2a) is proved.

Ker ~ = O,

But ~ maps

which

As a corollary

we have

(5.2b)

~,K

x E ~EKr which satisfy the

is the set of all elements

following conditions: (i)

<x,

ei>=

0

for all

i ~ I(n,r)

such that

Ti

has equal entries

Jn two distinct places in the same c61umn. (ii)

~

(iii)

= s(o)x %

for all

s(~)x~ -I = 0

o E C(T). any non-empty subset

J

of

~h+l(T), h ~ {l,2,...,r-l}

~ G(J) Proof (5.2a) shows that <x, R

>= 0

for

VX, K

s = 1,2,3.

consists of all

x E E~ r

such that

(5.2b) is an almost immediate consequence

S

of this, together with the fact that an "invariance" condition

<,>

is non-singular, and satisfies

67

(5.2c)

<x~, y> = <x, y~

for all

x,y 6 E K ,

n ~ G(r).

form the deflnitien of

<,>

>

This last condition is verified trivially

(see 2.7, example I).

It is now easy to see that "Weyl module"

-I

Vk, K

~oincides with Carter-Lusztig's

~± ", conditions (28), (29) of [CL, p. 211] are essent~ally

(i), (ii), (iii) of (5.2b).

Example..... I.

If

k = (r,0, "'" ,0)

(i), (il) of (5.25) are vacuous. transpositions all s ~ e t r i c

x

in

contravariant sense to the

over all

~1

e~ = e I

Pn

...e n

for

•

So

Assume

{v

Vk, K

Dr, K

: a ~ A(n,r)},

(v ,e~) = 6

r ~ n

x ~ = x,

for all

is the space of

EK~r. Therefore this module is dual in the

where

The non-singular contravariant form

so that

Conditions

Vk,K"

Condition (iii) says

rth symmetric power

has basis

is given by

[x ample 2. We write

Vr, K

I ~ ~.

(see 5.])

V r,K

v = (h, h +I) (h = i,..., r-l). tensors•

Example i).

we write

for all

~'~

{e ~ : ~ ( A(n,r)}

and that

of

EK va

(see 4.4, =

~ei,

sum

(,) : Vr, K × Dr, K ~ K

a,~ 6 A(n,r).

is a basis of

k = (I,...,I,0 .... ,0)

D

Here r,K"

with

r

l's.

V

for Vk, K. Condition (iii) of (5.2b) is vacuous, and (ir),K conditions (i), (ii) show that V is the space of all antisymmetric (ir),K whatever tensors in E,er K . V (ir),K is an irreducible module in ~ ( n , r ) , the characteristic of

K,

since its character

er(Xl,...,X n)

non-trivial expression as a sum of symmetric functions in Therefore

V

(Ir),K

Is isomorphic to the

rth

has no

Z(XI,...,Xn].

exterior power

68

D(ir), K =

~&rEK

ArEK ~ V

(see 4.4, example 2). which takes each

e

(ir),K

'fhere is an isomorphism

= e s

A ...4

ei

iI

(see 3.3) to r

(eil ® "'" ~ elr )( ~6G Z s(~)~). 5.3

The Carter-Lusztig Carter-Lusztig

is a cyclic module

basis for

Vk, K

have given a basis for [CL, pp. 216-219],

different proof of these results.

VX,K,

V~, K

It Is interesting that the Carter-Lusztig DX, K

If

X

given in (4.5a);

are connected by a certain unimodular matrix

(see (5.3)) which has appeared in work of D~sarmenien

Notation.

VX, K

We shall give here a slightly

basis (see (5.3b)) is not the dual of the basis of these bases of

and shown that

[De, ~.7~ ].

is any subset of the syum~etric group

Ix]

=

z

~,

{x}

G(r),

we shall write

=

n~X These are elements of the group-rlng even belong to

ZG(r).

(5.3a)

be the element of

Let

fg = eg{C(T)}

Proof For

lies in

Verify that y ~ RI U

as in (5.2a) y = e~{G(J)}. B = G(J)C(T) up

g

B

R2 (iii).

KG(r),

l(n,r)

of course.

K = Q,

given in (4.3b).

they

Then

Vk, K .

= O,

for all

there is no problem.

y E ~i U R 2 U R 3

Suppose then that

In the notation just introduced,

Using (5.2c),

<e~{C(T)},

{~, ~K}, each

(see (5.2a)).

y =

% s(v)eiv, v~G(J)

this reads

ei{G(J)} > = <e~, el{B}>,

= YC(T) -- see the proof of (4.6b).

as union of pairs

If

K

where

As in that proof, we break

being a transposition

6g

in

R(T).

Using (5.2c) again, and the fact that

<e%, ei(s(~)~

+ s(~K)~<)> = 0

for each pair.

e%<

= e% ,

Therefore

we have

= 0,

and this completes the proof of (5.3a).

Remark.

fg

is the element denoted by

Now let by

fg.

Since

As

V'

be the

K-space,

V'

V'

is

in [CL, p. 216].

SK(n,r)-submodule of

V~, K

is spanned by the elements

~i,jfg = (~i,jeg){C(T)}, Hence

~

~i,jfg(i,j

we see by (2.6a) that

K-spanned by the elements

which is generated E I(n,r)).

~i,jfg = 0

unless

b i = ~i,gfg(i ~ I(n,r)).

In fact we have the following much more precise statement.

(5.3b)

Theorem

[CL, Theorem 3.5, p. 218].

{b i = ~i,~fg

is a

K-basis for

generated as

Proof

Let

Vk, K.

SK(n,r)

In particular

(or

i,j ~ I(n,r).

that there is some ~ ~ R(T)

iR(T) of

h ~ l(n,r)

~ ~ G(r)

(see (4.3b)).

standard}

i.e.

VX ,K

is

f~.

From (2.6a) we have

~i,ge$ =

where the sum is over all

Ti

V' = Vk, K,

KFK)-module by

(5.3c)

if

: i ~ I(n,r),

The set

with

~i: e h , h such that

h = i~, f = ~ .

(i,~) ~ (h,g), But

g = ~

So the sum in (5.3c) is over the

i.e. such if and only

R(T)-orbit

i.

Now we bring in the form and calculate

(,) : VX, K x D~, K ~ K

introduced in 5.1,

70

(bi, (Tg : Tj)) = =

< ~i,ge~ ,

=

<

and

~(i,j)

i

=

~ eh , h~iR(T)

% s(o),

belong to the same Now suppose

% ~C(T)

o E C(T)

If

this implies

o = i,

(see 4.5), and if the matrix I

(~(i,j)),

= {k E I(n,r)

that

either

i > J

singular.

Note.

5.4

i

and

for all

and

J

i

that

jo

There

R(T)-orbit.

~(i) = ~(jo)

~(jo) > ~(j).

Let

~

denote

running over the set I

i,j ~ I ,

any total order

then

~(i,i) = i.

~(i,j) = (bi, (Tg : Tj)),

standard}

~(i,j) # O.

are in the same

If we give

and clearly

{(Tg : Tj) : Tj

A proof that

such

In any case we have

standard}.

{b i : T i

standard}

~

such

is a unimodular ~(i,j) ~ 0 Hence

and since

is a basis of

is a basis of

>

VX, K.

~ (,)

implies is nonis

Dk, K,

it

This proves (5.3b).

is unimodular is given by D~sarm~nien in [De, ~ . ~ ] .

Some consequences of the basis theorem For each

b i = ~i,~f~ of

o ~ C(T)

then (4.5d) implies

i = j ,

But since

follows that

is equal to

For by what we have shown above,

or

non-singular and

jo

i = j.

~(i) > ~(j) = i ~ J

triangular matrix.

~(i,j),

using (5.3c).

are both standard, and that

with

: Tk

s(o)ejo > ,

sum over all

such that

¢ ~ 1

using (5.2c)

R(T)-orbit.

Ti, Tj

must exist

< ~i,~eg{C(T)}, ej>

ej{C(T)} > ,

This last expression, which we denote (5.3d)

=

i.

From

i ~ I(n,r),

satisfies (5.3b)

it is clear by (2.3c) that the element

~i,i bi = b i , we deduce

thus

~ K, b i ~ V~,

where

is the weight

71

(5.4a)

Let

a (£(n,r).

{b i : i ~ ~,

In particular,

Then

Ti

Vk, K

has

K-basis

standard}

V k,K ~ = K.fg

(since

b~ = ~g,g fg

= f~).

A well-

The element

fg

known argument (see e.g. [J", p. 2]) shows (5.4b)

VX, K

has a unique maximal submodule

does not lie in character

Proof Vk,K,

~ax ,K '

~k,p,

By (5.3b)

V' =

FX,K = Vk,K/~ax ,K

has

Any proper submodule

M

The irreducible module

where

VX, K

p = char. K.

is generated by

since it does not contain

lie in

~ax ,K "

fg,

a proper

Z Vk,K, ask

sum of all proper submodules

M

f&.

has

M k= M N VX, K = 0,

K-subspace of of

and is the unique maximal submodule

VX, K

V~, K.

lies in

V',

hence is proper,

and does not contain

Since

DX, K

is dual to

VX, K

VX, K

M

Therefore the

third statement in (5.4b) now follows from the definition of (see 3.5, Remark (i)) and the fact that kI kn ~k = XI ... Xn + . . . .

and so

of

f~.

The

~k,p

has character

we have the following corollary to

(5.4b).

(5.4c) Hence

Dk, K

has a unique minimal submodule

minK , FX, K D~,

_min uk, K

and

min Dk, K

~

(Fk,K)°.

are isomorphic modules.

Proof.

The second statement follows from the first, and the fact that any

module

V

V°

in

~k(n,r)

(see (3.3e)).

If

has the same character as its (contravariant) dual V

is irreducible

this implies

V ~ V °.

72

Remark

Since

Fk, K

same is true of

has

minK . Dk,

k-weight space of dimension minK Dk,

Therefore

SK(n,r)-module

by, the element

k-weight-space

of

DX, K

is

1

(by (5.4c))

the

contains, hence is generated as

(Tg : Tg).

For (4.7a) shows that the

K (T~ : T~).

This proves (5.4d) below.

A

quite different proof comes from a standard argument for semisimple algebraic groups, using the fact (cf. 4.8) that the action of the upper and lower unipotent F = GL (K). n (5.4d)

(T~ : Tg)

of

@r V~,Q = {x ( EQ : < x, NQ > 0},

Vk, Z = E~Z

Lemma.

(i)

bi,Q

~r EZ ,

The sets

=

Vk, Z.

Vk, Z.

It

Vk, Q

is defined over

NO = Ker

is a

that,

lies in

Since (i) is a Q-basis of

Z-form of

Since

j ~ I(n,r).

