Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
495 Adalbert Kerber
Representations of Permutation Group...
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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
495 Adalbert Kerber
Representations of Permutation Groups II
Springer-Verlag Berlin.Heidelberg. New York 1975
Autor Prof. Dr. Adalbert Kerber Lehrstuhl D fiJr Mathematik Rhein.-Westf. Technische Hochschule Aachen Templergraben 55 51 Aachen/BRD
Library of Congress Cataloging in Publication Data
Kerber, Adalbert. Representations of permutation groups I-II. (Lecture notes in mathematies, 240, 495) Bibliography: p. Includes indexes. CONTENTS: pt. I. Representation of wreath products and applications to the representation theory of symmetric and alternating groups. i. Permutation groups. 2. Representations of groups. I. Title. II. Series: Lecture notes in mathematics (Berlin), 240, etc. QA3.L28 no. 240, etc. 510'.8s [512'.2] 72-183956
AMS Subject Classifications (1970): 05 A15, 20 C30
ISBN 3-540-07535-6 Springer-Verlag Berlin 9 Heidelberg 9 New York ISBN 0-387-07535-6 Springer-Verlag New Y o r k . Heidelberg 9 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 w 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. 9 by Springer-Verlag Berlin 9 Heidelberg 1975 Printed in Germany Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Preface
The d e s c r i p t i o n of the r e p r e s e n t a t i o n theory of w r e a t h products and its a p p l i c a t i o n s
are continued
in this second part.
In part I the emphasis
lay on the c o n s t r u c t i o n
matrix r e p r e s e n t a t i o n s
of w r e a t h p r o d u c t s
closed field.
In part
cible characters part
of the irreducible
over an a l g e b r a i c a l l y
II, I consider mainly the o r d i n a r y
of these groups, which were
irredu-
less important
in
I.
The c o n s i d e r a t i o n s presentations,
apply especially
to the s y m m e t r i z a t i o n of re-
so that we obtain quite easily famous results
Schur, Frobenius,
Weyl and van der W a e r d e n about the c o n n e c t i o n
b e t w e e n the r e p r e s e n t a t i o n theories tric groups.
of general
linear and symme-
They apply also to the theory of e n u m e r a t i o n under
group action so that we obtain the most theory,
important
results of this
which has been d e v e l o p e d mainly by Redfield,
de Bruijn.
of
This theory
torics and it yields
is nowadays
an essential part
the main e n u m e r a t i o n techniques
P61ya and of combinain graph
theory. These applications
and some related topics
are d i s c u s s e d here.
In the first sections the main r e s u l t s needed from part quoted,
so that this part
is in a sense also selfcontained.
I would like to express my sincerest who work in that
I are
thanks to many colleagues
field and in p a r t i c u l a r to the people w o r k i n g
at the "Lehrstuhl D f~r M a t h e m a t i k der RWTH Aachen" helpful and s t i m u l a t i n g discussions
for very
and cooperation.
Adalbert Kerber
Contents
Introduction CHAPTER
I:
. . . . . . . . . . . . . . . . . . . . . . . .
Characters
of w r e a t h
products
. . . . . . . . . .
1. R e p e t i t i o n and m o r e a b o u t c o n j u g a c y c l a s s e s of c e r t a i n wreath products . . . . . . . . . . . . . . . . . . . 2.
Representations
CHAPTER
II:
I!I:
inner
products
An a p p l i c a t i o n to enumeration under
4. E n u m e r a t i o n
under
5. E n u m e r a t i o n
of f u n c t i o n s
6.
Some
7. T h e References
products
and
their
group
cycle-indices construction
of r e p r e s e n t a t i o n s
....
c o m b i n a t o r i c s : T h e t h e o r y of group action . . . . . . . . action
. . . . . . . . . . . .
by w e i g h t
. . . . . . . . . .
. . . . . . . . . . . . . . . . . .
of p a t t e r n s
. . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
Subject-index
3
4
c h a r a c t e r s 22
An a p p l i c a t i o n to r e p r e s e n t a t i o n t h e o r y : S y m m e t r i z a t i o n of i n n e r t e n s o r p r o d u c t s of r e p r e s e n tations . . . . . . . . . . . . . . . . . . . .
3. S y m m e t r i z e d CHAPTER
of w r e a t h
1
. . . . . . . . . . . . . . . . . . . . . . .
62 63
103 104 128 142 156 167 174
Introduction
Having described representations a permutation
in part I a construction of the wreath product
group of finite degree)
closed field, we now consider The character yield directly
formulae
G~H (G a finite group, H over an algebraically
their characters.
obtained have various applications.
the characters
of several permutation
tions of G~H (if G is also a permuation terest in combinatorics, tiation
character,
the cycle-index.
H[G],
the way how the character
form for the groups
The applications
formulae
theorems
to representation
in G~H:
GxH ~ diagG*.H',
to H, and diagG* the diagonal
where G* which is
of the basis group. &
Hence a representation
in
theory are based on the fact
the basis group of G~H, H' a certain complement
isomorphic
to obtain
in question.
that G~H has a nice embedding denotes
the exponen-
[H;G]. Having obtained the
are derived gives a hint how to obtain enumeration weighted
representa-
it is in each case not difficult
Furthermore
They
group) which are of in-
e.g. the composition
[G] H and the matrix group
permutation
of the irreducible
F of G y i e l ~ a representation
&
~F of G~H
I, and the restrictions (if H ~ Sn) ,^as it ~ is described in part v "H" F := ~F $ diagG* of this representation
applied) A
lemma
of G and H, the ele-
centralizing.
(if the groundfield
shows a close connection V
F := ~F $ H'
yield representations
ments of which are mutually Hence Schur's
and
is so that it can be
between
the decompositions
^
n
of F and F. It shows how F, which is in fact equal to | F, if H ~ S n, can be "symmetrized"
with each irreducible
representation,
2
which is an irreducible
constituent
Doing this for G := GL(m,C), famous results tation theories
V of F.
F := idGL(m,C ), H := Sn, we obtain
about the close connection of symmetric
and general
There are also further and sometimes
between the represenlinear groups.
quite
surprising
applications.
Chapter I
Characters of W r e a t h Products
In the first section of this chapter the definition of the wreath product H
G~H
of a group
G
and a p e r m u t a t i o n group
of finite degree is recalled from part I
as well as the
results about conjugacy classes of wreath products of the form
GxS n.
To these results about conjugacy classes
some recent results
concerning their splitting over certain normal index
2
subgroups of
are added.
The second section contains the basic results of part
I
concerning the construction of matrix representations
of
w r e a t h products as well as some character
formulae w h i c h are
applied in the following chapters. For an example,
the ordinary irreducible representations
hyperoctahedral
groups
over certain normal detail.
$2~S n
as well as their splitting
subgroups of index
This corresponds
of
2
is described in
to the results of the first section
w h i c h concern the splitting of conjugacy classes and it covers the ordinary representation
theory of the series of Weyl groups.
The results on the splitting of ordinary irreducible representations is then applied to the evaluation matrices.
of decomposition
1.
Repetition
and more
wreath
products
If
is a group,
G
of s y m b o l s ordered (for
~n
pairs
short: G~H
For
:=
fw'
11 ..... nl,
or
:= G n x H = f' f'
l(f;w)
G
we a g r e e d
the r i g h t
upon
of c e r t a i n
acting the
of a map
on the
set
f
G~H
from
~n
and a p e r m u t a t i o n
set of all to
w ~ H:
and
elements
~ H
we d e f i n e
by
f-1
: ~n ~ G
: i ~ f(i) -I
f~
: ~n ~ G
: i ~ f(~-1(i)),
ff'
: ~
: i ~ f(i)f'(i),
n
reading
~ G
: i ~ I G.
products
of p e r m u t a t i o n s
from
to the left: ~'
: ~
n
"
h
n
: i
"
n(n'(i)),
we o b t a i n e d
v f,~,~, Using with
this
result,
the c o m p o s i t i o n
((f~)~, = f~,~).
it is e a s y to check, defined
G
I f : ~n ~ G A W ~ HI.
e : ~n ~ G Since
group
f ~ G n)
: ~n ~ G
e : 6n
classes
we c o n s i d e r
consisting
f : ~n ~ G
ff''
conjugacy
a permutation
(f;w)
such maps
f-l,
H
about
by
that
G~H
together
is a group,
the w r e a t h R r o d u c t o f
G
with
H.
Since V f'~
((f~-I )-I = (f-l) - 1
we o b t a i n e d
for the i d e n t i t y
inverse
(f;w):
of
G'~H = (e;IH)
The normal
product
G~H
of
and the
= (f-11;~-I)._
=
lalnlHl.
some i n t e r e s t i n g
subgroups
of
G%H.
subgroup G*
was c a l l e d
IG~ H
'
is
IG~HI Let us now m e n t i o n
~ 1
element
^ (f;w)-1
The order of this group
=: ~-~ )
:= l(f;1H) If : N n ~ GI ~_ G'-H
the basis ~roup of
of
n
subgroups
G ~ G i :=
Gi
G~H.
It is the inner direct
w h i c h are i s o m o r p h i c
to
G:
I(f;IH) I V J ~ i (f(j) = IG) I, I < i < n, n
G* = Gq x...x G n =
x G i. i=1
The subgroup
H' := J(e;~) which
is i s o m o r p h i c
to
H
I ~ ~ HJ
is a c o m p l e m e n t
G~H = G*H'
~
of
H,
G*:
A G* ~ G~H A G* n H' =
IqG%HI.
6 It is sometimes useful to describe by displaying the values of
(f;n)
more explicitly
f, i.e. to write
(f(1) ..... f(n);~)
instead of
Using
(f;N).
this
notation,
the
diagonal
o_~f the
basis group is: diagG*
:= {(f;IH)
j
f constantl
: l(g ..... g;IR) Multiplying
this by
H'
I g~
al ~ G .
we obtain a further subgroup of
interest: (diagG*)H' =
If
C(G)
C(G%H)
1.1
I(g ..... g ; " )
I f constant}
I g e G ^ ~ e H}
denotes the centre of of
G ~
= l(f;~)
Z G • H.
G, we obtain for the centre
G%H:
I1Gt
C(G~H) = I(f;1 H)
I f:N n ~ C(G) ^ f constant on each orbit of Hi.
More special subgroups Of G%H
can be defined using given
normal subgroups
of
in
1.2
G
M
of
G, N
H, which are of index
and H, respectively:
Def.:
M < G, N < H (i)
and
IG:M I < 2 >
G~H M := {(f;~)
It f ( i )
IH:NI,
then
~ M~ <_ G~H,
~ 2
7
(ii)
I
GXHMM N := l(f;~)
f(i
IG\M, if ~ e H\N
It is easy to check, that the following is valid:
1.3
If
M, N
are subgroups of index
tively, then
index
< 2
G~N, G%H M
in
1
and
~ 2
G%H~
in
G, H, respec-
are subgroups of
G~H.
Special cases of such subgroups have been considered in part I section
6.
To see this we use the following faithful permuta-
tion representation
~
of
G%H, if
9 : G%H ~ Stun : ( f ; ~ ) ~
and
H ~ Sn:
(j-1)m + i
!
1.4
G ~ Sm
/|
[ (~(j)-l)m + f(~(j))(i)
1
1<j<_n Then if we denote by
P+
the subgroup of even permutations
of a given permutation group
1.5
P, the following holds:
G ~ S m, then
(i)
if
m
~[G~H
(ii)
is even, we have:
G+
and if
] = ~[G~H] N A
m
mn
= ~[G~H] +,
is odd, then
+
~[G'H~+] = ~[G~H] n Amn = ~[G~H] +.
Hence e.g.
8
21m = ~[SmXSnA ] = ~[Sm~Sn ]+, m
2~m ~ ~[Sm~S~n ] : ~[Sm%Sn ]+. Am The subgroup
~[Sm~Sn]+
was denoted by
Sm~Sn+
in section 6
of part I. It should be mentioned, that the series of Weyl groups but not the exceptional Weyl groups
E6,E7,E8,F 4
cases of the groups defined above.
and G 2
are special
The Weyl ~roups are defined
to be certain permutation groups acting on certain subsets ("root systems") of euclidean spaces.
It can be shown, that
the following is valid (cf. e.g. Humphreys [I], 12.1, Stewart [I], chapter 7): 1.6
(i)
The Weyl group of type
(cf. I, p.29) (ii)
(n~
~[S2~Sn]
B n (n ~ 2)
is similar
as is the Weyl group of type
Cn
3).
The Weyl group of type
to
is similar
Sn+ 1.
The Weyl group of type
to
(iii)
to
A n (n ~ I)
D n (n ~ 4)
is similar
~[S2~Sn] N A2n = ~[S2~Sn]+ = ~[$2~S n
]. A2
Hence the following considerations can be applied especially to the series of Weyl groups.
Having mentioned certain subgroup~of
G~H, let us now briefly
recall some notation which was introduced when conjugacy classes of wreath products of the form
G~S n
We agreed on writing permutations disjoint cycles
w~
w E Sn
as
products of
as usual:
c(w)
1.7
were considered.
~
k~-fl
11= t]- ~=
(j~ ~(jr
(jr
~=1
k~-1 ~here in each of these cyclic factors j~
w~ = (j~...
(J~))
11
is the least symbol which is included and where the num-
bering of the cycles is uniquely determined by assuming Jl < J2 < "'" < Jc(w)" With these conventions, (f;w) e G%H, so that
1.7
f :
Nn
is uniquely determined. -~
If now
G, then the element -k~+1
1.8
g~(f;11):=ffw...f k _l(j~)=f(j~)f(~-l(j~))...f(w
(j~))~G
11
was called the cycle product associated with the ~-th cyclic factor
w~
of
w
with respect t__oo f.
We assume now, that
has countably many conjugacy classes and that
CI,C2,...
fixed numbering of them. If in
(f;11) ~ G~H
the permutation
w
is of type
Tw = (at(w) ..... an(W)), i.e.
if
ak(w)
is the number of cyclic factors
w~
of
G
is a
10 w h i c h are of length
k~ = k, I ~ k ~ n, let
1.9 be the cyclic factors of length
k
whose a s s o c i a t e d
cycle
products 1.10
belong to
C i, when occasion arises.