Take any

t h e p r o o f of

(5.3b)

j gives

Ti .

(see 5.1).

it has

Vk, Q.

<x,

Sz(n,r) ,

i 6 I(n,r),

(by (5.3b)), the proof of (5.4e)

we have T. J

Z-basis

hence that (i) is a subset of

x = ~k.b. ll

such that

Recall that

standard}

every element

x E E~r Z ,

~

the standard

~r,

We certainly have

k.1 ~ Q"

Ti

for each

Vk, Q

be a c h i e v e d i f we show t h a t

some

~Q

Z.

are both closed to the action of

is clear

~i,~ e~,Q{C(T)}

Z-span of (i). for

Vk, K

where

N Vk, Q

generates the irreducible module DX, K .

{bi, Q : i E l(n,r),

hence so is

will

triangular subgroups of

min

Dk, K

We show next that the family

Proof.

is stable under

See [St. p. 214].

The element

(5.4e)

(Tg : Tg)

is

x ( Vk, Z

is

(sum is over <x,

ej>

standard.

e.> = ~k. !,~ ( i , j ) , j 1

6

in the

Ti Z

standard)

for all

The c a l c u l a t i o n t h e sum b e i n g o v e r

s

But t h e " D e s a r m e n i e n m a t r i x "

~ = ~(i,j)),

in case

in

73

K = Q,

is integral and unimodular.

standard

Remark

E~;

Tj) follows

(5.4e) shows that

hence the inclusion

0 ~ l~, Z ~ EZ ~

in the category EZ~ ® K.

8K

(5.4f)

described.

as submodule of {EK ~}

is

Z-defined

From (5.4e) and (5.3b) it is

induces an isomorphism

bi, Q ® IK~

The family

If we tensor with

0 ~ V~, Z ~ K ~ E~r ~ K

VX, z ~ K

6 K : el, Q ® 1 K ~ el, K.

Z-submodule of

Z-split.

we showed that the family

8~ : Vk,Z®

which maps

is

We shall regard

In 2.6, Example 1

immediate that

Uence (5.4e) is proved.

is a pure

we get an exact sequence

MK(n,r ).

E~ ~r and maps

Ti).

(all

VX, Z = Sz(n,r)f ~.

Vk, Z = E~ r n VX,Q

K

~k i ~ (i,j) ~ Z

(all standard

It is clear that

any infinite field

by

ki ~ Z

So from

bi, K

{Vk, K } is

for all

K -+ VX, K.

i E I(n,r).

Z-defined by

The family of inclusions

So is the family of contravariant

Vk, Z

VX, K ~+ E ~r K

forms

From all this we deduce:

and the maps

6'K just

is also defined over

(')K : VX,K x Dk, K -+ K

Z.

defined in

5.1.

5.5

Contravariant

forms on

J.C. Jantzen (see

V%, K

~], CJ'] ....

)

has studied contravariant forms

on the Weyl modules for a simply-connected,

semisimple algebraic group;

particular,

SLn(K)

his results apply to the group

alteration to our case

F = GL (K) . n

,

in

and extend with little

In this section we shall give an

independent description of the contravariant

forms on the modules

V%, K .

74

We saw that element

is generated

f~ = e~{C(T)}

in the submodule form

V%, K

< , >

on

(5.5a)

of

E mr ,

Emr{C(T)} E mr ,

of

as

S(= SK(n,r))-module

and it follows

E mr

that

by the

VI, K

is contained

We have the canonical

and from (5.2c) we deduce

<x{C(T) }, y> = <x, y{C(T)}>

This allows us to define a "contracted"

contravariant

that

for all

v e r s i o n of

x, y g E mr

< , >

on

Emr{C(T)}

by

the rule

(5.5b)

If

x, y e E ~r ,

define

<<x{C(T)},

y{C(T)} >> = <x, y{C(T)} >

Any ambiguity

arising

from the fact that an element

expressed

x{C(T)}

= x'{C(T)}

as

is eliminated

by (5.5a).

contravariant

form on

symmetric, rule

(5.5b) gives

We might

also m e n t i o n

form on

If we restrict

V%, K ,

=

of

E mr ,

is a symmetric, it to

V%, K

which is moreover

= <e~,e~{C(T)}>

fields

x, x'

may be

~ s(o) oEC(T) << ' >>K '

K , is defined over

Z

we get a

non-zero,

<e~,e~o>

since

= 1

constructed

in

in the sense of

3.

Any contravariant factor, with

.

<< , >>

that the family of forms

this way for all infinite 2.7, Example

It is clear that

<>

Emr{C(T)}

for distinct elements

Emr{C(T)}

contravariant

of

<< , >> .

form

(

,

)

on

V%, K

For the contravariant

coincides, property

up to a scalar

(2.7d),

together with

75

the fact that values

VX, K = Sf~ ,

shows that

(f~,v) , v E VX, K .

with each

v

If

(

,

v s V%, K

)

is

determined by the

is decomposed as sum

belonging to the weight space

V%, K~

v = Ev~ ,

(~ c A(n,r)) ,

then

since weight spaces for distinct weights are orthogonal with respect to (

,

)

(see 3.4),

by the values space is

(f%,v)

K.f%

for

(f~,v) = (f~,vx) , v

(see 5.4a).

Therefore (since for all

we have

in the So

(

<> = i)

i.e.

%-weight space

,

if

)

(

,

)

x VX, K .

is determined

But this weight

is completely determined by (f~,f%) •

(f~,f%) = k ,

then

(v,w) = k<> ,

v,w e VX, K o

In the work of Jantzen which we have mentioned, and also in the earlier work of W.J~ Wong ([W],[W']) , a Weyl module submodule

(5.5c) space

(~W',

of

V .

In our case the result reads as follows.

Theorem 3B, p.362~).

M = {v g VX, K : <
M

M # V%, K , of

is that it provides a method of calculating the maximal

Vmax

submodule

Proof.

V

V%, K (~ ~ X)

of

I,K

since

The radical of

>>= O}

<< , >> ,

i,e.

coincides with the unique maximal

f% ~ax ,K

Since

M .

V~, K

by the contravariant property.

Therefore

M £~%ax -

lies in the sum V'

V'

is orthogonal to

.

v~%ax lies in ,K

M

Also

But we saw in the proof

,K

of all the weight-spaces X VX, K = K.f~ ,

we have

(~ax VX,K ) = <
the

V%, K .

is a submodule of

(5.4b) that

the importance of the contravariant form on

and the proof of (5.5c) is complete.

76

Example.

We shall calculate

of 5.2, Example

i).

one row, and so

Since

the form

I = (r,O,.,.,O),

C(T) = {i} .

to

Vr, K

of the canonical

{v

: ~ e A(n,r)}

and

< , >

E ~r .

form

Since

(r,~)

maxK , M = Vr,

1

has only

is just the restriction to the basis

I, the form is given by where

r~ f

C~n -

of this form is spanned by those

the integer

(notation

Relative

= (r,~).l K ,

C~I . . . .

divides

on

Vr, K

the diagram of

<< , >>

(r, ~)

M

on

Therefore,

given in 5.2, Example

= O (e # B)

So the radical

<< , >>

v

p = char K

for which

.

the irreducible

module

F r , K = V r,~~/vma~ r,~

(see

(5.4b)) has basis

(v

The

a-weight

the character with

(r,e)

+ M : ~ s A(n,r),

space of

Fr, K

of

is

Fr, K

~ O mod p .

~r,p

expansion

(5.5d)

(XI+°..+Xn)r

is

~

r,p

~ 0 m o d p} .

K . ( v ~ + M) =

IX ~ 1

°'°

Since the integers

the multinomial

we have the result:

is

(r,~)

=

~ esA(n,r)

,

for all

x~n n

(r,~)

~i

(r,~)X 1

(which is a polynomial

t h e sum o f t h o s e m o n o m i a l s

X l l ' " .xan n

'

e e A(n,r)

sum over all

.

Hence

~ e A(n,r)

are the coefficients

in

~n

... X n

over

Z ,

which have non-zero

by definition)

coefficients

77

when

(5.5d) is reduced mod

p .

The reader may deduce

case of Steinberg's

"Tensor Product T h e o r e m " ( ~ t ,

r = r0 + r l P

(O < ro,rl,... , ! p - l )

+ ...

i ~r,p(Xl,...,Xn) M = 0

=

symmetric

5.6

function"

Z-forms

of

Fr, 0 = Vr, K .

h =r

and

~ ~eA(n,r)

Of course in case

The character

X ~ (~i, p.14])

p = 0

is the "complete

.

V%,Q

D%,Q

are irreducible, mr

~Q : EQ

In fact the map

isomorphism.

: if

then

In this section we work over the rational V%,Q

p.218])

i

E ~ri,p(X ~ ...,X p ) . i>O ' n

and we can take

from this a special

For

~

÷ DX,Q

is certainly

field

Q .

The modules

and isomorphic

to each other

i n d u c e s a map

~:Vx,Q ~ DX, Q

a homomorphism, and i t

(see 5.1).

w h i c h i s an

is non-zero

because

(5.6a)

~(f ) =

Therefore

by Schur's

We would Z-form

Z ocC(T)

L ,

a

the argument

L % = Z. Yfz

Z-form of

essential

lemma

V%,Q

for some ,

and

the Z-forms

(since

lying in

V%,Q = Q'fz

O # y ~ Q .

(y-IL)% = Z . f

if we confine our attention

"normalized"

s(o)(T~:Tzo)

=

V%,Q

at the end of 3.3 shows that

of rank 1 '

Z oEC(T)

IC(T) I(T%:T%)

•

~ is an isomorphism.

like to describe

a free Z-submodule that

s(o) %(e%o ) =

by the condition

to

.

For any such

L % = L ~V~,Q

has dimension

It is clear that .

Therefore

Z-forms

L

i),

y-iL

is so is also

we shall lose nothing of

V%,Q

which are

78

(5.6b)

L % = Z.f

We already know two such normalized Sz(n,r).f %

Z-forms, namely

~r V%, Z = E Z {'~ V I , Q

=

(see (5.4e)), and

(5.6c)

XI, z = g-l(ic(T ) IDI,z) .

XI, Z

is a Z-form of

is a

Z-form of

VI,Q , because

DI,Q •

DI, Z

-

hence also

IC(T) IDI,Z

-

It is normalized, because

XI,ZI = ~-I(Ic(T) ID%I,Z) = %-I(z. IC(T) l.(r :T~)) = Z.f% , by (4.7c) and (5.6a).

Our aim is the following theorem.