The n o t a t i o n that
rj ik
aik(f;~)
1.9
is u n i q u e l y
determined,
is the least of the occuring ~ 2
if we again agree
symbols and that is
we have V I _< j < aik(f;~)
(r~k < r j+lik)"
Then ik" i,j,k
aik(f;~)>O The scheme
1 .11
T(f;-)
:= (aik(f;~)),
was called the type o_~f (f;~). same type
1.12 In case
i
row index
k
column index,
Conjugate
I < k < n,
elements are of the
(cf. I 3.7):
V (f;~),(f,;~') ~ GkH
((f;~) ~ (f';~')~ TCf;~)
H = Sn, the converse holds as well,so
= T(f';~)).
that two elements
of
G~S n
are conjugates
if and only if they are of the same
type:
1.13
V (f;w),(f,;w,)
In addition
~ G~S n ((f;w) ~ (f';w')~=~T
to these results which have been proved in part I
let us now answer the question, of
G~S n
A G~Snn, M
G~SnM , and of
of index
2
Since these three know
which
split into several conjugacy
subgroup
(f;w) = T(f';w')).
respectively, in
subgroups
where
are of index
conjugacy
of equal order, of
classes M
of
G%An,
denotes
classes of
a given
G.
that such splitting
jugacy classes
of these conjugacy
if the subgroup
U
G~S n
the centralizers
of an element
~ 2
classes
in
G~Sn, we
split into two con-
and that this happens considered (f;w)
if and only
is of index
2
and
of that conjugacy
class
G%S n C
(f;w)
in
G~S n
and
U
are equal:
CG~Sn((f;w)) Hence a consideration may provide
= Cu((f;w)).
of the centralizers
an answer to the question.
of elements
G~S n
Let us briefly recall,
what has been said in part I about the centralizer in
in
of an element
G~S n.
Centralizers
of conjugate
hence
implies
element
1.13
of type
elements
are conjugate
that to describe
(aik(f;w)),
subgroups,
the centralizer
we may assume without
of any
restriction,
12 that the map
f
at the points
r ik j
for each cyclic (cf. 1.9)
has value
IG
everywhere
(cf " I 3.22) "
factor
we have
except possibly
Thus we may assume
~Jik = (rfk'" ' .wk-1 (r~k))
(if
of
that
"
k > 2): B
so that if
1.14
aik(f;w)
V i,k,
> O:
I <_ j <_ aik(f;. ) (f(riJk) = gi0k(f;.)).
We define corresponding
maps
fJik : Nn * G
f(r~k), fJi k
Then the following
(f;')
s = r ik j
(s) := 1
1.15
if
by
G'
elsewhere.
is obviously
valid:
:
and i,j,k aik(f;~)>O
(ii)
the factors
pairwise For the factor
(f~k;.~k)
(f;w)
are
commutative.
(ffiJk;l~iJk]
(f~k;W~k) ~ (Grj
of
x G
we have
9
•215
~ G~S k
G k-1
)Sk ~ G~Sn"
13 Let us denote the corresponding element of
fik .ik j ; j Then 1 . 1 5
1.16
CG~Sn(f;w)
is a subgroup of G ~ S
by
-"
yields for the centralizer of
The centralizer
G~S k
of
(f;w):
(f;w)
in
G~S n
which is an embedding of
ik ik CG~SR(fl ;~I )~Saik(f;~)
X i,k aik(f;~)>O
To decide, whether such a subgroup of in a given normal subgroup
G~S n
is contained
U, we need a closer examination
of its direct factors. The centralizer of (f~k
ik I ik ;~1 ) = (gik (f;~)' IG' .... IG;~I
the
k
values of
was described explicitly in
1.17
CG~k(tl
I
)
f~k
3.19:
'~I ) = dlag(CG(gik (f;~))*)
Let us consider the elements of this group. We consider first the special case
(g,1, .... I;(I...k)).
14
Then the elements
of
diag(CG(g)*)((g,1 ..... I;(I...k)))
are
of the form
(c . . . . .
c;1)(g,1
.....
1;(1...k))1
(cg . . . . . cg,c . . . .
=
c;(1...k)l),
1-times where
c
denotes an element of
Hence the elements
of
CG(g).
CG~Sk(f~k;w~k)~Saik(f;w)
are con-
jugate to elements of the form
x::(
(cjg
....
where
cjg,cj
.....
p ~ Saik(f;~)
cj;(1...k) lj)
.....
1 ) g = f(rik
and
....
and
;p)~(G',,.Sk)~aik(f;~), cj
~ Ca(g),
I ~ j ~ aik(f;~ ). If we apply proof of
the a s s o c i a t i v i t y
2.29
in
I), then
of the w r e a t h product x
corresponds
(cf. the
to the element
11 y:=(o . . ,cjg .... ,cjg,cj ..... cj .... ;((I...k)
in
G~Skaik(f;~ ) .
This altogether
1.18
laik(f;~) , .... (I ...k)
yields the crucial lemma:
The centralizer
of
(f;w) ~ U ~ G~S n
tained in the normal
for all the pairs
subgroup
(i,k)
with
U
of
in
G~S n
aik(f;w)>O
G~S n
is con-
if and only if
we have that
;p)
15 all the elements laik(f;~) 11 rI I ;~)), ,...,(1...k) (...,cjf( ik ) ..... cjf(rik),Cj,...;((1...k)
where
cj ~ CG(f(r~k)),l
are contained in
~ j ~ aik(f;~),
and
p ~ Saik(f;~),
U.
In order to complete the proof of
1.18, we need only notice
that the normality of
that
U
implies,
1.14
can be assumed
without restriction. Let us apply this to several special cases.
1.19
The conjugacy class of
splits over
(i)
(ii)
G~A n
2 i k ~ aik(f;~)
The element
1.20
( ' ' ' ' c J ( r i k ) ' '1"
is contained in
~((1...k)
is contained in
and
= O.
. , c :jf ( r ~ k ) , C j, . ...~( ....
G~A n
in G~S n
if and only if:
2 $ k ~ aik(f;~) ~ I,
Proof:
1.21
(f;~) ~ G~A n ~ G~S n
,0
.k)
lj
,...;p))
if and only if the element laik(f;~)
11 .... , ( 1 . . . k )
A n ,where
~
;0)) ~ S n
denotes the permutation repre-
16
sentation
If
k
1.4 applied to
is odd, then
and only if
Sk~Saik(f;~).
1.21
is always contained in
An
if
aik(f;~ ) ~ 1, for otherwise the choice
11 ..... laik(f;~)
:= 0
and
p := (12)
would yield the
element ~(e;(12)) : (1,k+1)(2,k+2)...(k,2k) which is of form If
k
1.21.
is even, then
and only if
~ Sn\A n,
1.21
is always contained in
An
if
aik(f;w) = O, since otherwise the choice
11 := 1,12 ..... laik(f;w)
~((1...k),1
which is of form
:= O, p := 1, would yield the element
.....
1;1)
= (1...k)
~ SnkAn ,
1.21. q.e.d.
1.22
If
M
denotes a subgroup of index
2
in
conjugacy class of
(f;w) ~ G~SnM ~ G~S n
splits over
if and only if:
(i)
(ii)
G~SnM
G, then the in
G~S n
21k ^ aik(f;~) > 0 ~ C i ~ M, and
2~k ^ aik(f;w ) > 0 ~ C i
splits over
M.
17 Proof:
The element
1.20
is contained in
G~SnM
if and
only if the element
1.23
( .... cji( r Iik ) .... ,cjf(r~k '),cj, ...,cj .... ;(e;1))
i j-time s k is contained in
values
G~SnM, and this happens if and only if the
element , I lj ~. (cj)kfkrik j J
1.24
is contained in
M.
This must be satisfied for given c o ~ CG(f(rlik ))
and
cj := I
If
I f(rik) ~ M
and of exponents I f(rik) ~ M
Hence we need
f(rlik)
in
k
and each choice of
and
lj.
since one might choose
1j. = I
1.24. then we still need, that the elements
V(cj) k
I .25
J are contained in In case of odd
M. k, this needs
cj ~ M, i.e. it needs
CG(f(r~k))
~ M, i.e. it needs the splitting of the conjugacy
class
of
Ci
f(r~k)
If on the other hand k
is odd, then
1.23
over
M.
I f(rik) ~ M
and if
is contained in
Ci
splits in case
G~SnM. q.e.d.
18
1.26
If
M
denotes
a subgroup
of index
2
in
G, then
An
the c o n j u g a c y
G~S n
(i)
(ii)
splits
If
~(e;p)
of A G~S n nM
over
2~k ^ aik(f;w)
1.20
over
is always
is satisfied,
contained
= I A Ci
in
An G~SnM,
p ~ Saik(f;w),
> 0 ~ aik(f;w)
then for
have
and
M.
must be even for all
If this
in
if and only if:
> 0 ~ aik(f;w)
2@k ^ aik(f;w)
2~k
and
then
i.e.
= I.
aik(f;~ ) = I
11
~((1...k) Hence
(f;w) E G~SnM ~ G~S n
21k A aik(f;w ) > 0 ~ C i ~ G\M,
splits
Proof:
class
we have
~1) ~ Ak,
furthermore
V 11 .
that
(c 1)kf(r~k) ~ M. This needs
for odd f(r~k)
This altogether
k
and
a i k ( f ; ~ ) = 1:
~ M A c I ~ M.
yields
the n e c e s s a r y
condition
for the
we
19 splitting of the conjugacy class of 2+k A aik(f;w ) > 0 ~ aik(f;~) I.e.:
(ii)
is a necessary
(f;~):
= I A Ci
(put
11
c I ..... caik(f;N)
M.
k:
> 0,
condition.
If on the other hand we have for an even then we need
splits over
aik(f;w)
:= I, 12 ..... laik(f;~)
:= I)
that
:= 0,
f(rlik ) e G\M, i.e. we obtain
21k A aik(f;w ) > 0 $ C i 5 G\M, i.e.
(i)
is necessary
It is obvious
that ,if
all the elements
I .20
as well. (i)
and
(ii)
are contained
are satisfied, in
then
An G~SnM. q.e.d.
A special case is formed by the h y p e r o c t a h e d r a l G := S 2. 2
The types of elements
rows and
n
conjugacy class
c2
::
columns.
of
$2~S n
groups:
are matrices with
Let their first row belong to the
C I := 11S2 I
and their second one to
1(12)1.
Then we obtain the following
corollary
from
1.19,
1.22
1.26:
1.27
The conjugacy class of
(aik(f;~))
splits
(f;w) ~ $2~S n
of type
and
20 (i)
over
S2~A n
if and only if
(aik(f;w))
is
of the form
(ii)
(<1) o
(<1) o . . . )
(<1) o
(<1) o . . . .
over $2~S n
if and only if
(aik(f;w))
is
A2 of the form
(iii) over
0
*
0
*
0
0
0
0
A S2~S n nA 2
...)
if and only if
(aik(f;~))
is
of the form
0
0
0
0 ...
0
*
0
* .....
This corollary includes the results of Young on the conjugacy class of
$2~S n
(Young [I], cf. also Carter [1], [2], A2
Mayer [1], Taylor [1], [ 2 ] ) .
21 For a n u m e r i c a l
example
we consider
$2~S 4.
Its types
are
(4ooo) 13ooo) (2ooo I (l o o o)(oooo) 0 00
0
,
1 0 0 0
,
2000
,
3000
,
4000
(~oo) (~oo I (OLOO) (2OOO) (1000~ /oooo) (~o~o) (io~o (I o o o)(oooo) (o~oo) (OlO:I (oooo (ooo~) (~ooo) 0000
21
O0
0000 The types
,
1000
,
,
0 0 0 0
,
,
010
,
of
$2~S 4
split over
S2~A 4
200
QO0
0200
which are
,
0 1 0 0
,
11
,
0 0 1 0
,
1 0 1 0
,
,
0 0 0 0
,
0 0 1
.
characterize
(cf.
1.27
conjugacy
O0
classes
,
which
(i)):
(~o~o) (oo~oI (~ooo) (OOOoi 0000 The types
of
split over
, $2~S 4
$2~S 4
1000 which are
,
0010
characterize (cf.
1.27
, conjugacy
101
.
classes
which
classes
which
(ii)):
A2
(o2oo) (OOOl) 0 0 0 0
The types
split over
of
$2~S 4 A4 S2~S 4 A2
which
are
,
0 0 0 0
characterize
(cf.
1.27
9
conjugacy
(iii)):
I~176176176 (~176176176 1 0 2 0 0
,
0 0 0 1
.
22
2.
Representations
of w r e a t h p r o d u c t s and t h e i r c h a r a c t e r s
Let us n o w b r i e f l y r e c a l l w h a t has b e e n said in p a r t the c o n s t r u c t i o n of m a t r i x r e p r e s e n t a t i o n s
I
about
of w r e a t h p r o d u c t s
G~H. The s a l i e n t p o i n t in c o n n e c t i o n w i t h this c o n s t r u c t i o n w a s to n o t i c e
that
be extended,
G
that
underlying vector
For
F
over a f i e l d
sentation
of the b a s i s g r o u p can
and how this can be done.
Let us a s s u m e group
linear representations
is a l i n e a r r e p r e s e n t a t i o n of the K
space
with representation module V
M,
and a c o r r e s p o n d i n g m a t r i x r e p r e -
~ .
n ~ ~, we
form the
n-fold
t e n s o r p r o d u c t of
V
with
itself: n
%v. n-times
This v e c t o r d e n o t e by on
2.q
space over
K
yields a left
G*-module,
n ~ M, if we d e f i n e the o p e r a t i o n of
n @ v i := v I | .. ~ v ~ @ V i " n
(f;IH)
w h i c h we
(f;1H)
E G*
by
| v i := i | f ( i ) v i = f ( 1 ) v I ~ . . . | f ( n ) v n. i n
The c o r r e s p o n d i n g
r e p r e s e n t a t i o n of
G*
is d e n o t e d by
~F,
n
a corresponding matrix representation
~
is d e f i n e d by
23 n
2.2
~E(f;IH)
:= E ( f ( 1 ) )
x...x
E(f(n))
= (filk1(f(1))'...'finkn(f(n))), if
~
(g) = (fik(g)),
for
g e G.