(5.6d) ~

Verma IV, p.681]). Then

Proof.

contain

write

SZ

L

for

f% ,

Sz(n,r) ).

spaces

La

for

Z-form of

V%,Q

which

it contains

Sz.f ~ = V%, Z

(we shall

<> = <>

using the contravariant property of

SzL ~ L . ~ ~ I .

be any

On the other hand

= <<SzL,f~>> ~ <> , and the fact that

L

Vk, Z C_ L c_ X%, Z .

satisfies (5.6b).

Since

Let

So

But

f%

<< , >>

is orthogonal to all the weight

<> = <> = <>

= Z<> = Z . We have hereby proved that

L

Y%,Z = {y ~ V%,Q : <> ~C Z}

lies in the set

79

So it will'be

enough

to prove

follows

from the argument

let

be any element

z

DX, Q

~(z)

.~ kj 3

=

the sum being over

just given,

of

Y%,Z

"

taking

Using

That

X%,Z ~ Y % , z

L = X%, Z .

the basis

Conversely

theorem

(4.5a) for

Q .

Because

IC(T)I(T%:Tj) ,

j e I(n,r) z e YI,Z

such that

Our definition the case

is standard;

the

k. J

lie

(5.5b) gives

e Z ,

for all

a particularly

simple

i e I(n,r)

formula

.

for

<<

,

>>

in

char K = O , namely 1 <> =IC(T-~

(5.6f)

For if

T. J

we have

<> = <> i i

by

Y%,Z = X%,Z

we may write

(5.6e)

in

that

u = x{C(T)}

(5.2c),

following

(5.6g)

{C(T)} 2 =

for all

u, v E EmriC(T) j "- -~

as in (5.5b), we have

IC(T) I{C(T)}

.

We use

= <x,y{C(T)} 2>

(5.6f)

to

make

the

calculation:

<>

= I kj = E k. ~(i,j) J J J comes

of the D~sarm~nien T. , l

for all standard therefore

, v = y{C(T)}

and also

The last equality

standard

,

from the calculation

coefficient

~(i,j)

the unimodularity T. . J

the proof

Referring

of (5.6d)

.

preceding

Since

(5.6g)

of the Desarmenien to (5.6e),

is complete.

the definition lies in

matrix

this proves

Z

for all

shows that

that

(5.3d)

k° ~ Z J

z s X%, Z , and

§6.

Representation

6.1

The functor

theory of the s y ~ e t r i c

f:~(n,r)

÷ mod KG(r)

group

(r i n )

In this chapter we shall apply our results on the representations FK = GLn(K) , to the representation G(r)

.

IS] . in

The method is to use a process Suppose first that

A(n,r)

;

u-weight

r

space

l's). V~

determines

ES, sections

in

a functor

III, IVy)

Then there exists a weight

~

(notice that

f:MK(n,r)

that in case MK(n,r)

K = ~

are completely

which uses another functor,

in

representation of

MK(n,r)

profitable

MK(n,r)

theory of

G(r)

the

The correspondence

Sehur proved

(see

n < r

by this means he showed that

The proof which we have given later proof in

by an argument

~(r,r)

to

~',

p.77])°

(Is, pp.61-63])

~(n,r)

.

This

6.5.

Of course Schur used his functor about

.

,

hence are determined up to

Schur's

this time from

second functor will be described

;

~ , p.35].

handle the case

V s MK(n,r)

n ,

this funetor gives an equivalence

reducible,

(see

(I,I ..... 1,0 ..... O)

is a vector of length

KG(r)-module.

KG(r)

of this fact, see (2.6e), is essentially

make deductions

~

÷ mod KG(r)

and mod

isomorphism by their characters

ThenSchur was able to

of the symmetric group

invented by Schur in his dissertation

can be regarded as a left

Mc(n,r)

K

We shall see that for any module

between the categories modules

r ! n .

we denote this by

and contains

V ÷ V~

theory over

of

f ,

and its "inverse"

(see 6.2), to

his starting point was the known .

But since we have already got some knowledge

by the "combinatorial"

methods of §§4,5, it is also sometimes

to work in the other direction.

81

Let us keep Any m o d u l e

K, n, r

V ~ MK(n,r)

for any w e i g h t

fixed for the moment, can be r e g a r d e d as left

e ~ A(n,r)

regarded as left

and w r i t e

,

the w e i g h t - s p a c e

S(~)-module, w h e r e

S(e)

S = SK(n,r)

.

S-module, and therefore V~ = ~ V

(see 3.2) can be

denotes the algebra

~ S~

.

We

get then a functor

(6.1a)

f :M~(n,r) ÷ rood S(~)

w h i c h takes each 0:V + V'

in

S(~)

V s MK(n,r)

~(n,r)

is a

to

,

V ~ c mod S(~)

to its r e s t r i c t i o n

K-algebra with

$

,

and each m o r p h i s m

@~ :V ~ ÷ V ,~

as identity element.

If we choose some

G element

i e I(n,r)

(6. Ib)

w h i c h belongs

i = (i 1 ... 1 ~---~w- - ~

to

~ ,

2 2 ... 2 ~

for example

...

n n ... n)

w e m a y use the m u l t i p l i c a t i o n rules in 2.3 to show that as

K-space, b y the elements

~iz,i

follows that, for any elements if

~, ~'

' ~ c G .

~, 7'

of

is spanned,

F r o m the e q u a l i t y rule in 3.2

G ,

~iz,i = $i~',i

of

G .

So

S(~)

has

K-basis

if and only

{~i~,i } ,

over a set of r e p r e s e n t a t i v e s of the d o u b l e - c o s e t space

N o w suppose that

(6.1c)

S(~)

b e l o n g to the same double coset w i t h respect to the subgroup

G~ = {~ E G:i~ = i}

The e l e m e n t

,

r < n ,

and that

(6.1b) c o r r e s p o n d i n g to

u = (l,2,...,n)

Since the stabilizer in

G

e

e = ~

e

G~G

~

running

/G

is the w e i g h t d e s c r i b e d above. is w r i t t e n

l(n,r)

of this element is

.

G

= {I} ,

the algebra

82

S(~) has

K-basis

multiplication ~,~' in

{~u~,u : ~ c G} .

rule

G .

An elementary

(2.3b) shows that

We have therefore

application

~u~,u~u~,, u = g u ~ ' , u

an isomorphism

of

for all

of K-algebras

~

(6. Id)

S(~)

which takes isomorphism

~u~,u ÷ ~

KG(r)

for all

to be the functor

Hecke ring follows. set

K .

~(G,H) H(G,H)

H\G/H

By means of this

and mod KG(r)

can be identified.

the Schur functor

~, p.22~,

+ mod KG(r)

For the general case, where

over

.

f = f

with no restriction HK(G,G ~)

S(~)

we define

f:MK(n,r)

Remark.

~ ~ G = G(r)

the categories mod

With this identification

,

on

n, r)

S(~)

~

is any weight in

is isomorphic

We may follow Iwahori for any subgroup

has a free

Z-basis

of all double-cosets

of H

H

~,

A(n,r)

(and

to the Hecke ring

p.218~

and define the

of any finite group

as

{XA} ,

where

in

the product of elements

G;

A

G ,

runs over the in

this basis is given by

(6.1e)

where if Hv

XA XB =

¥

is any fixed element of

in the set

rule, see

~,

C g H\G/H

A-Iy ~ B . §~].

ZA'B'C XC '

C, ZA,B, C

(For an explanation

Alternatively

we may define

is the number of H-cosets of this artificial-looking ~(G,H)

to be the

83

e n d o m o r p h i s m ring of the subset r e g a r d e d as right

ZG-module.

Z G - e n d o m o r p h i s m of

EH]ZG

[H]ZG

HK(G,H)

R e t u r n i n g n o w to our case

6.2

~H]

(7 ~ G),

to

G = G(r)

becomes If

K

K-algebra

, H = G

,

the

is any

~(G,H)@ Z K .

we leave it as an

S(~) ÷ ~ K ( G , G )

is an i s o m o r p h i s m of

General theory of the functor

XA

~A] .)

to be the

prove that the K - l i n e a r map

~i~,i ÷ XG ~G

ZG , this subset b e i n g

In this i n t e r p r e t a t i o n

w h i c h takes

c o m m u t a t i v e ring, we define

exercise to

of

g i v e n by

K-algebras.

f:mod S + mod eSe

It soon becomes clear that m a n y p r o p e r t i e s of Schur's functor b e l o n g to a m u c h m o r e general context. need to be f i n i t e - d i m e n s i o n a l ) We define a functor the subspace

eV

E mod eSe .

If

f(0):eV ÷ eV' morphism.

of

Let

S

and let

be any e ¢ 0

K-algebra

be any idempotent in

f:mod S ÷ m o d eSe as follows. V

@:V ÷ V'

is an

eSe-module,

to be the r e s t r i c t i o n of

mod

@ ;

It is important to observe that

If

f

S ,

clearly

S .

V s m o d S , clearly

so we define

is a m o r p h i s m in

(it does not

f(V) = eV then w e define

f(@)

is an

eSe-

is an exact functor,

in

other w o r d s

If

(6.2a)

0 + V' ÷ V ÷ V'' * 0

0 ÷ eV' ÷ eV + eV'' ÷ 0

This is quite elementary. well known,

(I)

is an exact sequence in is an exact sequence in

The next proposition,

mod mod

S , eSe

then .

though easy and u n d o u b t e d l y

does not seem to appear in the literature

(I)

(a special case is

Our functor is a special case of functor described by M. A u s l a n d e r Communications indebted

in A l g e b r a

to J. A l p e r i n

for

I(1974),177-268; this

reference.

see p . 2 4 3 .

I am

84

g i v e n by Curtis and F o s s u m 6.2. appears, of

T.

p.402].

M u c h of the p r e s e n t section

sometimes w i t h different proofs,

Martins

(6.2b)

~F,

If

in the Ph.D. d i s s e r t a t i o n

~ ~a]) .

V g mod

S

is irreducible,

then

eV

is either zero or

is an irreducible module in mod eSe.

Proof.

Let

and also

W

be any n o n - z e r o

SW = SeW ,

to

V .

Hence

if

eV # 0 ,

w h i c h is a n o n - z e r o

eV = e(SeW) = ( e S e ) W ~ then

N o w suppose S-submodules

e S e - s u b m o d u l e of

Vo

IV

is an i r r e d u c i b l e

V ~ mod S , of

W .

V

and define

such that

the largest

S-submodule of

V

also define

a(V) = V/V(e ) .

eV.

Then

S - s u b m o d u l e of This proves eSe-module.

V(e )

eV o = O

-

W = eW ,

V ,

is equal

W = eV .

Therefore

This proves

(6.2b).

to be the sum of all the in other words,

w h i c h is contained in

is

V(e)

(I - e)V .