The most important fact is now n F
resp.
n ~ r
F
resp.
n ~ ~
n
~Ln
that this representation
can be extended to a linear representation of
G~H
(H
a permutation group on
11, .... n l) in a natural way.
=
n
To do this we extend the set of operators on group
2.3
G~H
@ V
to the whole
as follows:
(f;w) | v i := | f(i)v _ I
i
i
= f(1)v _ 1
(i)
(i)
|
f(n)vw_1(n). n
This yields a left
G~H-module
this module is denoted by
with underlying vector space
n ~ M.
| V,
A corresponding matrix repre-
n
~ ~'~ i s
sentation
tions
o b t a i n e d by a p p l y i n g s u i t a b l e
to the matrices
column p e r m u t a -
~ ~ (f;1) :
n
2.4
~(f;w)
:= (filk _1
(f(1))... fink _I (I)
'
Let us now evaluate the traces of these matrices. the dimension of the vector space of
V
and
= bil ~...| b i
I I ~ i~ ~ ml n
n
is a basis of
@ V.
If
Ibl,...,b m 1
V, then I| b ~
(f(n))). (n)
m
is a basis
24 Then
2.3
yields
(f;(1...n))
=
bi~
~
~
: | ~(~)b i
(1...n)l(~)
(El fiin (f(1))bi) @ (Zj fJil(f(2))bj)
l~i,j,...,k<m
fii
|174
(Zk fkin-l(f(n))bk)
(f(1))fji 1(f(2))... fkin_ l(f(n))bir
n
k.
This yields for the trace:
2.5
tr ~F(f;(1...n))=
~ filin(f(1))finin_1(f(J~))...fi2i1(f(2)) 1~_i~m = tr F(f(1)f(n)...f(2)) = tr ~ ( g ) ,
where
g
denotes the cycle product associated with
with respect to
(1...n)
f. n
In order
to
obtain
the
trace
E Sn, we need only notice on
| vi i
of
that
n ~F
of
2.3
for a general element says
that
(e;w)
acts
by just permuting cyclically the factors correspon-
ing to the symbols in the cycles. X
~N(f;~)
n ~F
This yields for the character
: c(~)
2.6
If
(f;w) e G%H,
=
W~k i,j,k aik(f;~)>0
T(f;~) : (aik(f;w)), ~ : T v=1 (el.
1.9), and
bi(f;~)
~V
:= Z aik(f;w), k
25
then
c(~)
n
F(f;w)
7]-
=
w=1
i,j,k aik(f;w)>O
: --[]- (• l
aik(f;~)>O
X Fi
where
conjugacy
denotes
class
Ci
A corollary
of
2.7
V g ~ G, w ~ H
(i)
2.6
the v a l u e
of
of
XF
on the
G.
is:
(X ~" ~ F(g ..... g;w)
=
n xF(gk)ak(~ 71k=1
n
(ii) (iii) Later on,
V I~ e H (X ~ F(e;1~)
= (fF)c(1~)),
v g ~ a (x ~ F(g ..... g;1) = xF(g)n).
2.7
will
We p r o c e e d
quite
sentation
F*
turn out to be v e r y useful.
analogously
w h e n we
of the basis g r o u p
of several maybe G,
= xF(1) c(n)
different
G*
start from a reprew h i c h is a p r o d u c t
representations
F',F",...
say: F*
:= F' ~... # F' !
Y
n'-times
J
~ F" ....~ F" L
9
n"-times
~... e
...
of
) ),
26
If
Sn, , Sn,,, ... denote
of the first
n'!, n'
n"!,
symbols,
can be extended
I applied of
F*
recall
this.
FI,...,
tations
to
G~(H
Fr
I
spaces
VI,...,
F1,...,
quite
where
system
Vr
modules
the F*
analogously.
the factors
over an a l g e b r a i c a l l y
representations
representation
consist
..., then
to be a finite
a complete
which
move at most
symbols,
to the case,
G
Sn
which
representations
denote
and i n e d u c i b l e
n"
of
N Sn,XSn,,•
K, assuming
corresponding vector
the next
are irreducible field
valent
permutations
this in part
closed
Let
...
the subgroups
group.
Let me
of pairwise
of
G
over
MI,...,
and c o r r e s p o n d i n g
K
inequiwith
M r , underlying matrix
represen-
~r.
Then each irreducible
K-representation
of
G*
is of the
form F*
:= F I
with representation
M*
:= M I
The u n d e r l y i n g
~... ~ F n =:
~ Fi, i
where
Vi
~... ~ M n =:
vector
denotes
IF I
Fr I
module
~ Mi, i
space
where
M i := M j,
if
is
V* := V I @ K ' ' ' ~ where
Fi
the u n d e r l y i n g
vector
Vn =: | Vi ' i
space
of
M i.
F i = F J.
27 If
nj
denotes the number of factors
equal to
Fj
9
for
Fi
of
F*, which are
Sn( ~ H)
consisting
1 < j < r, then m
TF* := (n I ..... nr) =: (n) was called the type o_~f F*. Denoting then by
Sn. J
of the permutations, factors
Fi
the subgroup of
which move at most the indices
equal to
F J, if
nj > O,
or
nj = O, we obtained for the inertia group G~H
(cf. I. 5.10):
2.8
G~HF.
:= l(f;~)
= {(f;~)
i
of
Sn. := lISn}, J G~HF.
of
F*
if
in
I F*(f;~) equivalent F* 1
I v i (F~(i)
= Fi)l
= G*(H R SnlX...• n )' = G~(H N S(n)) ,
where F*
S(n ) := Snl x...x Snr.
can be extended to a linear representation
as has been described above in
2.9
V (f;~)
~ G~HF.
for
(f;w) ~ G~HF.
((f;~)
implies
F*
of
G~HF.
2.3:
ir vi
:= i| f(1)v~-1(i)"
Fw(i) = Fi, so that
)'
f(i)v
I -
(i)
is defined. In this way we obtain a left
G~HF.-module
M*
with under-
28 lying vector space
V*.
If
Fj(g) = (F~k(g)), I ! J ! n, then a corresponding matrix representation is
~*(f;~) : (F~
2.10
Ik~ -1 (1)
....
F*
is irreducible since its restriction to
2.11 For
(f(1))... F~i (f(n))). nk~ -I (n) G*
is just
F*.
F* ~ G* = F*. the
2.12
character
If
of
F*
we o b t a i n
(f;w) ~ G~HF., where
from
2.6:
N
w =
i,' J, k
wJ ik
aik(f;~)>O
and
g~k(f;w)
wJk = ( r J k
...
is the cycle product associated with
"Wk-1(r2k)) , then
N-U xF*(f; ~) =
%s X
~gikkZ;
9
aik(f;w)>0
This formula was stated by Klaiber (Klaiber [I]) without proof.
If in addition to such an irreducible representation G~HF. = G~(H n S(n)) , F"
is an irreducible
of
H n S(n), we obtain a second irreducible
F'
of
G~HF.
by putting
F*
of
K-representation K-representation
29 2.13
F'(f;-)
:= F"(n).
Then the product F* @ F ' is a third irreducible
K-representation of
Clifford's theory of representations subgroups yields G~H
G~HF..
of groups with normal
that each irreducible
K-representation of
is of the form
2.14
F := (F* @ F')
t G~H.
This theory also yields a result complete system of irreducible
how we can obtain just a K-representations
of
G~H
(cf. I 5.20):
2.15
If
G
is a finite group and
closed field, then
K
an algebraically
F = (F* @ F') t G~H
runs exactly
through a complete system of pairwise inequivalent and irreducible
K-representations
F
of
G~H, if
runs through a complete system of pairwise not conjugate
(with respect to
K-representations
of
H') but irreducible
G*, and, while
F*
is fixed,
F" (cf. 2.13) runs through a complete system of
F*
30 pairwise
inequivalent
tations
of the inertia
In order to construct G~H
and irreducible factor
H N S,n,~J
the irreducible
of
F*.
K-representations
we need only know the representations
subgroups
K-represen-
H N S(n ) = H N Snl x...x Snr
of
of
G
of
and of the
H.
All properties
which remain valid under inner tensor product
multiplication
and induction,
G~H,
if they hold for the representations
(cf. I 5.39 - 5.42).
This together with
results about characters,
2.16
If the ordinary
integral
(real),
characters
A corollary
is
to conjugacy
I 3.16 G
If
characters
H N S(n )
of
are rational
G
H
and
of
G
of
of H.
as well
are rational
irreducible (real).
G~S n
referring is ambi-
of a finite group is
of its characters.
The ambivalency
and the ambivalency
H N S(n )
also yields
integral
is ambivalent,
For the ambivalency
to the reality
again:
2.12
then all the ordinary
G~H
G
I 3.14, which we now obtain without
classes:
valent as well. equivalent
of
of
of
e.g.
irreducible
as of the subgroups
of
hold for the representations
G~H
We also obtain
implies
the ambivalency
31 Since the ordinary are rational
irreducible
integral-valued,
characters
of symmetric
groups
we obtain as a special case of
2.16:
2.17
If the character then this holds
table of
table of
2.15
allow
G~H, when
of the subgroups
is rational
for the character
Besides these more theoretical together with
C
results
H 0 S(n )
table of
2.16
the evaluation
the character of
2.12
yields for the character
2.18
~(f~)~
H
G~Sn0
and
2.17,
2.12
of the character
tables of
G
as well as
are known.
of
Q~HF. (XF*|
integral,
F* ~ F':
~ XF"(~)
""
~
II
~rJ
~(g~k(f~))),
x
i,j,k aik(f;~)>O
so that furthermore
only an implementation
process
Gretschel
is needed.
have done this,
extending
so that now the character mn
< 15.
If
FG
denotes
of
H,
I
and Hilge
the results
(Gretschel/Hilge
of S~nger
tables
of
Sm~S n
a representation
of
G
introduced
in part
I
of the inducing
and
(S~nger
[1]) [I])
are known for
FH
the following
a representation abbreviation
32
n
(FG;F H)
:= ~ F G | F H.
2.18
yields for its character:
a.19
-N- FG bi(f;~) V(f;.) ~ O~H(• (~;FH) = x~H(~) [l(xi ) ) 1
This formula has been used by Littlewood
(Littlewood
[3], [4]),
I shall revert to this later on. Let us continue
this section with a numerical
example.
Let us consider the splitting of ordinary irreducible sentations of hyperoctahedral $2~S n
which corresponds
groups
$2~S n
over
repre-
S2~A n
and
to the splitting of conjugacy classes
A2 described in the precelding
section (for more details and A S2~Snn cf. Celik / Kerber / A2
for the splitting over
Pahlings
[1]).
Since
I[2], [1211
pairwise inequivalent over
is a complete
and irreducible
system of
representations
C (cf. I 4.27), the following is a complete
ordinary irreducible
representations
types of the basis group s
$2"
of
s-times The inertia group of
2] ; s,te2>0 A v
P
s+t=n~
B
t-times
s t ~[2] ~[I 2]
hence each ordinary irreducible
system of
S2~Sn:
t
~
S2
of pairwise different
2.201~[2];[12]:=!2];...;[2],[12];...;[I ~
of
is
S2*(SsXSt )' = S2~(Ss•
representation
F"
of the
33 inertia factor
S s x St
is of the form
denotes a partition of
s
t i o n of
s = O,
t (~
t)
(lf
and c o r r e s p o n d i n g l y The
[oJ~[~], where
(for short: ~
[a]~[~]
we put
s), and
[a]a[~]
:= [ a ] ,
if
~
a
a parti-
= [O]a[b]
:= [ ~ ] ,
t = 0).
ordinary irreducible representation of $2~S n which arises is
denoted by s
2.21
:=
This together with
2.22
t
[ 2 ] ~ [ 1 2 ] | ([a]~[~])')t S2~Sn.
2.15
yields Young's result:
A complete system of pairwise inequivalent ordinary irreducible representations
I(al~)
of
$2~S n
I s,t ~ Z~0 A s+t=n A a ~ S
is
A ~tl.
For an example I mention that the following is a complete system of pairwise inequivalent ordinary irreducible representations of
2.23
$2~$4:
1(4to),(ol4),(3,11o),(ol3,11,(221o),(o122),(2,121o),(ol2,12), (1410),(0114),(311),(113),(2,1
I1),(1 12,1),(1311),(1113),
( 2 1 2 ) , ( 2 1 1 2 ) , ( 1212),(12112 ) t. This system corresponds to the system of conjugacy classes of
$2~S 4
given in the preceding section.
We ask now for a complete system of ordinary irreducible
34
representations of
S2~A n
and
$2~S n
. A2
Clifford's theory of representations subgroups yields
of groups with normal
that for this we need only show, which
of the representations
(al~)
of
$2~S n
form pairs of
associated representations with respect to the normal subgroups of index
2
considered
(so that their restrictions
to the normal subgroup are equal and irreducible). maining selfassociated representations
The re-
split into two conjugate
and irreducible representations when restricted to the normal subgroup, these two irreducible constituents have to be described precisely, representations If we are given
say as representations induced by certain
of subgroups of the normal subgroup. (al~)
of
S2~Sn, then we obtain the associ-
ated representation by forming its tensor product with the alternating representation of
$2~S n
with respect to the
normal subgroup considered. The following is obviously valid:
2.24
The alternating representations
to
S2~An, $2~S n
representations
of
$2~S n
An , $2~S n , respectively, A2 A2
(Inlo), (Oln),
with respect
are the
(Olln), respectively.
35 Forming the inner tensor products we obtain the desired associated and selfassociated representations:
2.25
(i)
The ordinary irreducible representations of
which are associated with
(al~)
$2~S n
with respect to
An
S2~An, $2~S n
, $2~S n A2
(a'l~')
are:
= (al~) ~ (Inlo),
(~la) (~'I~') (ii)
, respectively, A2
= (al~) |
(oln),
: (~I~
(olln)-
|
The ordinary irreducible representation $2~S n
is selfassociated with respect to
$2~S n
, $2~S n
(al~)
of
S2~A n ,
An
A2
, respectively,
if and only if
A2
G
=
a'
A
~
=
~,
=
~',
a = ~', respectively.
(Recall from part I, p. 20, that c' denotes the partition which is associated with ~.) For example
36 2.26
(i)
The pairs of ordinary irreducible representations of
$2~S 4
S2~A 4
which are associated with respect to
are:
1(410),(1410)l,1(014),(011471,t(2,12t0),(3,110)t,
I (012,12), (013,1) l, 1(3117, (13 I1 ) t, t(113), (1 113) 1,
t(2127,(12112) }, l(2112),(1212)J. The ordinary irreducible representations of which are selfassociated with respect to
S2NS4
S2%A8 are=
(2210), (0122), (2,1 I17, (112,1 ).
(ii)
The pairs of ordinary irreducible representations of $2~S 4
which are associated with respect to
$2~$4A 2
are: I(410),(014)1,1(3,110),(013,1)1,1(2210),(0122)I, ~(2,1210),(012,1271, I(1410),(0114)I, I(311),( 1 1371,
1(2,1 117,(112,1)J,1(1311),(1113)t,1(2112),(1212)}. The ordinary irreducible representations of which are selfassociated with respect to
$2~S 4
$2~$4A 2
are:
(212), (12112). (iii) The pairs of ordinary irreducible representations of
$2~S 4
A4 $2~S 4 A2
which are associated with respect to
are:
37
~(41o) ,(Oll 4) l, I (o1~), (14 I~l, t(3,1 I0), (0i2,12) J, t (013,1), (2,12 I0)J, { (2210), (0122) }, t(31~ ),(~ 113) t, t(~ 13), (1311)t, {(2,1 I1),(112,1)}, {(212),(12ll 2) t. The o r d i n a r y
irreducible
which are selfassociated
representations
of
S2%S4
A4
with respect to
S2%S4A2
are:
(2112),(1212). As has been mentioned
above,
ciated representations presentation
the elements
restrict
of the normal
tions of selfassociated
of a pair of asso-
to the same irreducible
subgroup
of index
representations
of
re-
2, while restric$2~S n
split into
two representations. This yields various of
symmetry properties
of the character
$2~S n, some of which have been described
I 6.13 - 6.15).
in part
table
I (cf.
It allows also to derive a great deal of A
the character
table of
S2%An,
$2~S n
and
S2~Snn A2
A2 the character
table of
S2~Sn.
But we are also interested are the remaining subgroup
entries
considered?
of characters
from
in the remaining of the character
problem:
What
table of the normal
i.e. we would like to know the values
of the irreducible
tions of the selfassociated
constituents
representations
of
of the restric$2~S n.
38 An answer to this question would imply an answer to the question:
are the values
of the characters
of
S2~An,
$2~S n
, A2
S~ z
A
n
nA2
rational
The values integral
integral?
of the characters
in general,
of
S2~A n
this is obvious.
even a strictly complex character stituents
+ [2,1]-
irreducible
of
(cs
S2~A 3
has
The irreducible
I 4.54)
yield
con-
the
representations +'
~[2] S2~A 3
For example,
table:
[2,1] ~ A 3
3
of
are not rational
+
| [2,1]-
with strictly
= (2;2,1-)
complex-valued
characters. A
Nevertheless,
the characters
of
$2~S n
and A2
have rational groups
integral
of type
Dn
is a splitting
values.
It is in fact known
have character field for these
presentations
inequivalent
tables over (cs
I would like to show this directly system of pairwise
S2~Snn A2
that Weyl
Z, even that
Curtis/Benson
by constructing
and irreducible
may
[I]).
a complete
ordinary
re-
of these groups.
For this we need only construct tions of selfassociated with the normal
subgroup
the two irreducible
the constituents
representations
of
of the restric-
$2~S n.
We start
S2~A n, so that we have to show
constituents
of the representation
how
3g
2.27
(alp) ~ S2~An,
can be o b t a i n e d sentations
where
a = a'
as r e p r e s e n t a t i o n s
induced
of a suitable
subgroup
To do this we first a p p l y in order to c o n s t r u c t representations. sentations
of
F*
repre-
S2~A n. S2~A n
system of its o r d i n a r y
irreducible
of these
constituents
representations
s t ~[2] ~ [ 1 2 ] ,
g r o u p of
2.29
F*
where
irreducible of
repre-
2.27.
of the basis group
s,t ~ Z70
and
s+t=n.
is
S2~(SsxSt0An ) ,
so that we n e e d to know the o r d i n a r y of
from c e r t a i n
are a g a i n the r e p r e s e n t a t i o n s
:=
The i n e r t i a
~ = ~',
to the w r e a t h p r o d u c t
We then ask w h i c h
irreducible
S2~A n
2.28
a complete
are the i r r e d u c i b l e
The o r d i n a r y $2"
2.15
of
and
irreducible
representations
S s x S t N A n.
F r o m the c o m p l e t e
system
2.30
IEa]~[~] l a ~ s A ~ t l
of o r d i n a r y
irreducible
at once a c o m p l e t e if
s
is
As
exclude
or or
t
representations
system of r e p r e s e n t a t i o n s
is less than or equal to At~
these
of
We may then c o n s i d e r
cases by a s s u m i n g the
series
of
that
we o b t a i n
S s x St 0 An
I~ for then
so that we n e e d only a p p l y
trivial
S s • St~
I 4.54.
S s x St n An Let us
s,t ~ 2.
40
2.31
A s x A t < S2 x St n An < S x St . 2 2 s
It is obvious,
that e x a c t l y
2.32
[a]~[#],
of
2.30
the e l e m e n t s
a = a'
where
are s e l f a s s o c i a t e d
with respect
In order to obtain the i r r e d u c i b l e
2.33
[a=a,];[~=~'] 2.32
the f o l l o w i n g
(recall
2.34
[a=a']~[~=~']
Since
everyone
[a]~[~]
in
to
S s x St n An .
constituents
of the r e s t r i c t i o n
~ S s x St n An
of the r e p r e s e n t a t i o n is v a l i d
~ = ~',
and
we use
2.31
that
and n o t i c e
that
s,t ~ 2):
SAsXA t = [n]+~Es]++[n]+~[~]-+[~]-~[~]++[~]-~[~]-.
of these
S s x St,
two of them induce
four i r r e d u c i b l e
Frobenius'
constituents
reciprocity
the i r r e d u c i b l e
induces
law implies
constituents
of
2.33
that in
S s x S t n A n. A consideration following
2.35
of the r e p r e s e n t i n g
equivalences
(use
matrices
then shows the
I 4.55):
[a]+~[~] + ~ S s X S t n A n ~ [m]-~[~]- t S s x S t n A
n,
[ a ] + ; [ ~ ] - # S s X S t n A n ~ [a]-~[~] + I S s X S t n A n-
H e n c e we can c o n c l u d e
that the f o l l o w i n g
holds:
41 2.36
If
n=s+t, where
s,t,~2, and
a = c ' ~ s,~=~'~ t,
then [a]~[~]
$ Ss•
n = [a]+~[~]+~Ss•
+ [a]+~[~]-~Ss•
is the decomposition of the restriction
[a]~[~]$Ss•
into its irreducible constituents.
This allows us to construct a complete system of ordinary irreducible representations of the inertia factor of the representation
2.28
in
S2XA n .
In order to obtain just a complete system of ordinary irreducible representations of sentations
F*
S2%An~ we would like to know which repre-
has to run through.
factors of just two kinds, namely to notice
that
that for
n ~ 4
A
is
n
(n-2)-fold
(so that
Since [2]
F*
and
consists of
[12], it suffices
transitive.
n-(n-2) ~ n-2)
F*
This yields
needs only to
run through a complete system of irreducible representations of
$2"
of different types.
The cases
n = 1,2,3
are easy
to handle. Hence we are left with the question, which of these irreducible representations of
S2xA n
2.37 Since the matrices of
are the irreducible constituents of
(a=a' I~=~') ~ S2XA n. (aI~)
are up to a sign for each n o n -
n
42 zero box just the matrices of [a][~] = [a]~[~]
~ S n = [a]+~[~] + ~ S n = [c]+~[~] - ~ S n
(cf. I 5.24), we see that since irreducible ~ _ ~ s t
constituents
(~[2]~[12]~([a]+~[6]+
of
2.36
2.37
holds,
the desired
are just
(~+1~+) +
~ SsXStglAn ) ' )
@ S2~A n =:
S2%An
~ SsXStnAn)')
f S2XA n =: (c+Ip -) ~ S2~A n
and s
t
(~[2]~[12]~([a]+~[6]if and
s
and
t > I.
(a-10)
If
~ S2~An,
and analogJously
if
~=0
or
t=1, they are
(a+l I) ~ S2~A n s=0
or
(01~-) f S2"A n, (II# +) {' S2"A n
s=1 : and
and
(5+10) ~ S2~A n
(o.-11) ~ S2~A n
(01~ +) ]' S2".An and (116-) t S2%A n.
For example:
(2210)
$
(0122)
$ S2~A4
S2%A4 = ( 2 2 + 1 0 )
= (0122+)
+ (22-10),
+ (0122-),
(2,1ll)
$ S2~A4 = (2,1+11) + ( 2 , 1 - 1 1 ) ,
(112,1)
~ S2~A4 = (112,1 + ) + ( 1 1 2 , 1 - ) .
Thus) the following system is a complete system of pairwise inequivalent ordinary irreducible representations of
$2~A4:
43
2.38
t(410) ~ S2%A4, (014) ~ S2%A4,
(2,1210) ~ S2%A4,
(012,12) $ S2~A 4, (311) ~ S2%A 4, (I13) $ S2~A 4,
(212) $ S2%A4, (2ll 2) $ S2"A 4, (22*10), (22-10), (0122+), (0122-), (2,1+11), (2,1-11),
(112,1+), (112,1-)1.
These results together with the results of section conjugacy classes of ter table of
S2%A n
S2~A n
1 about the
allow the evaluation of the charac-
(which in general contains complex numbers).
What can be said about the characters of
$2~S n
? A2
It is known, that their values are rational integral. like to derive this)using the above results. we need only consider the representations
2.25
(cla)
I would
says of
that
$2~S n
and to obtain the irreducible constituents of its restriction
2.39
(ala) ~ S2~SnA 2
say as representations induced by certain representations of suitable subgroups. To do this, we apply Clifford's theory to the normal subgroup
2.40
$2" 0 S2~SnA 2 = {(f;ISn ) I ~ f ( i ) i
= IS2} _< $2~S hA2.
Every ordinary irreducible representation of this group is of the form
2.41
s t 2] ~[2]~[1 ~ $2" fl $2~S n
. A2
44
Since
(cla)
arises from s
s
n
~[2]~[12],
s := ~ ,
we need only consider, which of the irreducible of
$2~S
representations
arise from nA 2 S
2.42
S
~,[2]~[12]
~. $ 2 "
r] $2",,S n
. A2
The inertia group of this representation all the
(f;w) e $2~S n
2.42
consists of
which satisfy for each A2
(f';1) e $2" n $2~S n
: A2
n
2.43
sgn
s
s
2]
f'(j) = ~[2];[I
s
s
(f';1) = ~[2]~[12](f;w)(f';1)(f;w) -I
= sgn~f~(j). s+1 This holds if and only if
2.44
If
n
S
S
is even and
w e ~[Ss~S2] ~ S n.
s := 2' the inertia group of
~[2]~[I 2] $ $2" n $2~S n
in A2
($2" N $2~S n
Hence we obtain
)~[Ss~S2]'
$2~S n
is A2
= $2~ ( ~[Ss~S2])A2
A2 S
The restriction
S
.
of ~[2]~[I 2] to S 2 0 $2~S n
can be extended A2
to S2~(~[Ss~S2])A 2 as follows:
45
n S
S
~[2]~[ 12] ( f ; ( g ; P ) )
:= - I T
sgn(f(i)).
i=~+1 The check is very easy. The inertia factor is
( S 2 . ( m [ S s ~ S 2 ] ~ / (S* N $2~S n
A2
) ~ SsNS2,
so that the above extension produces the two irreducible representations
2]
(o;2),
and S
S
#[2]#[1 2] | (a;12) '
of the inertia group S2~(e[Ss~S2 ])A2.
We notice that
(c;2) = [a]#[m] | [ 2 ] '
, (C~;12)
= [a]#[c]
@ [12] '
yields s s (#[2]#[I 2 ] | ([~]#[a])')
=
@ L
((c;2)
+
~ S2~(~[Ss~S2])
(a;12))
J
extended to S2~(~[Ss~S2])A 2.
'
$ S2~(~[Ss~S2]IA2
46 We have obtained 2.45
If
n
of
(ale)
n
is even,
s := ~, and
$ S2~S n
a e
, then the decomposition
into its irreducible
constituents
is
A2
(aiR) $
s
(ala)+
(ala)+ +
S2%S : nA 2
::
, where
s
(~[2]~[12]
~ $ 2 - I] $2~S n
@ (c;2)')
f $2~S n
A2
(ala)_
:= (~[2]~[12]
$ S2" N $2~S n
, A2
@ (a;12) ') ~ S2%S n A2
This together with
2.46 Every ordinary
2.25
. A2
yields:
representation
of $2~S n
can be written A2
over ~ (and hence of these groups, rational member
integral
over Z), i.e.
the ordinary
Hence
(Celik/Pahlings/Kerber irreducible
[I])
table of each
is rational
by Benson and Curtis
[I]). The preceding
the characters
of type D n have
the character
of the series of Weyl groups
(Benson/Ourtis
generalized
of the Weyl groups
values.
This result was obtained way
so that especially
integral.
in a different
direct derivation
can be
so that we obtain e.g. A representations of G~SnM and G~SnnM '
47 M a subgroup of index 2 in G, G abelian.
The preceding theorems on the splitting of conjugacy classes and of ordinary irreducible representations modular representation theory,
can be applied to
too. For if we have evaluated
the decomposition matrix of $2~S n, say, we may ask for the A
decomposition matrices of S2~A n, $2~S n we n e e d
t o know t h e
splitting
of both
and $ 2 ~ S ~ , for which A~ A~ ordinary irreducible
representations and p-regular conjugacy classes. Let us consider an example: $2~S 4 ,p := 2.