We

Then we can make a functor

a:mod S ÷ m o d S ;

notice that if into

V'

(e)

0:V ÷ V' hence

'

@

is a m o r p h i s m in

mod S ,

induces a w e l l - d e f i n e d map

then

@

maps

V

(e)

a(@):a(V) ÷ a(V')

.

The virtue of this functor,

is that it gets rid of the part of each m o d u l e

V

f ,

in

w h i c h is annihilated b y f(V)

(6.2c)

.

and does this w i t h o u t d e s t r o y i n g a n y t h i n g

E x p r e s s e d precisely, we have

Let

V a mod S .

induces an i s o m o r p h i s m

T h e n the n a t u r a l map

f(cq.):f(V) ÷ f(a(V)). ,s

~v:V ÷ a(V) = ~/V(e )

85

Proof.

Clearly

f(~v)

f(V)

= eV,

is o n t o

zero

since

V(e)~

Our next which we

can

the

objective

Se

also

If

~:W ÷ W'

a right

Ker

of

~V

to is

f ( ~ v ) = eV f ~ V ( e )

is an i s o m o r p h i s m .

functors

partially,

= Se

meS e W

in

Let

W

,

S-module

from

as i n v e r s e s

mod

S .

h(W) in m o d

We g e t

proposition

c mod

for

eSe

.

W

e mod

(it is a left

eSe-module,

the n e x t

(6.2d)

And

f(~v )

is to d e f i n e

is a m o r p h i s m

is a m o r p h i s m Moreover

Thus

.

the r e s t r i c t i o n

mod

to

eSe

f .

to

As

mod

first

S , attempt

definition

is a l e f t

and

is j u s t

= e.a(V)

.

at l e a s t

h(W)

Since

which

f(a(V)) (l-e)V

serve,

employ

,

eSe

.

ideal

of

is w e l l - d e f i n e d eSe

,

then

in this w a y

shows

that

h

Then

e.h(W)

S ,

and

h(~)

is a left

= ~Se

a functor

,

* h(W')

eSe ÷ m o d

inverse"

and

S-module.

H ~:h(W)

h:mod

is a " r i g h t

= e m W

of c o u r s e )

to

f

S. .

the m a p

~ w + e m w(w

Proof. Thus

that there

for all 0 = q(e and

gives

e.h(W)

= e(Se

the m a p

check that

E W)

defined

= w

, w

E W

.

This

is p r o v e d .

.

= eSe

takes

eSe-map.

is a w e l l - d e f i n e d

m w)

eSe-isomorphism

m eSe W)

above

it is an

s E Se

(6.2d)

an

W

onto

Then

q:Se if

establishes

w

e.h(W)

m eSe W = e ~ W

To p r o v e map

W

e.h(W)

that

;

the

,

is s u c h injectivity

.

as stated.

it is e l e m e n t a r y

it is i n j e c t i v e ,

~eSe W + W g W

= f(h(W))

such that of

that

first q(s

notice

m w)

e m w = 0 the m a p

to

,

= esw we

get

w c e ~ w

,

,

88

h

The trouble w i t h the functor i r r e d u c i b l e module

W

is that it u s u a l l y takes an h(W)

to a m o d u l e

w h i c h is not irreducible.

H o w e v e r we have

(6.2e)

If

W ~ m o d eSe

is irreducible,

m a x i m a l proper submodule of

Proof. Thus

Write

an

V = h(W)

a(V) # 0 ,

N o w let

V'

h(W)

.

.

T h e n by

,

is contained in

Definition.

by

Let

V(e ) .

h*

for all

By f ,

W c mod eSe

irreducibles

(6.2f)

Proof.

If

V .

If

eV' # 0

then

eV = e.h(W)

e.h(W)

. So

V .

eV'

, being

(recall

Then eV' = 0 ,

i.e.

(6o2e).

denote the functor

ah: m o d eSe ÷ mod S ,

so that

= h (W) /h (W) (e)

.

f(h*(W)) ~ W

for all

to irreducibles.

V c mod

There is an

f(a(V)) ~ f(V) ~ W°

is a proper submodule of

eSe-module

This proves

is the unique

is irreducible.

a contradiction.

(6.2c) and (6.2d) this functor i.e.

a(h(W))

(6.2d)) is equal to

h*(W)

to

V(e )

e S e - s u b m o d u l e of the i r r e d u c i b l e

h(W)(e )

(6.2d) and (6.2c),

be any p r o p e r submodule of

V' ~_. SeV' = S(e m W) = h(W) = V , V'

Hence

w h i c h shows that

e.h(W) = e m W ~ W

then

S

is

h*

,

like

W ~ m o d eSe

h , .

By

is a right inverse (6.2e)

h*

takes

We have finally

irreducible and if

eV # 0 ,

S-map B:h(eV) = Se ~eSe eV ÷ V ,

then

w h i c h takes

h*(eV) ~ V

s m ev

87

to

sev ,

equals

V

for all

s e Se ,

because

V

proper submodule hence

of

v ~ V .

is irreducible. h(eV)

.

But

the only maximal proper

Therefore onto

B

The image of

induces

is

So the kernel of

eVc

mod eSe

submodule

an isomorphism

B

of

of

SeV , B

is a maximal

is irreducible

h(eV)

is

which

by (6.2b),

h(eV)(e)

,

by (6o2e).

h(eV)/h(eV)(e ) = a(h(eV))

= h*(eV)

V .

Taking

(6.2g)

all these facts, we arrive at our main theorem.

Theorem.

modules Then

together

Let

be a full set of irreducible

in

mod S ,

indexed by a set

{eV%:

% E A' }

is a full set of irreducible

Moreover

Remarks

if

V%, K ,

then

for any

V ~ mod S

eV t O

that

Let

it will be useful

eSe-module

= 0 )

Vmax ,

modules

Hom S (Se, V) ~ eV

(see

to notice

A' = {% g A : eV%# O} .

DR, V

p.375]).

eV m a x

if

V

image of

V ~ mod S

proper

of

is Se .

modules

has a unique

is either equal

to

eV

submodule

(i.e°

of the

The proof is easy.

In the same context we shall use the following:

symmetric bilinear

.

(isomorphism

to the Carter-Lusztig

or else it is the unique maximal

eV .

mod eSe

Therefore

is a homorphic

that if any

then

in

.

When we come to apply the Schur functor

e(V/V max)

denotes

A •

h*(eV%)

if and only if

maximal proper submodule

3.

V%~

It is well known

irreducible,

2.

% s A' ,

I.

K-spaces),

Again,

{V : % s A}

form on

the restriction

V

such that

of this form to

the proof is an easy exercise.

(eV,

If

(l-e)V) = 0 ,

eV , then

rad(

(

,

)

is a

and if (

, ~ = e.rad(

, , )

)e

88

6.3

Application

I. Specht modules

and their duals

In this section we shall apply the general special

case of the Schur functor

any infinite We take

field,

S = SK(n,r)

identify

eSe

~u~,u ÷ w , for any

with

and

n , r

,

e = Sw

KG(r)

for all

V e MK(n,r)

Notice

correspondence We shall write partitions

Recall

of

are fixed integers = ~u,u

w c G = G(r)

A = A+(n,r)

the effect of X

,

that

DX, K

of

that

%

X

f

is

and

takes

f(V) = eV = V ~

on the modules

of

r A

Z s(o) ~EC(T) c£'i~'

all

%-weight

space

of

f(Dx,K)

is the

= ~l(n,r)

w-weight space

space

o ~%

D%,KW

.

as the set of all A

K-span of the elements

(T£:Ti)

%AK(n,r)

r ~ n)

is a fixed element of

is the

D%, K ,

are in one-to-one

(because

We saw (p°55) that these bideterminants

w-weight

K

r ~ n .

(6.1d), which

A+(n,r)

and think of

From now on,

(T£:Ti) =

the (left)

such that

(see 6.1 for notation)

Notice

with the partitions

(p.54)

Here

.

that the elements

r .

÷ mod KG(r).

by the isomorphism

Our aim is to calculate VX,K

f:~(n,r)

theory of 6.2 to the

of

i e l(n,r)

all lie in the right

~(n,r) D%,K

.

,

.

But by definition

and therefore

lies in

89

(6.3a)

of

%AK(n,r) ~ = E o AK(n,r) o E%

%AK(n,r)

.

Elementary calculations based on formulae (4.4a,a')

and (6.3a) show that

if and only if

%~(n,r) ~ =

v' s vR(T)

I K.c ~sG %,u~

Since

C~U~ = C~u~V

(see 4.5 ), there is an isomorphism of

K-spaces

(6.3b)

%AK(n,r) ~ ÷ KG[R(T)]

which takes

c

~(T)]

u~ +

,

left ideal of the group algebra the other hand

%AK(n,r)m ,

SK(n,r)-module

%AK(n,r) ,

(T%:Ti)

such that

~ G

i c w

and

KG[R(T~

hence is a left

KG-module.

f(D~,K)

On

KG-module by means of (6.1d). c

%,uv

to give

C~UT~

It follows

KG-isomorphism.

T.~

has

K-basis consisting of all

is standard.

The elements

i = u~ (~ c G) .

to

to the left KG-submodule

i

in

The isomorphism

o~C(T) and so it takes

is a

~-weight space of the left

which by (4.4a) is equal to

f(D%, K) = D%, K

(T%:Tu)

NOW

acts on the element

can be written, uniquely, in the form (6.3b) takes

~ c G .

becomes a left

at once that (6.3b) is a left

By (4.7a), p.59,

KG ,

being the

To be explicit, the element ~C%,u~ = SuT,u o e%,u~ ,

for all

(left ideal)

90

ST,K = KG (C(T) } ~R(T)]

of

KG .

We shall define

ST, K

to be the Sg e c h t m o d u l e

corresponding to the bijective

X-tableau

T .

(over

K)

(This is a little

different from the original definition of Specht;

for an explanation

of the latter, and of the equivalence of the two definitions, see ~, p.9d.)

(6.3c)

We have now the

Theorem.

~{C(T)}~R(T) i

ST, K

has

K-basis consisting of the elements

such that

Tu~

is standard.

is an irreducible Tx ,

X-tableau full

set

DX, K

Dm x,K is

(by (4.7a)), NK(n,r)

If we choose for each

SX, K = STX ,K ,

.

in

given by (4.7a)

X s A

ST, K

a bijective

{Sx, K : X s A}

Then the this

last

present

look of

is

f(Dx,K)

the module

E~ r

and s i n c e

a full

statement

case

at

already quoted.

by ( 4 . 7 b )

{DX, K : X ~ A}

a subspaee

in

set

is a

g SX, K

VX, K .

and t h e r e f o r e

= (E~r) m .