In part I the decomposition matrices of S 2 and S 4 with respect to p = 2 were evaluated, we obtained (I, 7.12, 7.16):
A
F1
2.47
1
[1 2 ]
V
V
FI
F2
1
0
1
1
0
1
1
1
1
0
[4] [3,~] [2 2 ] [2,12 ] [14 ]
Hence there are exactly two 2-modular irreducible representations of S2%S 4, namely
^
2.48
V
F I := (F I;F I)
A
and
V"
F 2 := (F I; F 2)
In order to obtain the decomposition matrix of $2~S 4 we need evaluate the multiplicity of F i in ( ~ ) ,
a 2-modular re-
48
presentation of $2~S 4 corresponding to (alS). Since (recall p = 2):
(~i~) : ([a]~[~])'
~ s2~s 4
we obtain V(f;~) ~ S2~S 4 ((alS)(f;~) : [a]Es] (~)), so that the following holds for the multiplicity of F i in (al~):
V
2.49
((ols),
F i) = ([a][s], Fi),
The multiplicities
i = 1,2.
([a][B], ~i ) can be obtained with the
aid of the Littlewood-Richardson-rule
(I, 4.51).
E.g. [2,1][I] = [3,1] + [22 ] + [2,12 ] together with 2.47 and 2.49 yields
((2,111),F i)
([3,1] + [2 2 ] + [2,12], Fi}= / 2' i = 1
[
3, i
2
In this way we obtain for the decomposition matrix of $2~S 4 with respect to p = 2:
49
2.50
F1
F2
1
0
1
1
(41o) (3,11o)
0
1
(22io)
1
1
(2,121o)
1
0
(141o)
1
0
(o 4)
1
1
(o 3,1)
0
1
(0 22)
1
1
(0 2 , 1 2 )
1
0
(o 14)
2
1
2
3
(2,1 I1)
2
1
(1311)
2
2
(212)
2
2
(2112)
2
2
(1212)
2
2
(12112)
2
1
2
3
2
1
(113) (112,1) (1113 )
50 With respect to p := 3 we obtain quite similarly:
1 1
[410) (22lo) (1410) (3,~10) (2,12lo) (ol4) (0122) (0114 ) (ol3,1) (012,12)
1 1 1 1 1 1
1 1 1
2.51
1
(311)
(2,111) (1311 )
1 1 1 1
(~13)
1 1
1
1
(1 12,1) (1113 ) (2t2) (2112) (1212) (12112)
(In fact both these matrices can be completed fairly easy in order to obtain the generalized
decomposition
matrices of
$2~S 4 with respect to p = 2,3). Let us derive the decomposition
matrices of S2~A 4 from
2.50, 2.51. S2~A 4 contains three 2-regular classes,
for the conjugacy
class of (e; (123)) splits over S2~A 4. Hence one of the two
51
2-modular irreducible representations FI, F 2 of $2~S 4 splits over S2~A 4. Since F I is one-dimensional, it must be
F 2 l s2,~ 4
,-*
F2:
F~ + F z
Clifford~ theory yields part of the decomposition matrix by cancelling one element of each pair of associated ordinary irreducible representation of $2~S 4. We obtain in this way:
F I ~ S2~A 4
2.52
F~
F~
1
0
0
(4Jo)
1
1
1
1
0
0
1
1
1
(3,1 Io) i $2~- 4 (ol4) i $2~- 4 (ol3,1) ~ S2"~A4
2
1
1
(311)
$ s2"~ 4
2
2
2
(212)
~ S2".A 4
2
2
2
(2112 ) $ S2~- ~
2
1
1
(113)
$ s2"~. 4
1, S2"A4
It remains to evaluate the decomposition numbers of the +
irreducible constituents (cl~)- of +
(22 I0)
~ $2'~--4
:
(2210)
+ (22"10)
(0122)
~. S2",,,,A4
=
(0122+) + (0122-3
(2,111) @ s 2 ~ 4
=
(2,r
+ (2,1-11)
(1 12,1) ,~ $2'~ 4
=
(112,1+)
+ (112,1-}.
52
Since F 2 = (2210), the constituents
(22~0)
of
are irreducible.
F 2 @ S2~A 4 so that
(22]0)
+
= F2 9
Using 2.52 we are able to evaluate characters
of $2~A4,
the matrix of Brauer
it is the matrix
(4 0 0 O) 0 0 0 0
(1 0 1 + 0 0 O)
(10
0 1 0)0 0
1
1
1
i~-1
1 +
1
1 + iff
if
2.53
Using this,
We may number
some characters
we obtain the decomposition to p = 2. It is the matrix
1 2
~-
2 2
i~ -I
of $2~S 4 and S2~A 4 as well as 2.50 matrix of S2~A A with respect
53
FI ~ $ 2 ~ 4
2.54
1
0
0
1
1
1
0
1
0
(4;o) ~ Se~A4 (3,110) ~ Se~A 4 (2210)
0
0
1
(22Io)
1
0
0
1
1
1
(014) ~ s2~ 4 (013,1) ~ s2~ 4
0
1
0
(0122+ )
0
0
1
(0122- )
2
1
1
1
2
1
1
1
2
(311) ~ S2~A4 (2,1'11 ) (2,111)
2
2
2
(212)
} S2~ 4
2
2
2
(2112)
i $2~A4
2
1
1
(113)
$ S2~A 4
1
2
1
(112,1 + )
1
1
2
(112,1-~
In order to evaluate the decompostion matrix of $2~$4A 2 with respect to p = 2, we notice first that $2~$4A 2 possesses exactly two 2-modular irreducible representation since no 2-regular class of $2~S 4 splits over $2~$4A 2. Hence both F I ~ $2~$4A 2 and F 2 ~ $2~$4A 2 are irreducible. We thus obtain the decomposition matrix directly from 2.50. It is the matrix
54
F 1 $ S2~$4A 2
2.55
F 2 $ S2~$4A 2
1
0
(41o) ~ Sa~S4A 2
1
1
(3,110) $ $2"~$4A2
0
1
(2210) ~ S2NS4A 2
1
1
(2,1210)$ $2",$4A2
1
0
(141o) $ $2"$4A 2
2
1
(311 ) ~ $2~$4A 2
2
3
(2,1 11 ) ~, $2",,$4A 2
2
1
(13il)
1
1
(212)+
1
1
(2t2)
1
1
(2112) ~ $2",'$4A 2
1
1
1
1
(1212) ~ $2"$4A 2 (12112)+
1
1
(12112)_
A4 The decomposition matrix of $2~$4A 2 turns out to be
~, $2".$4A 2
with respect to p = 2
55 A4 F 1 ~ $2~$4A 2
A4 F 2 ~ $2"~$4A 2
0
A4 ~ $2",.$4A2 A4 (3,1 10) ~ $2",.$4A2 A4 (2210) ~. $2",,$4A2
I
A4 (2,1210) ~. $2",,$4A2
(410)
A4 J, $2",$4A 2
I
0
(1410)
2
1
(311 )
2
3
(2,111) ~ S2~4A 2
2
1
(1311)
~. S2%S4A2
2
(212)
A4 ~ $2""$4A 2
A4 ~
$2~$4A2 A4
2.56
A4
L
(2112)+ 1
(2112)_
2
A4 (12112) ,~ $2",,$4A2 (1212)+ (1212)_
This done one may ask for the generalized decomposition numbers which complete the decomposition matrix (see part I). Numerical results concerning can be found in Celik/Pahlings/
Kerber [1].
58
There is also a theorem concerning generalized
decomposition
numbers
the question when the are rational
Sufficient
for this is that the values
characters
on p-singular
(Reynolds
2.57
elements
integral.
of the ordinary
are rational
integral
[I]). This together with 2.18 yields
The decomposition if the character
numbers
of G~H are rational
tables of G as well as of the inertia
factors H 0 S(n ) have rational This holds in particular of G is rational
integral
entries
only.
for G~S n if the character
table
integral.
2.46 yields the following
2.58 The generalized
integral,
corollary:
decomposition
numbers
series of Weyl groups are rational
of each member
integral.
of the
57 With this we have shown some applications construction products
of irreducible
as it is described
of the
representations
of wreath
in 2.15.
2.15 says that the irreducible
representations
of GkH are
of the form
F = (F* | F') # G~H,
where
F* denotes the extension
of F* of G* to its inertia
group G~HF. as it is described by 2.9. the factors
of F* are equal:
in the case w h e n all
say to the representation
D of G,
then F* is just the representation
(D; IH),
where
IH denotes
representation of H. In the n same way as the irreducible representation @D can be extended n to @D = (D; IH), we can extend @(DI+D2) , where D1, D 2 are arbitrary
the identity
(i.e. may be reducible)
representations
of G to
G~H.
Hence the question arises (DI+D2;
IH).
following 2.59
for the decomposition
of
It will later turn out to be useful to know the
result of F. S~nger
(hitherto
If D I and D 2 are representations (which may be reducible)
n
~
@(DI+D 2) = D (@ D I @ k=O
unpublished): of a finite group G
over a field K, then n-k @ D 2) ~ G ~ S n.
58
Proof:
Let
vector
space
~
D i be Vi,
_~(DI+D2)
~(MISM2)
afforded
afforded
underlying
by
where
each
a left
Wk = VI @
...
~ VI @
left
= ~ k=O
...
k-
@ VI |
KG~Sn-module
w k,
KG~Sn-module
| V I @ V2 @
and
... n
| V2
- k
~ VI ~ V2 @ VI @ V2 @ J
...
e V2 @
n - k -
@ V2 | VI | I
...
J
k
@
...
k
@ V2 D
| VI
i
n - k
(V I
...
i
n - k -
I
(e;c)
I
~ V1 @ V2 @ VI | VI @ V2 @
2
...
@ V2 I
I
k-
...
L
I
i
@
underlying
space
k summands
=
M i with
n
:= ~ ( v 1 $ v 2)
W k is
module
the
vector
n
w
the
i = 1,2.
is
with
by
@ VI
@ V2
@
...
n - k
@ V 2)
1
59 where L is a complete system of representatives oZ' the left cosets of S k • Sn_ k in S n. Hence Lemma (4A.I) in Curtis/Reiner
[I] yields that W k affords the representation
of G~S n which is induced by the representation afforded by the module with underlying vector space
V1
@
...
@ V1
| V2
@
...
|
V 2.
But the module with this underlying vector space affords the representation n-k D1 I
*,
D2
~
of G~S k
q
!
of G~Sn_ k
!
of G%(S k x Sn_k)
q.e.d.
An example is provided by 3
,~ ( [ 2 ]
+ [12])
f ~ + ([2]~[12]~,[12] = (2;3)
3
= ,~ [ 2 ]
+ ([2]qk[2]T[12]
(~ ( [ 1 ] ~ [ 2 ] ) ' ) ~ '
e~ ( [ 2 ] , ~ [ 1 ] ) ' ) ~ $ 2 " , . S
3 $2"~S 3 + ,~[1 2]
+ &[2]~:[12] f S2',..S3 + [ 2 ] ~ [ 1
] I' s2-s 3 + (12
3).
Having obtained a result on the decomposition of the extension
3
6O n
of the reducible
representation
~ (DI+D 2) to G~S n let
us conclude this section with a hint to another way of producing
representations
of G~H from representations
DofG.
The method described above arose from the desire to apply Clifford's
theory of representations
of groups with normal
subgroups where one has to start from irreducible representations of the normal
subgroup in question.
begin with irreducible
Hence we were
representations
forced to
of the basis group
which are just the outer tensor products F* = @ F i of irreducible representations
F i of G (if we assume the ground-
field to be algebraically
closed).
If we are not forced to obtain irreducible
representations
of G~H we may start from reducible representations
of G* as
well and there is in fact a way to do that and apply an extension process quite
similar to 2.3 which yields an in
general reducible representation
of ~ H
which will turn out
to be useful later on.
Let F G denote a representation let V denote the representation module.
of a group G over a field K and space, M the representation
If n is a given natural number,
then we may form the
n-fold outer direct sum ~ of M with itself. The underlying n vector space is X V and the operation of G* } G%H (H } Sn):
2.60
(f; I H)
(v I ..... v n)
:= (f(1)v I .... ,f(n)v n)
It can be extended to G%H in a fashion quite similar to 2.3:
61
2.61
(f;~)(Vl, .... Vn) := (f(1)v~_1(1) ..... f(n) v ~-I (n) )"
n
We denote this module by $ M, the afforded representation b y $ F G. n
It is easy to check that the character values of $ F G are as follows n
SFG
2.62 X
~
FG
)i i
~(i)--i
If e.g.
~ is the natural representation of G < Sm, then
it is not difficult to see that $ ~ induces on the natural basis
~(0 ..... O,ej,O ..... 0) 11 ~ ej ~ m} (where
acts on C m = <<el,...,em>> by ~(g): ei|~ eg(i )) a permutation group similar to @[G~H], where ~ denotes the permutation representation 1.4.
Chapter
II
An application to representation Symmetrization
theory:
of inner tensor products of representations
The results of the preceding chapter are applied to the theory n of symmetrizing the n-fold inner tensor power | F G of an ordinary representation
F G of a group G with ordinary irre-
ducible representations
[~] of S
Some applications
n
are discussed and the case G := S m is
considered in more detail 9
63
3.
Symmetrized
Let
FG
a field
inner products
of representations
denote a linear representation
of a group
K
M
w i t h representation module
vector space
V
G
over
and u n d e r l y i n g
and a corresponding matrix representation
F G 9
In section number
2
n
vector
we have seen, how an additionally
leads to a left n | V
space
sentation
n ~ M
w i t h underlying n ~ FG
which affords the representation
G* = G I x...x G n < G~Sn, n ~ ~G
G*-module
given natural
and how a corresponding
can be defined.
how this r e p r e s e n t a t i o n
n ~ FG
matrix repre-
A n d we have seen in section 2,
can be e x t e n d e d
to
a representa-
n
tion
of
n
~ FG
of
G"~n, which
is
of the
same d i m e n s i o n
a corresponding matrix r e p r e s e n t a t i o n was denoted by
as
In this section I w o u l d like to show, that it can be very ful to consider, restrictions
for a given
FG,
of the c o r r e s p o n d i n g
~ FG,
n ~ ~G"
use-
suitable natural numbers and n ~ FG
representations
or
n
FG
to certain subgroups of
suitable groups which contain
An interesting (cf. Serre f FG
G*
or even
a.