S = SK(n,r) no r e s t r i c t i o n

on

From the E Hr ,

on

formula

we s e e

n , r)

that

for

DX, K

follows

has

at

is non-zero

By d e f i n i t i o n

f(Vx,K)

(2.6a)

If char K = 0

= Vm X,K

which gives

any weight

then

character

of irreducible

(6.3c)

•

f(E ~r)

then

then

KG-modules.

irreducible

Now l e t ' s is

char K = O

The first statement comes by applying the isomorphism (6.3b) to

the basis of

since

and w r i t e

of irreducible

Proof.

each

KG-module.

If

modules

S in

once from (6.2g), for

all

(see is

X ~ A .

(5.1b))

a sub-space

the action

~ ¢ A(n,r)

this

of (and with

of

91

(Emr) e =

~ E mr

=

E

K.e i

i ~

So in particular of

(Emr) ~

for all

has

{e

: ~ g G} .

u~

KG-module

is given by

T , ~ ~ G .

Therefore

there

The structure

Teu~ = Su~,ueu~

is a left

= eu~

'

KG-isomorphism

(Emr) ~ ÷ KG ,

takes

e u~

By (5.3b), standard}

.

(see 5.3.).

+ ~ ,

for all

(5.4a) weJknow

Recall

that,

If we put

i = u~

bu~ = eu~[R(T) ~

{C(T)}

(6.3d)

to the element

v rR(T)]

to the left

KG-submodule

We have

that

V%, K

has K-basis

i c l(n,r)

in formula

H~

U~

b i = $i,%f% = ~i,%e~{C(T)}

is carried by the isomorphism

KG . Therefore

of

: ~ s G , T

(5.3e) we get ~u~,~e% = eu~[R(T) ]

This element {C(T)}

,

{b

~ K V%,

is carried

(left ideal)

K

KG .

~ ~ G .

for any

Hence

of

K-basis

as left

(6.3d)

which

(Emr) ~

= KG[R T)]

the following

theorem,

whose proof

is entirely

analogous

to that of (6.3c).

(6.3e) ~(T~

Theorem. {C(T)}

is an irreducible

ST, K

such that

has Tu~

KG-module.

K-basis

consisting

is standard. If we choose

If

of the elements char K = O

for each

X e A

then

S--T,K

a bijective

92

X-tableau

T%

and write

~

= S %,K

full set of irreducible

The

modules

module

ST, K

is in fact dual

the contravariant

of this form.

~ ~ G ,

when we regard

form

, D%, K~

that these KG-modules form, by means

the calculation

(V,~u,u~

=

described

as

=

property

in 5.3

in 5.1.

,

But this becomes

KG-modules

to exhibit

the

(2.7d) gives

, ~u~_l,u d)

(v

(~v, d) = (v,~-id)

by means of (6.1d),

are dual to each other.

given

are dual to each

and this shows

Naturally we can transfer

(6.3d),

to give an invariant

to do this,

the following

this

form

and also to apply explicit

version of

form in question.

The

~ , ~' e G , vT

The m a t r i x relative

d)

We lave it to the reader

Theorem.

the tableaux

)

to the Specht

are dual to each other under

KG-modules

There is an invariant bilinear

for all

,

of the isomorphism~(6.3b),

ST, K x ST, K ÷ K

the invariant

(

v ~ V%, K , d e D w ,K

V%, Km

a

=~ a " , p-460 ].

V%, K , D ,K

The contravariant

(~u~,u v , d)

(6.3f)

(in the usual sense)

the modules

w (see 3.3) V%, K, D%, K

restriction

is

• %,K

ST, K - this was first proved by G.D. James

other under

for all

: ~ ~ A}

'

KG-modules.

We can give another proof:

Therefore

then T%,K

where

and

form

~

ST, K

, ST, K

( , ):ST,KXST,K

~, = Es(o)

,

are row-equivalent.

(~ ,~,

: ~T , ~'T standard)

ordering

of the standard

÷ K

such that

sum over all

~'dT

to a suitable

are dual to each other.

is unipotent ~T .

s C(T) such that

triangular,

93

Remarks.

I.

Since

identified with the basic tableau 2. of

The m a t r i x the

e,

3.

(~ ,~,)

Desarmenlen

=

matrix

~(u~,u~')

For if we take

with

~

.

in the last theorem is just that part

(~(i,j))

Therefore

to

corresponding

to

ST, Z = ZG{C(T) }[R(T)]

these last are

and have

Z-bases

{~R(T)]{C(T) } : T u~

standard}.

6.4

Irreducible

Application

Throughout r ,

can be

u

1,j

e w ;

in

II.

fact

is replaced by

are integers

, -ST, Z = ZG~R(T~{ C ( T ) }

{~{C(T) }[R(T)]

that

satisfying

K

: Tu~

ST, Q ,

standard}

,

char K = p .

has finite

r ~n

Z .

(6.3b),

Z-forms of the QG-modules

K G(r)-modules,

this section we assume n

K

then we may check that the isomorphisms

D ~X,Z ' VX,Z ~

respectively,

and that

Tug

in this section remain true when

K = Q ,

respectively. ST, Q

appearing

and

T ,

T

.

All the results

(6 • 3d) take

the tableau

u = (1,2, .... r) ,

.

characteristic

p ,

In 5.4 we constructed

a

+ full set

{F%, K : X s A (n,r) = A}

Apply the Schur functor

(6.4a) subset of

Let A

A

Of course

and we have by (6.2g)

be the set of all partitions

consisting

{F ~~,K : ~ E A' }

f ,

of irreducible modules

of those

%

of

such that

is a full set of irreducible

in the next theorem.

~(n,r)

.

the theorem

r ,

and let

F~ # 0 ~,K

A'

be the

Then

KG(r)-modules.

this still leaves open the crucial question:

The answer is contained

in

what is the set

A' ?

94

(6.4b)

Theorem

3.2]).

(Clausen

The set

A' of (6.4a)

= (Xl,X2,...,Xr,O, for which all O

and

p-i

Proof.

~C£,

... )

We must

of

the integers

module

min D%,K

show that

Therefore for all

X~ # 0

X

which is zero unless

X

is

~i,~ o (T~:T~)

on

i.e. lie between

in (6.4c)

iG = w

the elements

i.e. iT

this module

X is column

K-space

by the elements

but with by

the X •

p-regular.

by

(T%:T%)

.

~i,j o (T :T~)

,

By (4.4b)

=

~ .

E hcl

Ei,j(Ch,~)(T~:Th)

,

Therefore

by the elements

The element

is the weight

We denote

SK(n,r)-module

.

j~

FX, K ,

as

Z ~i,~(Ch,~)(T~:Th) heI

i g I .

elements

p-regular"

not with

if and only if

as

K-spanned =

to work,

is generated

it is spanned

~

which are "column

hi - %2 ' X2 - X3 ' "'" ' %r

~i,j o (T~:T~)

for all

EJa', Theorem

of those partitions

(see (5.4c)).

i,j c I = I(n,r)

(6.4c)

r

James

.

By (5.4d),

where

consists

It will be convenient

isomorphic

Lemma 6.4, p.184],

(6.4c)

containing

such that

=

(T%:Th)

i .

i e ~ .

So

If

X~

is

i e ~ ,

for all

are all distinct.

,

~-weight

lies in the

iT = iT' => ~ = ~' , (~ e R(T))

Z h~iR(T)

K-spanned then

~,~' So

space

G

s G .

X~ ,

by those

acts regularly In particular

95

(6.4d)

If

i E ~ ,

then

that

is the group of all elements

Suppose preserve

the set of columns

permutation each

e

maps Since

t

of the basic

0q of

is a p e r m u t a t i o n T

x(s,t)

to

IWql = %

q

- %

has length x(S,eq(t)) q+l

,

the order of

'

(see 4.3) that

(T%:Ti) = (T%:Ti0)

So by breaking

up the sum in (6.4d) IHI .

If

%

that

%

enough to show that

(6.4e)

p-regular,

into

~u,~ o (T%:T%)

T .

which

Such a

of all

of (4.2a) and all

for such

, t ~ Wq

(%1 - %2)~(%2 - %3 )' .... as product i s I

p-singular, i.e.

R(T)

t ->- 1

H-orbits,

of determinants

and all

~ c H .

we see that it is IHI

is divisible

by

X~ = 0 .

To prove the other half, we assume

and show that # 0 .

X~ # 0 .

By (6.4d)

and

For this it is (4.3a),

is equal to

E E ocC (T) TeR(T)

s (o) c%o,uT

There is a unique element entries

for all

is zero,

one half of (6.4b).

is column

~u,~ o (T :T%)

,

of

Wq

is

(T%:T i)

is column

hence every term (6.4d)

This proves

of

e

.

where

s -> 1 , H

(T%:Ti)

el,e2,... ,

In the notation for all

from the expression

p ,

%-tableau

of the set

q .

Now it follows

divisible by

~ ~eR(T)

can be specified by a sequence

q -> 1 , col

that

H

~i,~ o (T~:T%) =

in each column of

T~ ,

~ E C(T) namely

which reverses

the order of the

9G

~:x(s,t)

For

example

if

TZ =

We

shall

If

~

y E G = ~o

M = 4)

.

columns are

]

2

2

2

2

2

3 4

same

elements

2

2

3

3

1

1

1

3

2

2

4

1

1

the R(T)

that

TZ

row.

places

of

is p r i m e

c£~,u

in

to

in

(6.4e)

entry

also

in

,

hence

T%~

that

, M-2, TZ~

,

~T

= ~

satisfying

is

The

in

T~

,

Tzo

,

all e n t r i e s

also

H

q>l

in

TZ~

hence

iK # 0

just

is hence

are

in the

M

in

T~,q

consider

arguments The

M are

the p l a c e s

similar group

to of

order

,

glven

, and

there

all e n t r i e s

z = J . has

zero.

tile e x a m p l e , M

Next we

and by

1

T = T ,

(in

entries

that

)q

M

TZ~ T ,

.

clearly

then

implies

say

all

((%q-%q+l) ~

argument

s(~)w,

,

is n o t

= c%o,uT, This

in turn,

= TZo

(6.4e)

.

in

...

in

c~r, u

TZ~ T = T~o M

=

cz~,u

that

the e n t r i e s

.

T%~

£~y = Zo , uT = UT

M-I

p

of

such

Since

as

conclude T e R(T)

are

and h e n c e But

1 ,

the m a x i m u m

, .

1

coefficient

w =

which

t g Wq

2

by e n t r i e s

given

and

4

, T e

first

s -> 1 ,

for

4

that

t e WM

occupies that

1

Consider

in the

in the

1

In

,

we h a v e

1

such

.

(7,5,2,2)

1

prove

~ s C(T)

some

% =

÷ x(q+l-s,t)

shows

that

so the p r o o f

of

the

coefficient

(6.4b)

is c o m p l e t e .

97

This theorem has some interesting lemma c o n c e r n i n g

the

left

module for the algebra

ideal

S{

of

w

S(~) = ~ S ~

consequences.

,

S ,

First we need a

which

is

also

a right

and hence, by (6.1d), a right

KG-~aodule.

(6.4f)

S~ w E mr

S~

has

K-basis

given by

and a right

{~i,u : i e I(n,r)}

~i,u + ei

(i g l(n,r))

By 6.2, Remark i, w

,

2.7, Example

V

is

Vw # 0

and hence of

(6.4g)

If

E mr .

V ~ M~(n,r) to

(James

module of

E mr

But both

E mr

S = SK(n,r)-map

is irreducible,

a submodule

~a',

of

1

ST,K = KG[R(T)~{C(T)}

of 6.3.

<< , >>

Emr{C(T)}

on the space

(E~r) w {C(T)} ,

E mr

be irreducible.

is a homomorphic

and

V

image of

are self-dual

Vw # 0

(Notice,

F%, K

is column

(by

if and only if

we a s s u m e

is isomorphic

r

< n

to a sub-

p-regular.

In 5.5 was defined a contravariant .

.)

the "dual" Specht module

Restrict

and then transfer it to

(6.3d).

V

then

Theorem 3.2])

if and only if

V e MK(n,r)

We have therefore

Next we have a theorem concerning

isomorphism

K-isomorphism

is a left

Now let

if and only if

I, and (5.4c), proof).

isomorphic

Corollary

The

KG-map.

The proof of (6.4f) is routine.

S~

.

this to the

KG{C(T)}

The result is a symmetric,

w-weight

form space

by means of the

invariant form on

KG{C(T)}

98

which we denote by if

~, ~' s G ,

(

,

)

and which is specified

,

then

I s(~-l~ ') (6.4h)

(~{C(T)}

In 5.5 we considered and showed

unique maximal

if

-I

, ~ C(T)

,

, 7r'{C(T)}) = 0

V%, K ,

by the formula:

if

-i , ~ C(T)

the form obtained by restricting ((5.5c))

submodule

that the radical

~ax,K

now to apply the Schur functor,

of

V~, K .

.

<< , >>

to

of this form is the

It is a routine matter

and use Remarks

2,3 of 6.2 to prove

the following.

(6.4i)

Let

restricting ST, K

be the invariant

of

below),

%

is column

then the radical of ST, K ,

From

form on

the form given by (6.4h).

if and only if

p-regular, _-~nax ST, K

( , )

Then

( , )

p-regular.

( , )

K)

automorphism

of

KG

elementary

given by

is non-zero %

device.

B(~) = s(~)~

on

is column sub-module

.

(6.4i) we may deduce a w e l l - k n o w n

by the following

If

obtained by

is the unique maximal

_-max ~ T , K / S T , K ~ f(F

and

S--T,K

theorem of James Let $ ,

denote

for all

(see (6.4k),

the

~ c G •

K-algebra Let

K S

denote

the field

~k = s(~)k ~(M)

,

for

K ,

regarded

~ E G , k s K .

is also a left ideal of

6(M) ~ MmKK s

as one-dimensional

which takes

KG ,

Then if

M

,

by the action

is any left ideal of

and there is a

m m 1K ÷ 6(m)

KG-module

for all

KG ,

KG-isomorphism m s M .

It is trivial

99

to check that

B

maps

{C(T)} ,

respectively, where

T'

of

% )

T .

r

conjugate to

The bilinear form

[R(T)]

is the

%'-tableau

if

[R(T')] (%'

,

{C(T')}

is the partition

obtained by "transposing" the (6.4h) on

KG{C(T)}

syrmnetric, invariant bilinear form the formula:

to

~, 7' g G

( , )

%-tableau

is translated by on

KG~R(T')]

: {1

if

~-l~'e

0

if

B

to a

specified by

then

,

R(T') ,

(6.4j)

Moreover

B(~T,K) = ST,,K

= ST, K mK K s

(6.4k)

(James ~a, Theorems ii.I, 11.5]).

~-tableau, where

the invariant form on (6.4j).

Then

( , )

~

is

submodule

Remark.

~

ST,,K

is a partition of

~

ST,,K

is p-regular if

p-regular, then the radical of max

ST,,K of

ST,,K

'

the module

D~

(IJa, p.39])

module

in

[Ja', §i]

Let

r .

Let

if and only if

and

~'

( , )

T'

be a ( , )

be

~

is

is column p-regular). is the unique maximal

max

ST,,K/ST,,K = f(F ,,K) ~ KS

Comparison with the notation of James in

Dx

ST',K

obtained by restricting the form given by

is non-zero on

p-regular (by definition, If

which shows incidentally that

.

and so (6.4i) translates as follows:

Theorem

bijective

-

-I , ~ R(T')

is isomorphic to

is isomorphic to

between James's two families of irreducible

~a]~

shows that

f(F~',K) ~K Ks

f(F ,K) .

The

So the connection

KG-modules is

100

(6.4~)

D ~'' ~ D% H K K

,

for all column

p-regular

% .

s

The importance (6.4i),

of James's

or of the equivalent

is that it gives a satisfactory

(isomorphism p-regular

classes

~ ,

D%

of) irreducible is isomorphic

of a dual Specht module f

theorem,

~I,K

gives an independent

has a one-sided interesting

h

using the isomorphism ~ S~ W ~ KG(r)

namely

h,h g{w

of (6.1d).

of the

for each column

to the unique irreducible

quotient

We have also seen that Schur's DIm

f(Vl, K)

the functor

is related

First we re-define

labelling

KG-modules:

connection,

inverse,

that

"

"natural"

theorem

h

.

as functors

~ Emr

of

in 6.2.

used by James

of from

for any

V

V e ~(n,r)

such that (6.2d)

(6.4m) Define w e W .

that

Let the

W

V(~)

that for each

,

takes

{~

,

S-submodules

to

eu ,

U

it follows

h(W) ~ = e u ~ W .

be any left ideal of

Ker y = h(W)(~) .

(6.4f)

[Ja'].

W E mod KG we define

is the sum of all

Since

S-map 7:h(W) ÷ EmrW

Then

h*(W) & EmrW

U~ = O .

,

in

and the isomorphism

h(~'$) = E mr mKG W , h (W) = h(W)/h(W)(~)

where

It is

from mod KG(r) ÷ MK(n,r)

(6.4f),

This means

The Schur functor

defined

to a construction

functor

by .

KG ,

regarded

as left

KG-module.

y(x ~ w) = xw ,

for all

x s E ~r ,

Therefore

y

induces

an

S-isomorphism

101

Proof.

Suppose that

written

v = eu m w

But we have e ~ = e u

V ~h(W)(~)

If M .

for some

0 = y(v) = e w u

(~ ~ G)

u~

V = Ker y . w

.

form a

s W

since

This implies

K - b a s i s of

e KG u

of

V~

can be

V ~ ~~ h ( W ) m = e u m W w = 0 ,

.

.

since the elements

Hence

i.e.

V = 0 ,

is not zero, it contains some irreducible submodule

Since the ~ -weight space of M .

M~ ~ 0

,

v

.

y(h(W)(~))

true of

Any element

by

But

M ,

(6.4g)

.

h(W)(w )

is zero,

as i r r e d u c i b l e submodule of

This c o n t r a d i c t i o n implies

the same m u s t be E ~r

,

satisfies

h(W)(~)C

V ,

and the

rest of the proof of (6.4m) is immediate.

Since left

KG

is a Frobenius a l g e b r a

KG-module is i s o m o r p h i c to some left ideal

(61.6)]).

If we combine this w i t h

c o n s t r u c t i n g some irreducible James

~a']).

F ,K

for

Example.

%

S-module where

that

v~

EmrW p

of

e v e r y irreducible KG (~R,

column

W = K~G]

is an ideal of G .

If

is a submodule of does not divide

terms

r = q(p-l) + s .

KG,

and as left

By (6.4m),

i E ~ g A(n,r)

is the basis element of

q

isomorphic to

p-regular.

E ~ r w = Emr[G].

w h e r e there are

p.417,

(this m e t h o d is due to

Of course we can get in this way only modules

V

Vr, K

r,K '

,

h (W) then

and has weights So

It is easy to see that p-I

,

KG-module

W

affords

is isomorphic to the ei~G ] =

IG Iv

,

described in 5.2, Example I.

IG [ = ~l!...~n ~

is the highest such weight.

given by

W

,

(6.4m) and (6.2g), we have a way of

SK(n,r)-modules

the trivial r e p r e s e n t a t i o n of

So

([CR, p.4203)

~ ,

for all

h (W) ~ F%, K ,

~

such

where

%

% = (p-l,...,p-l,s,O,...,O),

and the n o n - n e g a t i v e integers

q,s

are

102

6.5

Application

III.

The functor

MK(N,r)

In this section we fix our infinite integer MK(n,r)

r > 0 . ,

as

such that

n

We consider varies.

N > n .

functor

of 6.2.

the irreducible in a very is based

and

modules

satisfactory

in

N > n ,

I(N,r)

I(n,r)

is the subset

With this convention, SK(N,r ) .

For

(6.5a)

SK(N,r )

for which

two elements

of all

are positive

integers

,

to those in

to characters

(see

,

whose

(6.5b)).

as a subset of

~,

It p.61].

I(N,r)

with components

components

can be regarded

from

and behaves

in his dissertation

I(n,r)

i

"mod S + mod eSe"

MK(n,r)

i = (il,...,ir)

of those

, i

all lie in

as a subalgebra

of

has basis

SK(n,r)

Sk,%

the categories

a very easy way of passing

given by Schur

i,j e I(n,r)

~i,j'

gives

EK(N,r)

j , 1,j s I(N,r)

and we can identify $i,j

d

SK(n,r )

and also the

case of our general

we may regard

consists

n

+ MK(n,r)

way with regard

on a construction

Since namely

The functor

N ,

> n)

a functor

d = %,n:MK(N,r)

as a special

K ,

between

that

We shall produce

which can be viewed

field

connections

Suppose

÷MK(n,r)(N

,

with .

of type

the

K-subspace

For the rule (6.5a),

spanned by those

(2.3b)

for multiplying

has the consequence

that

E N n .