G~S n
as subgroups.
example is provided by a proof given by Tate
of each ordinary irreducible
of
as well as to induce to
[I]) of the well known theorem,
finite group
C(G)
G~S n
G
divides the index
that the dimension
representation IG: C(G) I
F G of a given
of the centre
64 In order to prove
this, we form,
for a g i v e n n a t u r a l n u m b e r
n n ~ ~,
the
representation
as a r e p r e s e n t a t i o n
n
~ FG
space,
if
of
V
G* ~ G'XSn,
denotes
which has
|
the r e p r e s e n t a t i o n
space of F G. A subgroup
of
G*, w h i c h has not been m e n t i o n e d
U::
{(f;1)
in section
I,
is n
I f: Nn
i:I n
l(g1'.... gn;
=
The order
of this
subgroup
lul The e l e m e n t s of
of
U
I)
I gi~C(G)
rr
^
gi
i=1
=lal"
is :
Ic(G)l n-1
act on the g e n e r a t i n g
elements
v I |174
vn
n | V, v i E V , as follows:
(gfl'''''gn; Since
gi~C(G),
~i = l e '
I) v I |174
v n = glvl
|174
givi = ~ivi , ~ieC, and since
so that
the
irreducible
gnVn .
~ T g i = 1G, n
representation
~ FG
of
n U
acts t r i v i a l l y
on
| V: n
V (fit) e U, v I | 1 7 4 Hence
U
v n e | V,
is c o n t a i n e d
tain an o r d i n a r y
((f;1) v I | 1 7 4
in the kernel
irreducible
of
representation
n ~ FG
vn = v I |174 so that we ob-
F
of
G*/U
putting n
F((fd)u) The d i m e n s i o n IGIn/IC(G) In-1
of
F
: : ~ F a ( f d ). is
(fFG) n, and it divides
of the r e p r e s e n t e d
group,
the order
i.e. we have
by
Vn).
6s
V n~N(JG : C(G) In
I
(fFG)n which
shows,
that
)
~ ~ . Z
IG : C(G)I/f FG
,
is a natural number. q.e .d.
n
An example using We considered
~ FG
a finite permutation
a representation tion group
FG
H ~ Sn
over a field
and induction was discussed
K.
of
group
G, say
in volume
I.
G ~ S m, and
G, together with a second permuta-
with a representation We formed
~ FG
of
FH, F G
G~H
and
and
FH
F~, defined by
!
V (f;~) Since
1.4
e G~H
yields
(FH(f;n)
an embedding
:= FH(,)). of
G%H
into
Smn,
the
inner
n
tensor
product
following
(FG;FH)
:= ~ F G @ F~
allows
to define
the
representation:
n
3.1
F G Q F H :=
(Fc;F H) of
t Smn = ~ F G | F~ r Smn.
ia-H]
This in general
reducible
representation
the symmetrized
outer product o f
FG
If on the other hand we restrict we obtain the following
interesting
of
and
~ FG
Smn FH
was called
(cf. I w1675,6).
to the diagonal
representation
of
of G:
G*,
66
n
3.2
:=
~ FG
which is in general Notice
n
n
~ F G ~ diag G* =
~ F G ~ diag G*,
reducible.
that n
3.3
n
V g ~ G (| FG(g)
=
# FG(g ..... g;IH)).
The question arises which are the irreducible
constituents
n
of
| F G.
This problem can be attacked aid of the representation degree
n,
e.g.
the famous
of Schur,
Reiner
[I],
[I], Weyl
w 67, Kerber GL(n,C)
[8]).
on
GL(n,C)
: F G := idGL(n,C),
using the ordinary
H ~ S n, K over
We notice
groups [2],
a field and
of
FG
G
[I],
that
theories [2],
[3], cf. also Curtis/ the identity
representation
and obtained
representation
that
(cf. Schur
They considered
as an ordinary
Hence let us assume,
H
and Weyl,
between the representation
[I],
mapping
G
van der Waerden
linear and symmetric
van der Waerden
groups
A special case of this is in fact
there is a close connection of general
solved) with the
theory of permutation
H:= S n.
discovery
(but not always
important
theory of
is a group,
of results by
S n.
n
a natural number,
to be a linear representation
of
K. first,
that the composition
law of
G~H
implies,
67 that the elements of
3.4
of
diag G*
commute w i t h the elements
H':
V g~G,~H
i.e.
((g ..... g;~)=(g ..... g;IH)(e;~)=(e;~)(g, .... g;IH)),
diag G* ~ CG~H(H')
Hence the c o r r e s p o n d i n g
A H' ~ CG~H(diag
elements
G*).
of the image of this repre-
sentation commute as well:
3.5
n n nr~ V g e G,~ ~ H (~FG(g ..... g;~) = ~FG(g ..... g;IH)~FG(e;w)
= ~FG(e ;w) ~FG(g ..... g;IH).
Hence by putting n
3.6
(i)
v g ~ a (FG(g)
:: ~FG(g ..... g;IH)),
V ~ ~ H
:= ~FG(e;w)),
n
(ii)
(FG(W)
we obtain from a given r e p r e s e n t a t i o n
FG
of
G
and a natural
^
number
n ~ ~
a representation
FG
of
of
their
G
and a r e p r e s e n t a t i o n
v
FG
of
H
so
that
the
elements
n
3.7
^
^
n
FG = |
F G.
commute:
v
u g ~ G,w ~ H (~FG(g ..... g;~) = FG(g)FG(~)
Notice that the following holds:
3.8
images
= FG(W)FG(g)).
68 The crucial fact is
3.7.
It shows, that we can apply a
corollary of Schur's lemma when the groundfield
K
(cf. Boerner
[2], I w 8), in the case
is algebraically
closed and
charK ~ IHI. v
Let us assume
that this is valid.
completely reducible,
It implies
that
FG
is
so that we can choose a basis of the n
representation
space
| V
matrix representation
~G
of
FG
which yields a corresponding
in its completely
reduced form,
say
9.~i(~) v
3.9
V ~eH
=
I
n
(FG(W) = ~FG(e;w) =
t ~ niDi(~) i=1
=
o
"0
]
ni-times
"
Di(~)
$ (inl • i "
ni>O where
D I,...,D t
denotes a complete
valent and irreducible ni
is the
system of pairwise inequi-
K-representations
of
H
and where
ni-rowed identity matrix.
The corollary of Schur's lemma now implies that the same basis n of ~ V yields a corresponding decomposition of FG" If fi denotes the dimension of
Di
then there are matrices
P G [] Di(g)' which satisfy the following:
3.1o
v g ~ a (~a(g) =
~$((~a m~z~g).
• ~fi)'
ni>O These matrices
FG[]~i(g)
form a matrix representation
of
G.
69
The corresponding
3.11
If
K
FG~Di
representation
is algebraically
closed and
of
G
hence satisfies:
charK X JHJ, then V
to each irreducible constituent
Di
ponds a c e r t a i n
reducible)
(and i n g e n e r a l
of
FG
there corres-
constituent
^
F GE] D i
of
product of
FG
which we call the symmetrized
FG
and
(Robinson [5]).
inner
D i, following Robinson's notation
It satisfies n
^
v
V g e G,w ~ H (~FG(g, . . . . g ; ~ ) = FG(g)FG(n) =
(FG[:I Di(g)
~
• Di(~))).
i ni>O The dimension of
Di
in
vF G
F G [] D i
and it occurs
denotes the dimension of
is the multiplicity
ni
" fl-times in
fi
D i, so that we obtain
~G =~0fi (Fc[]Oi). i ni>O This yields
(apply
3.8):
n
| FG = ~ . f i i
ni>O
^FG, if
(FG~D
i).
of
70 Before with
specializing
on
some p r e l i m i n a r y
ters to become
One of the main
and
remarks
a bit more
of sy~mnetrization
irreducible
G
H let us continue concerning
acquainted
section
corresponding
with this useful
characconcept
of representations.
questions
which
representations
arise
Di
charK = O, we may use characters tions,
this
of
is the question, H
occur
and their
in
which
[G"
If
orthogonality
rela-
w h i c h yield V
3 9 12
n i = (FG,D•
= 7W7
FG
x
( ~ ) ~ D ~ ( ~ -1
v
In o r d e r t o e v a l u a t e 3 13 -
u g ~ G,~ ~ H
~G (w), we use
2 . 7 , which g i v e s
(x~FG(g ..... g;w)
II l(FG(gk) ak(~) k=1
so that especially
n~
V
3.14
Hence
3.15
(Since
Y ~ ~ H (xFG(~)
3.12
~Z-~H
has ordinary
every natural surprising
= (fFG)c(w)).
implies
(FG'D) = 7 ~
G
= x~FG(e;w)
number
corollary
m ~ ~, the number
representations
m, we obtain that
of d i m e n s i o n
as a b y p r o d u c t
for a subgroup
H
of
m
for
the maybe Sn
and
71
I UHmc 7~7~ is a nonnegative
(w) ~ (~)
integer for any ordinary irreducible
character
of H.) 3.15
gives us
3.16
If
FG
and
is an ordinary representation
D
an ordinary irreducible
permutation group
H
trized inner product
of a group
representation
of finite degree, FGE]D
G
of a
then the symme-
exists if and only if
~, (fFG)c(~)~D(~) ~ O. ~EH Hence for example
FGE] IH
always exists.
For its dimension
we obtain 3.17
f FG [] IH
Taking
H := S n
3.18
v I ~ = (~a 'IH) = 7WT
I ~ ~
of
Sn:
H
we get f FG [] [n]
Analogously
(fFo)c(~)
(fFo)c(~) Sn
we obtain for the alternating
representation
[1 n]
72 3.19
If
FGE] [I n]
exists, then it is of dimension
1 I~FG F] [In] = ~-!
~Snr
fFG)c (w)
For example
fF G []
FG [12] = ~I ((fFG) 2 - fFG) = (f2)"
In fact one can show [1 n ] 3.20
fFG
and that
FG[] [I n]
that FG = (fn)'
if
FG~] [In ]
exists if and only if
return to this question of existence of Then we shall also discuss, when
3.11
exists,
n ~ f
FG~
Di
FG
.
We shall
later on.
F G [] D i is irreducible.
allows an evaluation of the character of
F G ~ D i.
We
need only apply the orthogonality relations to the equation
3.21
x~FG(g . . . .,g;w) .
~ X FGF~ Di (gk Di (~), i ni>O
which is an immediate consequence of We multiply both sides by irreducible the elements relations and
~DJ(w-1), where
K-representation of ~
of
3.11.
H.
Dj
denotes an
By summing up over all
H ,we obtain from the orthogonality
2.7 (i):
73 3.22
If
K
is algebraically closed and
if
F G[] D i
exists, its character reads as follows:
V g ~ G (XFGr:] Di (g) = ~TI ~ H
If
FG[3] Di
charK f IHI, then
~Di (~-I )k~IXFG(gk)ak (w) ).
does not exist, then we have n
v g
a
Insertion into
3.23
:
Di | F G = E f (FG~] Di ) yields
n
V g e G (xFG(g) n =
1
"
~,fD i z ni>O
k=1
H := Sn,
Then we obtain for an ordinary representation sufficiently high dimension) and each
~ x ~ G ( g k ) a k (w)
~ e H
Let us again consider the special case
3.24
o).
FG
K := C. of
G
(of
g e G:
xFGE][1] (g) = xFG(g),
(i)
(ii) xFG El[2] (g) = 89 (xFG(g)2 + xFG(g2)),
(iii) X FG
[] [I2]
(g) = 89 (xFG(g) 2 - xFG(g2)),
(iv) XF o ~ [3](g) = ~ (xFG(g)3 + 3XF0 (g)xF0 (g2) + 2xFG(g3)), (v)
XFG~] [ 2 , 1 ] ( g )
= ~ (2xFG(g)3
_ 2xFG(g3)),
74
(vi)
3.24
FG [] [1 3] X (g) = ~ (xFG(g) 3 -
3X
and so on (use the c h a r a c t e r
tables
provides
tables.
a useful m e t h o d
It is w e l l known,
sentation
G,
G
K-representation
of
of
of
of s y m m e t r i z e d to subtract
repre-
classes
for each
constituents FG
number
of d i f f e r e n t
such a f a i t h f u l
the c h a r a c t e r s 3.24
is a faithful
the i r r e d u c i b l e
occurs u n d e r
gn
3.22,
of c h a r a c t e r
G
which
3.11,
FG
groups).
then every i r r e d u c i b l e
of the c o n j u g a c y
evaluate
of symmetric
a finite group,
orders
lies,
F G(g)x G(g2) + 2xFG(g3)),
for the e v a l u a t i o n
that if
n ~ F G, n = 1 , 2 , . . . , t , t t h e
Thus if we are g i v e n
F
of
g ~ G
values
representation G
of
X
FG, the
as w e l l as the class in
and some
n ~ ~, then we may
of the tensor p o w e r
n | FG
and use
to break them up by f o r m i n g the c h a r a c t e r s
inner p r o d u c t s
FG[=I [a],
known irreducible
characters
a~n. of
The m e t h o d is G
if they are con-
FOrq [a] t a i n e d in of
G
X
and check,
(cf.
G
in this way, Esper
other
concerning
character
further
irreducible
characters
say by a m a n - m a c h i n e
interaction
program
or not. to seperate
[1]).
These r e s u l t s the
the r e m a i n i n g
is i r r e d u c i b l e
It is often p o s s i b l e of
whether
have been obtained
h a n d we s p e c i a l i z e the
presentations
connection of
H := Sn
on
by specializing O
we o b t a i n
between the and
on
If
on
famous results
ordinary
G := G L ( m , C )
H.
irreducible if
we p u t
re-
75
FGL(m,C ) := idGL(m,C ) : GL(m,C) For in this case it turns out, is i r r e d u c i b l e
if it exists.
that We
~ GL(m,C)
: g ~ g.
FG[~] [a] = i d G L ( m , g ) E ] [a]
shall r e t u r n to this
later
on.
I w o u l d like to c o n t i n u e this of van Z a n t e n and de Vries have p o i n t e d
out,
section with
(van Z a n t e n / d e
how t h e o r e m s
some of the r e s u l t s Vries
concerning
[I]).
the n u m b e r
They of solu-
tions of the e q u a t i o n
3.25
xn = g
for a g i v e n e l e m e n t natural number the p a r t i a l
3.26
n
g
of a finite group
G
denote a finite group,
and
n
a natural llx
number.
g
I x ~ G ^ x n = gll
is at least one o r d i n a r y
sentation
FG n @ FG
a fixed e l e m e n t
G
which
contains
of
G
If
then there
of
on
n | F C.
of
Let
irreducible
and a g i v e n
can be d e r i v e d u s i n g the above r e s u l t s
reduction
so that
G
IG
+ I, irreducible
satisfies
F G + IG
repreand
at least once as an
constituent.
Proof:
We assume
on the contrary,
ducible
representation
FG
that each o r d i n a r y
different
irre-
from the i d e n t i t y
repre-
76 n
sentation
IG
of
G
is such that
~ FG
does not contain
IG, i.e. we assume that
V FG * IG (~T ~ An a p p l i c a t i o n
of
3.11
XFG(x)n : o)
yields
(take
H := Sn):
,~
~n
(1)
x)
=
0),
x~G
na>O so that we obtain for each partition
a ~ n
with
n a = (FG,[a]) > 0:
1 ~
(F a + Ia A nc>O ) ~ T ~ T
Let us compare
aX
FGE1[~](x)
these results with the character
X~FG (x . . . . .
x;(1...n))
(2)
= O.
value
FG(xn )
:
(3)
[a]
: ~xFG[VI
(x)~a((1...n))
~ n
na>O (use
T~
3.13
and
3.21).
(3)
]
yields
,~a((1...n))
~
(x),
Gl.-n
na>O so that an application
of
(2)
yields
V FG + IG (T~ ~'~xFG(xn) = 0). x
If now
r(n,g)
~
(4)
~
denotes the number of
n-th roots x
of
77 g ~ G, then it satisfies the equation
x e G
xFG(xn) = ~-~ir(n,g)TFG(g), g ~ G
so that by the orthogonality
relations we obtain
' x~---~GTFG ( xn) X F G(g-1),
r(n,g) = T~T ~
(5)
g where
the
first
summation
is
irreducible representations Equation
(4)
over
FG
all
of
now shows, that
IG
the
ordinary
G. yields the only
contribution to this sum, so that we get r(n,g) =
I ...I-.CT.IGI : 1,
which contradicts our assumption. q.e .d.
Remark:
3.27
to prove this we used the following
If
G
is a finite group with an ordinary irreducible
representation
(i)
xFG(gn
FG, then for each
) =
>
n ~ ~, g E G:
' FG[] [a] ,~c((1...n))x (g),
~ n
na>O
(ii)
xFG(g) n = ~
,' fcxFG[] [a](g),
~mn
na>O
where
v
~ ( ~ ) ( f F G ) c(~)
I
n a := (FG,[a]) = ~.,
~E
Sn
78 It seems remarkable to notice that this implies
3.28
If
G
is a finite group with an ordinary representation
FG, then for each natural number
~GFG ~71 g X
(g)n,
as well as
n
we have
I
g~G
xFG(gn)
are rational integral.
Proof:
Since 1
7~
~-~
~
geG
FG~ ~n
~
~gJ
n =
(eFa,Ia) '
the first part is trivial. For the second part we use
3.27 (i)
which implies
Cw (gn) = I - n g a ( ( l " ' ' n ) )
~<
(g)"
na>O We know, that
1 g ~ a F G ~ ] [a](g) = (FGE] [~],IG) ~ Z,
Y~Y as well as that
Ca
has all its values in
2
(0
is a splitting
field, cf. I). Having proved this, van Zanten and de Vries consider the result of applying the mapping
3.29
o : G ~ G : g ~ gn, n E N
fixed and prime to
which induces a permutation of the ordinary irreducible
IGI,
79 characters of G, so that together with F G there is an ordinary irreducible representation F G of G with character 3.30
V g ~ G (XF~ (g) := MFG(gn)).
Van Zanten and de Vries proved that 3.31
If
G
is a finite group with an ordinary irreducible
representation
FG
and if
n, a
and
C
FG
are as is
n
described in
3.29, 3.30, then
~ FG
contains
FG
as an irreducible constituent.
Proof:
Otherwise we have (use 1 ~ O = ~ g ~ G
= T ~ 1T
~
FG
3.27 (ii)):
n FG n (g) X ( g )
~ f C ~ G
xFG E][a] (g)xFG(gn)'
na>O what implies, that for each partition
I
~ G
a
of
n
with
XFGE] [a](g)xFG(gn) = O.
Thus also
1 T~
FG n a FG [] [a] (g))~'-~, ((1 .n))'X (g) = 0 "" a~n na>O
~( g~G
so that an application of
3.27 (i)
gives us
na>O:
80
~FG(gn)2
= 0,
which is impossible. q.e.d. For further results concerning the number of solutions of the equation
I
3.25
~GFG X
as well as the integers
(g)n
and
g~G
~
X FG (gn),
the reader may consult the paper of van Zanten and de Vries.
We have got some nice results about characters of symmetrized inner products,
but there are still some fundamental
to be answered,
for example we would like to know at least FG~] D i
some special cases, where
is irreducible and
we would prefer a more direct answer than of the existence
of
questions
3.16
to the question
FG~] Di.
These problems can be attacked by looking closer at the definition of
FGE]Di. n
The definition was, that the which affords
H-invariant
ni-times the representation
V
subspace of Di
of
H
9
(n i := (FG,Di)) affords
fl times (fi := dimension of
a certain representation
of
denoted by of
n | V
| V,
F G[] D i.
D i)
G, and this representation was
In other words:
into its homogeneous
is also a direct decomposition
The direct decomposition
components with respect to of
n | V
with respect to
H Gt
81 n
and the homogeneous component of type respect to
H
yields a left
pairwise equivalent left
Di
of
@ V
with
G-module which splits into
G-modules.
fi
The representation
afforded by each one of these direct summands is denoted by F o [] o i .
It can be shown (cf. Boerner [I])
that F G ~ D i is afforded
9 n by e1(@V), if e i denotes a primitive idempotent of KH such that KHe l
affords D i. Let us furthermore denote the centrally primitive idempotent which generates the homogeneous component of D i in KH by e i, i.e. we put fi ~ H ei := T~T
3.32
x
Di
(~-I) TT. n
Hence the homogeneous decomposition of H
~ V
with respect to
is n
3.33
n
| V = 9 (e i e V).
,,
i
The sum can be taken over a complete system of pairwise inequivalent and irreducible
K-representations
v
n i := (FG,Di) = 0
3.34
of
H, since in the case
n
FG[V]Di
we have
exists if and only if
Let us consider the case 3.16
has shown, that
3.35
If
FG
e i | V = IOl.
K := C, H
F G [ ] [n ]
Thus
n e i @ V ~ IO1.
:= S
exists.
n"
What about
FG[?] [In]?
is an ordinary representation of dimension
f FG
82 of a group
n ~ f
FG
G, then
FGI--~ [In ]
The dimension of
exists if and only if
FGE] [I n]
is
(fFa)
if
n
it exists.
Proof: The centrally primitive idempotent corresponding 3.36
e
(I n)
= ~
~ ~S
to [1 n] is
cww . n
Hence
FG[~ [ln ]
exists if and only if
(~ ,' %~)
3.37
n
{ v 4 ioJ .
Sn
If
le I ..... e FG I f
is a basis of leil |174
V, then
e i I I ~ i~ ~ f
FG I
n n
is a basis of for
each
basis
vector
FG[7]__[I n] does not exist if and only if | ei
we
have
(w~ ~ 'S ' n ~ W ) | ~ e.z~ = ~ e 'S ' n cn |~ eiw_1 (~) : O.
3.38
(i)
such
| V, so that
If
fFG < n, then in each such basis vector
least two factors
e i~
Thus the transposition
are equal, say (kl)
e ik
at
eil.
satisfies
V w e S n (w | el9 = w(kl) @w e i9). Since
and
| ei~
(I)
83
~ Sn we see, that (ii)
fFG
w ~ ~n
(I)
implies
If on the other hand (fFg
_
1)...(f FG
-
n | V = lOJ.
(Z r f
FG
> n, then there are
n+l) basis vectors
pairwise different factors
| ei~
with
e . , and all the other basis
vectors are mapped onto the zero vector by left multiplication
(see (i)).
The basis vector
~ ei~
is mapped onto
with pairwise different factors
,
ei~
n r
| ei ~ | V ~ ~-I (~)
~ Sn
as are all the other basis vectors with the same set lei~ I 1 ~ ~ ~ nJ
of factors up to a sign FG image space has dimension (fn):
(~
3.39
'S n
r
~1.
n| V : ~) = (fF G ) > O, n
Hence the
if
fF G
>
n.
q.e.d. 3.40
If
FG
is an ordinary representation of dimension
of a group
G
and
~ = (a I ..... a h) ~
exists if and only if
Proof:
h ~ f
This theorem contains
FG
3.35
n,
then
and it is surprising, 3.35.
FG
FGQ] [a]
.
that it is proved mainly by an application of
f
84 To show this we use the fact that the centrally corresponding primitive
to [~] can be expressed
idempotents,
in CS n which affords afford
primitive
as a sum of orthogonal
each of which generates
a minimal
[~]. Since all these minimal
[m] are isomorphic
idempotent
to CSne ~ (recall
left ideal
left ideals which
from part I that e~ =
a
~1~1
)' we need only show that the following
3.41
~.~ c ~fa n
I
I ~v#
FG
lOlr
To prove this, we use the fact
is valid:
that if
Va, = x Sat 0 a
then
a and that the factors commute,
side of this equation
so that
1
3.35
on the right hand
yields,
3.42
j-~2
w ~ Sa, J
,,
n
p E
a~
that
()
p E
And the argument
s' Cpp) | a~
# IOl~h<__ f
FG
in the proof of 3.35 shows clearly
that this is equivalent
to
3.41. q.e.d.
What can be said about irreducible
F G[] D i, if
representation
Di
of a subgroup
is an ordinary H
of
Sn ?
85 n
FGE]D i
exists if and only if
only if
Di
v
v
P G.
FG
on
occurs
the
irreducible
is just the restriction
n @ V, so t h a t
occuring
under
e i r V # I0], i.e. if and
[a]
Di
occurs
if
constituents
of a representation and only if
there
is
of of
Sn
an
which satisfies
$ H,D i) > O.
3.43 Hence we have v i (FG,D)
3.44
= ~([a] g~n na>O
$ H,Di)~I~.
This reduces the problem to a question about the relationship of the representation
3.45
If
FG
and if
theories of
Sn
is an ordinary representation
H
irreducible
is a subgroup of
representation
Sn
n
satisfying
If this happens,
h ~ f FG
D i, then
ct~-n
h
([a]
of a group
FGE ] Di
G
exists
a = (al,...,a h)
([a] ~ H,D i) > O.
the dimension of
~--'~,
m
and
H:
with an ordinary
if and only if there is a partition
of
and
$ H,Di)lu~.
FGE] Di
is
86
In the case when
D i, D k
are irreducible representations of
over an algebraically closed field that
FG~
Di
presentation
as well as FG
K
FG~] D k
of a group
with
charK ~
JHJ
exists for a given
H
and so
K-re-
G, then it is of course reasonable
to define. FGE] (D i + D k) := F G ~
D i + FG~
D k.
This may be generalized as follows:
3.46 Def.:
If
G
is a group, H a subgroup of Sn,
algebraically closed field with two K-representations
K
an
charK @ JHI and F G, F H are
(may0e reducible)
of G, H, respectively,
so that for at least one irreducible constituent D i of F H the symmetrized inner product exists, then we put
F G[] F H := ~ ,
(FH,Di)(FG~] Di) i
summing over all i for which
FG[] D i exists
(i the indices
of a complete set of pairwise inequivalent irreducible constituents D i of FH).
This allows us to state the following theorem (Clausen [1]): FG 3.47
If F G is an ordinary representation of dimension
f
of a group G, and D i an ordinary irreducible representation of H ~ Sn, then, if
FGE] Di
exists, we have
87
F G [] D i =
([a] $ H , D ~ ) F G ~
[a].
a~n
h
n
@ V = e~ ( 9 (e a @ V)) = 9 e i e a ( ~ V) ~n a~n
Proof:
n
=
~, ([a] ~ H,Di)(e ~ | V). ~wn i q.e .d.
Hence the d e c o m p o s i t i o n
3.48
| Fa :
, fa(PcD
~a])
a~n h
into the symmetrized products w i t h ordinary sentations
of
S
n
irreducible
is at least as fine as is the d e c o m p o s i t i o n
into symmetrized products w i t h ordinary irreducible tions of any
representa-
H ~ S n.
To the remaining I only mention
repre-
question,which
of
that in the case
FG[~Di
are irreducible,
G := GL(m,C),
H := S n, K := C
and F G := idGL(m,C ) each existing
3.49
F G ~ ] [a ]
a = (a I .... , a h ) ~ n
is irreducible.
:
GL(m,C)
~ GL(m,C)
is irreducible,
and
h ~ m =
: g ~ g,
i.e. that we have
(a) := idGL(m,C)~ 7] [a]
88 This is one of the crucial facts concerning the connection between the representation theories of symmetric and general linear groups.
The reason for this remarkable result is that
to this representation correspond two subalgebras of n End C (| cm), one of them a homomorphic image of a homomorphic image of
CGL(m,C)
CSn, the other
and so that the latter is
the enveloping algebra of the centralizer of the former (cf. Boerner [2], Clausen [I], Curtis/Reiner [I], Weyl [4]).
[I ], Dieudonn@/Carrell
This fact and various consequences are well
known (cf. the quoted literature) and basic for the methods which especially D.E. Littlewood uses in his development of the representation theory of general linear and symmetric groups. They will be described in detail in another part of these lecture note s. The main tool of Littlewood is the consequent use of certain polynomials which are closely related to the characters of the representation 3.22 (a> 3.50
(a).
yields for the character of If
of the representation
GL(m,C): m,n ~ N, a:= (a I ..... an) ~
V g~GL(m,C)(g(a)(g)
The trace
~(a)
tr(g k)
= ~ !I
n, h<_m, then
n tr (gk)ak(~) ). D ga(~) "IT ~S n k=1
is the value of the polynomial
89
m xk ~ O[xl ]~ 1 i=1
~ at
(r162
the matrix
where
r
'Xm] ''~
are the characteristic roots of
g.