103

if

i,j,k,Z

product p,q

e I(n,r)

~i,j~k,£

do belong

to

it would be if

,

then the coefficient

is zero unless both I(n,r)

,

$i,jSk, ~

as by

follows. a

=

If

then this coefficient

map

~ = (al,...,an)

(al,...,an,O,...,O)

~p,q

p,q e I(n,r)

were computed

We define an i n j e c t i v e

of any

~ ÷ a

,

while

if

is the same as

in

SK(n,r)

.

of

A(n,r)

into

we d e f i n e

A(N,r)

e A(n,r)

,

Then

image of

A(n,r)

: Bn+ 1 = ... = ~

= O} .

the

in the

e A(N,r)

a

under this

map is the set

A(n,r)

Notice

that if

i

= {B e A(N,r)

belong

So another description G(r)-orbits

of

of

I(N,r)

to a weight A(n,r)

(6.5b)

e = E~B ,

This is an idempotent e ~i,j = ~i,j ~i,j e = ~i,j

or zero,

I(n,r)

element

sum over all

of

SK(N,r ) , according

or zero, according

,

then

i e I(n,r)

.

is that it is the set of those

which lie in

Now define the following

B E A(n,r)

as as

of

.

SK(N,r)

B e A(n,r)*

and it is clear i c I(n,r) j ~ I(n,r)

once that

eSK(N,r)e

:

= SK(n,r ) •

(using

(2.3c))

that

or not, and or not.

It follows

at

104

From 6.2

(taking

S = SK(N,r))

which takes each some agreeable

(6.5c) which

If

V ~ MK(N,r)

properties,

V c MK(N,r)

lie in

A(n,r)

to

~V

to

then

Z

in

B e A(N,r)

E A(n,r) statement

eV = ZV B ,

N

in

or not;

This functor has

.

n

variables

~eV

XI, .... X N)

eV $ = e ~$V = V $

the first statement

~ ~ A(N,r)

(which is a

X I , . . . , X n)

is related

by the formula

= ~v(Xl ..... Xn,O ..... O)

then

comes from this,

summed over those

the character

variables

~ev(Xl,...,Xn)

If

Y~(n,r)

which we now describe.

over

(a polynomial

Proof.

eVe

Consequently

symmetric polynomial

a:MK(N,r)÷~(n,r)

we have a functor

.

or zero,

according

of (6.5c) follows.

together with the definition

as

The second

of a character

(see 3.4).

Next we look to see how our functor behavas and

VX,K

(6.5d) the

FX, K .

Lemma. X-tableau

Proof.

X

n+l

places

be distinct,

Let T. i

X s A+ (N,r)\

x,K '

A(n,r)*

and

i ~ I(n,r)

.

Then

is not standard.

we must have

satisfies Xn+ 1 # 0 .

in its first column. since

D

For this we need a lemma.

= (XI,X2,...,X N)

X ~ A(n,r)

on the modules

i ~ I(n,r)

.

X I ~ X 2 ~ ... ~ XN , So the

X-tableau

But the entries Therefore

T. i

Ti

and since has at least

in this column cannot all is not standard

(see (4.5a)).

105

(6.5e)

Theorem.

D%, K ~ VI, K

or

other words

Proof. if

FI, K

Then

eXl = 0

Suppose

X

= Dl, K

or

Vl, K ,

such that the

i E B

implies

Conversely, eXl # 0

by

eX l # 0

,

T. i

~

denote

,

and that

of

is standard

Since

n

parts.

~ s i(n,r)

Then

to the number of

((4.5a),

(6.5d) that

FI, K

In

non-zero

is equal

X%

any one of

~ s i(n,r)

has more than

we deduce from

eX l = O .

X%

if and only if

~ ~ i(n,r)

l-tableau

then from (6.5c) that

and let

the dimension

i s I(n,r)

eF%, K = 0

,

if and only if

first that

i ~ B

we have

+ % s A (N,r)

Let

(5.3b)).

Since

Xl = O ,

and

is a factor module

of

Vl, K ,

also.

suppose

(6.5c),

that

~ c i(n,r) *

for any of the modules

We know that XI

X ~l # 0 ,

in question.

hence

This completes

the proof of (6.5e)°

If

~ e A+(N,r)

I ~ A(n,r)

,

is that

then a necessary ~ = p

for some

and sufficient ~ s A+(n,r)

.

condition

that

We know from (5.4b)

+ that

{FI,K:I

s A (N,r)

is a full set of irreducible

modules

in

So (6.2g) and the theorem just proved give the first statement

MK(N,r)

in (6.5f)

below.

(6.5f) MK(n,r) character

p

{eF ,,K : p s A+(n,r) .

In fact formula,

(including

is a full set of irreducible modules

eF , K ~ F ,K valid

p = O)

for

(isomorphism

all

+ ~ e A (n,r)

,X n)

=

,

in

MK(n,r))

and for

.

every

...

~p,,p(X I,

...,Xn,O

,...,0)

Hence the characteristic

:

n,p(X I,

in

.

.

106

Proof.

The second and third statements

eF , , K

and

F

K are both i r r e d u c i b l e modules in

character with leading

Remarks

i.

following

~I ~n X I ...X

term

A direct proof lines.

Let

that

E(n)

The inclusion

E(n) mr C

of a

E(N) mr

and to induce an isomorphism

2. onto

N > n > r ,

+ A (N,r)

have at most e A+(n,r)

(6.5g)

categories.

hd ,

E(N)

el,...,e n ,

with basis V ~,K

el,...,e N . to

eV~*,K

'

Similarly we may show

eF ,,K

.

~ ~ ~*

.

For any

+ I g A (N,r)

r

non-zero

parts.

Then

(6.5f)

N > n > r ,

are naturally

MK(N,r)

equivalent

the functor described to

r ,

for some uniquely

d = dN, n

= MK(r,r)

a functor

take for

W e MK(n,r)

of

dN,n:b~(N,r) classes

can defined

~ M K(n,r)

of irreducible

In fact we have the stronger result:

[Co, p.7]).

h

% = ~*

+ A (n,r)

from

being a partition

the sets of isomorphism

(6.5g), we must produce

dh

,

shows that the functor

the functor

In particular

gives a b i j e c t i o n

Hence

for example

each

K-space with basis

can be shown to take

in these two categories.

If

To prove

(i), 3.5).

can be made along the

F ,K

K-space

F ,K

the map

induces a bijection between modules

a

having

eD ~*,K

D, K

If

NK(n,r)

(see Remark

eF ,,K

denote

which we regard as subspace

that

follow from the fact that

defines

for all

an equivalence

N ~ r .

h:MK(n,r ) ÷ MK(N,r)

to the appropriate

of

identity

We leave it to the reader to verify

such that functors

(see

that we may

in 6.2, which in the present

case takes

107

h(W) = SK(N,r)e

We might mention then the

that

(6.5g)

K-algebras

Do,

p.34]) .

6.6

Application

has another

SK(N,r)

IV.

Some

mSK(n,r )

and

on decomposition MQ(n,r)

context,

numbers.

÷ MK(n,r)

and prove

a general

If

V ~ mod S

and

of

V%

to

V% ,

(6.6a)

in

V .

in mod

% e A ,

in any composition

series

V = VO b V I ' ) . . .

Here

Lemma. n%(eV)

If

result

theory

of

of T. Martins

EMa]

reduction

S

n%(V)

{V% : % g A}

he a full set

is an algebra

over any field.

the composition

is the number of

multiplicity

of factors

isomorphic

V

-)Vz = 0 .

be an idempotent

to be the set of those

Let

denote by n%(V)

e

eSe"

in 2.5.

where

That means,

Now let

(6.6b)

S ,

(see

numbers.

Then we apply this to the modular

We start with a piece of notation. modules

N > n > r ,

are Mot ira equivalent

our "mod S--)mod

which was described

of irreducible

If

on decomposition

In this section we first extend 6.2 to a "modula#"

formulation:

SK(n,r)

theorems

W .

% g A

of

S ,

such that

I e A' ,

is the composition

then

and define

A' ,

as in (6.2g),

eV% # O .

n%(eV)

multiplicity

= n%(V) of

,

eV%

for any

V ~ mod S .

in the eSe-module

eV .

108

Proof.

From

By (6.2a),

(6.6a) we get a series of eSe-modules

e(Vj_I/V j) m eVj_i/eV j

Removing

those terms

Vj_I/V j

is isomorphic

composition

series

Now let properties IC)

C

K

,

element of

SC

~:R ÷ K .

SK = SR ~ K

i.e.

,

for

f.k = ~(r)k

theory of algebras

(ii)

reduction of

R

Let

SC

SK

mR ,

(see

~,

VR

K

and

with the

(iii)

SR

as

is an

If

{U~, K : d s A}

VK

via

w ,

representation

the categories (cf. 2.5),

as

or "admissible

is the R-span of some C-basis

SK-mOdule;

"R-order"

{u I m iK,..,u b R IK}

reduction"

[~o, 48.1(iv),

the identity

R-module

connects

can be found an "R-form"

as left

contains

there is a

R. Brauer's m o d u l a r

of "modular

or

R

contains

thus

is regarded

~ANT, p.lll])

§6, p.256],

{Vx, C : X s A} ,

C

C-algebra with finite basis

closed:

r c R , k ~ K).

can be regarded VC .

and

for which

(in particular, R ,

be a

1 .

at once.

K-algebra with K-basis

by the process

(i)

of

j = 1 .....

we are left with a

S R = Ru I * ... e Ru b

is a

V C ~ mod S C

,

be a subring of

ideal domain

(IB]; see also

V R , i.e.

SRV R ~ V R

VK = VR m K

(6.6c)

mod

In each

R-lattice"

i.e.

~ s A\A'

and is m u l t i p l i c a t i v e l y

means

follows.

and

such that the set

Then

and

eV.j_l = eV.j ,

The lemma follows

be fields,

m

SC

for

for some

eV .

(here and below,

mod

Vw

eSe-modules),

is the field of fractions

{Ul,...,Ub}

SC .

for which

is a principal

ring-homomorphism

in

to

for

C ,

(i) R

, (ii)

eVj_ 1

(as

eV = eV 0 ~ e V l ~ ... D~eVz = O.

of

p.299]).

VC ,

and

Then

is called a modular

109

are full sets of irreducible modules we define for each

% ~ A ,

Let

e = eR

in

n~(V%,K)

be an idempotent

m o d eSce

Similarly,

, namely

e K = e R m 1K

{eKU~, K : ~ e A'}

eSRe

R-module,

,

eSRe

reduction

from

decomposition

the

with

numbers by .

Proof.

be an

eSce-module

eKSKe K . to

in a modular

VR

e ~ SC ,

A' = {% e A : eV~, C

K-algebra

in

C-algebra SR .

mod eKSKe K ; .

#0}

S K = SR m K ,

mod eKSKe K ,

namely

eSce

-

this is because,

as

For the same reason, we can we have a process

let us denote

Of course

of modular

the corresponding

these are defined only for

these decomposition

numbers,

and

is very simple and satisfactory.

[Ma]).

R-form of

V R = eV R • (l-e)V R eV C ,

Since

where

in the

Therefore

d%~(eSe)

(T. Martins .

,

modules

The connection between

d%~(S) = d%~(eSe)

Let

in the

defined previously,

Theorem

sun,hand of

U6, K

dx6 = d%6(S)

A' = {6 ~ A : eKU~, K # 0}

R-order

mod eSce

~ e A'

dx6 (S)

(6.6d)

number

SR .

{eV~, C : ~ c A'}

is a direct summand of

eSRe ~ K

,

respectively,

By (6.2g) we get a full set of irreducible

is an idempotent

where

is an

identify

X e A'

of

in the ring

and so we get a full set of irreducible

Now

mod S K

VX, C

we can apply the theory of 6.2. modules

SC ,

6 g A the dec£mposition

to be the composition m u l t i p l i c i t y reduction of

in mod

Let

% e A' ,

V C = V%, C This implies

and also that the

6 e A'

Then that

eKSKeK-mOdule

eV R eV R

.

Then

is a direct is an

eV R m K

R - f o r m of the can be

110

identified

with

multiplicity of

U~, K

of

in

eKV K ,

where

eKU6, K

in

VK .

R = Z ,

and define

K

~:Z ÷ K

any infinite ~(n) = n.l K •

given in 2.3.

the categories

and

MK(n,r)

Identify

respectively,

Corresponding

C = Q

for all SK

to the sets

with mod SQ

Denote

the decomposition

numbers

which

appear

(6.6c)

Fix

SK(n,r)

by the isomorphism

and

(6.6e) map from (i)

Theorem. A(n,r)

mod

n, r

p ,

SK

with

and let

MQ(n,r)

in the general

case we take

+

these

,

{F%, K : ~. g A (n,r)}

.

sets are indexed by the same set

numbers

by

d%u = dl

(GLn)

.

These

A+(n,r).)

are the same

in the formulae

~%,o(XI,...,Xn)

of 3.5, Remark

characteristic

n s Z .

+

in this case,

multiplicity

as in 2.4.

{V>~,Q : ~ e A (n,r)}

(It happens

the composition

(field of rational

field of finite

S--SQ(n,r), S Z = Sz(n,r) Identify

(6.6b)

(6.6d).

of (6.6d) we take

by

By

is the same as the composition

eKV K

This proves

For our applications numbers),

VK = VR m K .

=

E+ s A (n,r)

d%~,p(Xl,''',X n)

(i).

Suppose into

that

A(N,r)

da6(GL n) = d ,B,(GLN)

,

N > n , given

for any

in

and let 6.5.

~ ÷

Then

~,B ~ A+(n,r)

•

be the injective

111

(ii)

d~,~(GL N) = O ,

Proof.

(i)

and let

(6.5e),

X e A+(N,r)k~(n,r) *

is a direct application

e = eZ

e e SZ ,

for any

and

be the element of e K = e m 1K

the sets

A',A'

A+(N,r) /~ A(n,r) *

of (6.6d). SQ

Take

defined

is the element of

and

p e A(n,r)*

SQ = SQ(N,r)

as in (6.5b). SK

defined by

, etc.,

Clearly (6.5b).

By

which appear in (6.6d) are both equal to

So we take

~ = ~ , , p = B*

in (6.6d),

and then use

(6.5f). (ii)

Since

% ~ A(n,r)

,

composition m u l t i p l i c i t y eF ,K

in

eKV%, K ,

Part

decomposition

(6.6f)

Here

Fp, K

A+(N,r) . identical

V%, K

numbers

In other words

dx~(GLN)

= O ,

for all

n

r and

are contained

K ,

the

in the m a t r i x

E A(r)

which can be identified with the set of all

~ = (~l,...,%r) and the map

So (6.6e)(i)

the

is complete.

d% (GLn)

,

By (6.6b)

is the same as that of

(i) of this theorem shows that, with fixed

+ A(r) = A (r,r)

N > r ,

in

eKV%, K = 0 .

p c A(n,r)

(dx~(GLr))%,D

partitions Then

of

because

and the proof of (6.6e)

Remark.

then by (6.5e),

of ~ ÷ ~

r .

Assume induces

first

n = r

a bijection

of

in (6.6e). A(r)

shows that the d e c o m p o s i t i o n m a t r i x for

(up to this bijection)

with

(6.6f).

Next take

N = r .

onto GL N Then

is

112 *

n ~ r

and the map

those

% c A(r)

matrix

the submatrix

of (6.6f)

to partitions

Theorem

for

than

bijectively n

non-zero

n

in this case we merely

Here

partitions

r .

in

of

~%,K

"

E SQ ,

(see (6.3e),

(6.6g)

A(P)(r)

which are column

Let

eK = ~ (6.4%)

showing

dx6(G(r))

~ SK .

,

~a',

the matrix

given in James's

article

[Ja'~.

3.~).

S = SQ(n,r)

If and

those columns

and

,

p-regular multiplicity (r # n) ,

The result

then

~ ~ A(P)(r)

•

and char

of etc.,

D~ ~ eKF6, K

r ! n ,

2 < r < 6 ,

To see this,

KG(r)-modules

the composition

each).

shows

G(r) is

p-singular.

6 c A(P)(r)}

S%,Q e eV%,Q

% c A(r)

for

which

which

group

suppress

the set of all column

preceding

Theorem

(6.6f)

:

(6.6d) with

We have

for all

QG(r)-,

{D 6

denote

and the remarks

(James

= d%6(G(r))

,

denotes

Now we may apply

Theorem

d%6(GLn)

~ s A(r)}

with

all rows and columns

for the symmetric

that we have full sets of irreducible

respectively.

So the

parts.

numbers

to partitions

:

parts.

(up to this bijection)

by repressing than

onto the set of

a simple proof of a theorem of James,

of (6.6f)

{~.,Q

Tables

obtained

having more

of (6.6f), which refer

e = ~

i (n,r)

is identical

n

of decomposition

also a submatrix

D6

GL

(6.6d) gives

that the matrix

recall

takes

which have not more

decomposition

refer

+

~ * ~

is as follows.

K = 2,3

are

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ANT

E. Artin, C. Nesbitt and R.M. Thrall, condition,"

Bo

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M. Clausen,

Letter place algebras and a characteristic-free approach

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P.M. Cohn,

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C.W. Curtis and T.V. Fossum,

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C.W. Curtis and I. Reiner,

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Essai d'une th~orie g~n~rale des formes alg~briques,

M~m. Soc. Roy. LiSge 17 (1892), pp.156. De

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appendix to "Th~orie combinatoire des invariants

G.-C. Rota,

S~ries de Math. pures et appl. I/S-OI,

Universit~ Louis-Pasteur, Strasbourg, 1977. DKR

J. Desarm~nien,

J.P.S. Kun$ and G.-C. Rota,

Invariant theory,

Young bitableaux and combinatorics, Advances in Math. 27 (1978), 63-92.

114

Do

L. Dornhoff,

"Group representation theory",

Part B, Marcel

Dekker, New York, 1972. G. Frobeniu$,

Uber die Charaktere der syrmnetrischen Gruppe,

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H. Garnir,

Th~orie de la representation lineaire des groupes

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J.A. Green,

Locally finite representations, J. of Algebra

41 (1976), 137-171. Ha

W.J. Haboush,

Central differential operators on split semi-simple

groups over fields of positive characteristic,

Publ. I.H.E.S.

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G. Hisman, of

Representations of general linear groups and varieties

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W.V.D. Hodge,

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W.V.D. Hod$e,

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J.C. Jantzen,

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T. Martins,

P

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I. Schur,

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J.-P. Serre,

Groupes de Grothendieck des schemas en groupes

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R. Steinberg,

Lectures on Chevalley groups, Yale University,

New Haven, 1966. Sw

M.E. Sweedler,

T

J. Towber,

Young symmetry, the flag manifold, and representations

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J. of Algebra 61 (1979), 414-462.

Tu

H.W. Turnbull,

"Hopf algebras",

W.A. Benjamin, New York, 1969.

"The theory of determinants, matrices and invariants",

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116

We

H. Weyl,

Theorie der Darstellung kontinuierlicher halbeinfacher

Gruppen durch lineare Transformationen, Math. Zeit. 23 (1925), 271-309; 24 (1926), 328-376, 377-395, 789-791. We

H. Weyl,

"The classical groups",

Princeton University Press,

Princeton, 1946. W.J. Wong, p , W'

Representations of Chevalley groups in characteristic

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Irreducible modular representations of finite Chevaliey

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A. Young,

On quantitative substitutional analysis (3), Proc.

London Math. Soc. (2) 28 (1927), 255-292.

INDEX

affine group scheme, affine ring,

13, 25, 30

decomposition number, defined over

8

antisymmetric tensor,

67

Z ,

12, 23, 30, 34

D~sarm~nien matrix,

68, 72, 79

diagonal subgroup, basis for

D%, K ,

bideterminant,

dominant weight,

55

15, 27, 47, 107

37 36

51 equality rule,

canonical form,

33, 65

21

equivalent categories,

25

Carter-Lusztig basis,

68

equivalent representations,

Carter-Lusztig lem~a,

57

exterior power,

Carter-Lusztig module, character,

40

-- formal,

40

-- natural,

finitary function,

Garnir relation,

5

57

6, 20

coefficient function, coefficient space,

6

column stabilizer,

51

comodule,

65

42

coalgebra,

38, 55, 67

Hecke ring,

6

hyperalgebra,

ii

induced module,

9

composition multiplicity, contravariant,

82

107

64

invariant matrix,

6, 28

31

-- dual,

32

James module,

61

-- form,

32, 73

James's theorem,

99

3

118

K-space,

semigroup,

3

KF-bimodule,

34, 49, 53

KF-isomorphism, K F-map,

3

-- irreducible,

semigroup-algebra, Specht module,

3

3

KF-module,

91, 97

standard

tableau,

symmetric theorem,

Modular

reduction,

Modular

theory,

Morita

14, 26, 108

multi-index,

partition, -- column

107

-- ring of,

36, 94 p-regular,

place-permutation,

-- A-rational, -- matrix,

3

row stabilizer,

Schur

function,

Schur functor,

41, 45 42

symmetric

group,

18, 80

symmetric

power,

31, 54, 67

s~mmetric

tensor,

tableau,

50

-- basic

X- ,

-- standard,

weight,

67

51 55

36

weight-space,

8 51

Weyl group, Weyl module,

Schur algebra,

41, 45

7

6

-- polynomial,

41

94, 112 19

37 36 17, 50, 65

II, 21 47 80, 82, 88

theorem,

77

-- elementary, -- monomial,

18

representation,

function,

i0, 25

equivalent,

55

tensor product

-- complete,

109

3

90, 99

-- dual,

Steinberg's

44, 71

Martins

3

Z-form,

12, 14, 26, 77

77

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