Hence if we put I la~:=~.,
3.51
~ w~S
n ak(~) ~T o k k=1 ,m
~a(~) n a
= aHn~ ~a
n
I
where the sum is taken over all types (for short:
a H n) and where
a ~a
ordinary irreducible character
'
a = (a I .... ,an)
of
n
denotes the value of the ~a
(of
conjugacy class of elements of type called Schur-functions
ak
k__Ti'kak l ak!~
(for short:
a.
[a])
of
Sn
on the
These polynomials are
S-functions). We shall
meet these polynomials again in a seemingly quite different region of mathematics,
namely the theory of enumeration under
group action which will be discussed in the following sections. Other expressions of 3.51
~a I
can be used by comparing them with
in order to obtain polynomials the coefficients of which
turn out to be character values.
It can be shown for example,
that the following holds (cf. Boerner [2]): cj+m-j det(x i ) 3.52
Comparing this with
la~ =
3.51
det(xim_J).
we obtain a famous formula:
90
3.53
("Frobenius' ~
formula")
7
det (x~j+m-~
n
1
ak Ok ~a ~ ak , ,m k=1 k -a k . c
det(x m-j)
= a~n ~
If we take e.g. m := 3 and ~ := (2,1) we o b t a i n the p o l y n o m i a l 2 2 2 2 2 2 XlX2+X2X1+X2X3+XlX3+XlX3+X2X3+2XlX2X3 The right h a n d
1 .(2,1)
Since
(2,1) , o,o O O
+
a comparison
shows,
characters
of
~ 1,1,0)
= O,
can be e x t e n d e d
This is useful
~
= 1" to evaluate
and important,
but
Iot
to an i s o m o r p h i s m
representations
ring of symmetric
functions.
Hence
representations
considering
the r e p r e s e n t a t i o n ~ i n D.Knutson
r e s u l t the
of the S y m m e t r i c
symmetric
of symmetric
groups
question
Theorem
(Knutson
groups
all onto a
one is free in
or the c o r r e s p o n d i n g of this and c a l l e d
of the R e p r e s e n t a t i o n
[I], p.
Theory
135).
But let us continue
this section w i t h a f a s c i n a t i n g
prising
of 3.47.
application
contains
in order to derive r e s u l t s
gave a m o d e r n t r e a t m e n t
"Fundamental Group"
from a ring w h i c h
of all finite
on o r d i n a r y
this
t2,1)
~ 0,0,1)
can be u s e d in order
groups.
independent,
is, that the m a p p i n g [aJ
polynomials.
(2)
"
(2) yields:
12,1
= 2,
of symmetric
the o r d i n a r y
0,1)03
(1) and
how S - f u n k t i o n s
the main thing
+
sums al,m,...,On, m are a l g e b r a i c a l l y
(3,0,0) T~is
(I)
side of 3.53 is
the p o w e r
if nSm,
I 3 = ~(~ - a3)
In fact we w o u l d
and v e r y
sur-
like to show using
91 3.47 that the ring of nxn matrices polynomial
over C satisfies the standard
identity of degree 2n. o. Kostant was the first to
show that this can be done by an application (cf. Kostant
of symmetrization
[I]) and I think that in particular this example
shows that symmetrization
is in fact very useful in order t~
discover deep symmetries
in various mathematical
structures.
We apply 3.47 to the case G := GL(n,C) , F G := idGL(n,~),H
:= A2n+l,
Di
:= [n+1,1 n ]
(recall that [n+1,1n]$ A2n+1 = [n+1,1n] + + [n+l,ln] -, cf. I 4.54). We state that the following holds
(cf. Amitsur/Levitzki
3.54 If A1,...,A2n are n• the following [A I ..... A2n]
over C, then they satisfy
"standard polynomial :=
[I]):
identity":
~ cwA (I) ... Aw(2n ) = 0. 1~-$2n
Proof: (i) We first prove that the mapping r (e;~(1...2n+1)~-1):
2n+lcn |
~
2n+I |
w~S2n+1 which maps v1|
onto cw(...| v
w~S2n+1
-1 w(2n+1...1)
~''') (i)
is the zero mapping. In order to prove this, we notice first that i d G L ( n , C ) ~ [n+1,1 n] does not exist (use 3.40). Hence 3.47 yields that both
92
idGL(n,C)[] r[n+1,1nj~ corresponding
do not exist.Expressed
in terms of the
centrally primitive idempotents
e
+ [n+l,ln]
-
this reads as follows: e
+ ,ln] -
[n+l
2n+I @
~n =
IOJ,
so that we also have (e
_e [n+l,ln]
+
Applying Frobenius' groups
[n+l,ln]
2n+I ~ ~n = IOl
-
theorem on the characters
of alternating
(I. 4.55) to this situation we see that this difference
of idempotents
is a multiple of
D pp~ (2n+I)+ ((2n+I) • (I...2n+1) (ii)
)
the two
A2n+1-classes
splits).
This yields
We now prove
nxn-matrices
over
( D
T~ (2n+I)-
that
. . .2n+1 )w-1 )
into which the class of our first statement.
tr[A1,...,A2n+1]
= 0
for arbitrary
C.
tr[A I, .... A2n+1] = tr
~ w~S2n+1
r
= ~ cw tr Aw(I) = Dcw (cf. proof of
r
~$2n+I
2.5).
A~(1)
... Aw(2n+1)
... Aw(2n+l)
tn~--~ tr @idEnd(V) (A I ..... A2n+I ;w(1 ..... 2n+1)~-I)
Hence
tr[A 1 ..... A2n+1] = tr#idEnd(V)((A I ..... A 2 n + l ; 1 ) o ~ and this is 0 since
(i).
(e;~(1..2n+l)w-))
93
(iii)
We are now in the position to prove the statement.
In order to do this we first remark that $2n+I = S2n. ((I...2n+I)) (S2n
may be considered as stabilizer of the symbol
$2n+I).
This together with
0 = tr[A1,...A2n+1]
in
2n+I
gives us
(ii)
cw tr Aw(I)
=
... Aw(2n+1)
weS2n+1 T tr AaT(1) ~S2n
... AaT(2n+ I)
c Te((1...2n+1))
= (2n+I) tr A2n+1-
~ c a Ao(1) a~S2n
... Ao(2n )
= (2n+I) tr A2n+I.[AI, .... A2n ]. Since this holds for arbitrary nxn-matrices
A2n+1, the state-
ment must be satisfied. q.e.d.
After having considered this interesting application,
we are
left with the question how symmetrized inner products
split.
We would like to discuss this question for the Case when both factors are ordinary irreducible
representations
groups, i.e. we ask for the splitting of irreducible
of symmetric
[a] [] [~]
into its
consituents.
This problem has not been solved completely but there is a formula which reduces this problem to the splitting of symmetrized inner
products of form
[~][~ [r], r~_n, if
This formula is quite analogous to formula
~n.
I. 6.20, where the
94 splitting of outer symmetrized products
[ a ] ~ [8]
duced to the problem to split products of form
was re-
[a] (~[r].
We first state that the following holds (cf. Robinson [5], 3.62): 3.55
([a][~[nl]) | ([a][~[n2]) = [a] F]([nl][n2]).
Proof:
If
a ~ m, w~S m, then the character of the left hand
side is X[a][] [nl](~) X[a]~] [n2](~ ) = 1
~,
~I ~a(~k)ak (p)
nl ! P~Snl k=l 1 nl !n~.v
1
~,
n2
C~(~ l)al(a)
n2! a~Sn2 1=1
n I+n 2 ~'--'-~ ~ ~a(~k) ak(T) T=po
(I)
=
SnlXSn2 The character of the right hand side is [a] N ([nl ][n2]~
1 = (nl+n2) , 9
n +n 2 1 ~ [nl][n2] 'k~~ ~a(~)ak (~) (nl+n2)' = 9 $ nl+n 2 S (nl+n2) ! IC nl+n2 (~) N SnqXSn21 n ~ 2 ~a(~)ak (*)
~Snl+n2 nl!n2!
This is obviously equal to
IcSnl+n2(~)l
k=l q.e.d.
(1).
This formula yields at once a determinantal expression for [a]~][~].
If instead of the multiplication 9 we use inner ten-
sor product multiplication (we indicate this by writing
I - I|
95
we obtain
(Robinson
3.56 Hence
[5], 3.63):
[~]EZ][~] = I E a ] F l [ s i + J - •
|
for example
[a][':q[2] [~][][2,1]
[ a ] [ ~ [3]
@
=
I = ([m]l~l [ 2 ] )
[~]FI[I]
| ([a]l~
[1])
- [a][~] [ 3 ]
= ([~] [] [2]) ~ [a] - [a] [] [ 3 ] .
In order to prove more general
3.57
3.56 one needs
in fact a result which is
than ~-55- One needs
([~] [ ] [6]) e ([a] [ ] [~]) : [a] ~ ( [ ~ ] [ ~ ] ) .
This fact can be proved in the same way as 3.55. ponding out.
character
formulae
If we agree upon 3.57,
The corres-
are longer winded hence
I left it
then 3.56 is obtained by applying
I 4.41 which yields
[~] = l [ ~ i + There are various
J-i]l.
other results
on the decomposition
of
[a][][~]. If F denotes
an ordinary
irreducible
representation
group G, then we may put the question, F ~ ] [I 2] contain the identity interesting
to notice
of a finite
whether F [] [2] and
representation
IG of G. It is
that an answer to this question
characterizes
96 the kind of F (one calls F to be a representation kind if and only if F is equivalent
of the first
to a real representation,
if this is not the case then F is called a representation second or third kind, respectively, character
~ F has real values
For a well known result
of the
if and only if its
only or not,
respectively).
says that the following
holds.
If we
put
3.58
OF :=
then we have
3.59
TGT
gEG
(cf. e.g. Felt
F is of the first,
[I],
second,
(3.5)):
third kind,
respectively,
if and only if c F = I, -I, O, respectively.
The complete context.
proof of 3.59 does not fit in very nicely in this
But we can show at least that c F is in fact an invariant
and how it can be expressed If F c denotes
in terms of symmetrized
the representation
contragredient
products.
to F, i.e.
we put 3.60
u g ~ G (FC(g)
-'''(tF(g-1)) the
transpose
:= tF(g-1))
of F(g-1)),-- then we have for its
character:
3.61
We notice
v g
a
first that
= 4
(g ) ).
if
97 3.62
(F,F C) = (F @ F, IG),
so that we obtain in terms of symmetrized products:
3.63
(F,F c) = ( F ~ [ 2 ] ,
IG) + ( F ~ [ 1 2 ] ,
IG) E I0,1}.
This implies (use 3.24 (ii) and (iii)):
3.64
c F = (F[] [2], IG) - (F[] [12], IG) ~ I0,I,-I}.
3.61 shows that F is of the third kind if and only if (F,F c) = 0 so that 3.63 yields
3.65
F is of the third kind if and only i[
o: ~ c ~ , i~= ~ ~ ,
i~ c~~ E ~
: o)
If F is of the first kind, then it is well known that we may assume that the representing matrices are orthogonal, so that
(g) = FC(g). This yields (we ~ut ~(g) : (~ik(g))): I ~,4F(g2): ~ = T~7 g~G =
~ g~G
~, fik(g)fik i, k
1
i, k =
tr( ~ (g) f (g))
g~G
1
~7
=
I 7~7 ~
1.
gEG
(g-l)
98 The last equation follows from the orthogonality relations (Felt [I], (1.9)).
It is not as easy as that to show that
second kind implies c F = -1. A corollary of 3.58 - 3.65 is:
3.66
Let F denote an ordinary irreducible representation of a finite group. Then (i) F is equivalent to a real representation if and only if (F~
[2], IG) = 1, and (F~] [12], IG) = O.
(ii) F is not equivalent to a real representation but has a real-valued character if and only if (FFI[2],
IG) = O, and ( F ~ l [ 1 2 ] ,
IG) = I.
(iii) The character of F is not real-valued if and only if
(FEll2],
te) = ( F E I [ 1 2 ] ,
I a ) = O.
This is a characterization of the kind of F in terms of multiplicities
of IG in the inner symmetrized products
F [ ~ [2] and F ~ [12].
Since Q is a splitting field for ~n' we obtain e.g.
3.67
V a ~ n (([a][]
[2],
[n j)
= I a ([ct] [ ] [ 1 2 ] ,
[n])
= 0).
99 We have seen that a finite group G has R as splitting field if and only if for each of its ordinary irreducible representations F the following holds:
( F [ ] [2], IG) + O. Hence it might be useful to ask for the multiplicity of the identity representation in various symmetrized inner products. Several physicists pointed to groups G with the property that for each of their ordinary irreducible representations F, IG is not contained in F ~ ] [2,1]. They called such groups simple-phase groups (van Zanten/de Vries [I], Butler [I], Butler/King
[3]).
Groups which are not simple-phase groups are called non-simple-phase
groups. Correspondingly an ordinary irreducible
representation ~' is called
a simple-phase representation of G
if and only if
( F B [2,1], and a non-simple-phase
za) = o,
representation of G if and only if
(F[] [2,1], IG) + O. An application of 3.24 (v) yields 3.68
mF := ( ~ B [ 2 , 1 ] , z G )
1 = T~T
(for finite G):
~,
(4
F(g)3 -
dF(g3))
g~G
Let us consider the case F := [a], ~
n.
An application of the example following 3.56 yields
100
m[a ] = ([a] @ ([~J~[2]),
In]) - ([a]~l [3], [n]),
where we may substitute [a] e [a] - [a] ~
[ 12 ] for [a][] [2]
so that we obtain (regard 3.67):
3.69
3 m[c] = (@ [oj, in]) - ([~]EI [3], [n]).
This shows, how the character table of S n and the decomposition of [a]~] [3] can be used in order to check whether [a] is simplephase or not. It was shown by Derome
(direct verification)
that S 6 is a non-simple-phase group
(cf. Derome [I]). Butler
and King give many further examples
(Butler/King [3]).
As far as I know
the most complete tables with decompositions
of symmetrized inner products [a] ~ [8] were given by Esper (Esper [2]), if a ~ m, 8 ~ n, he gave such tables for all the partitions for all a and 8
a and ~ with 2 ~ m ~ 10 and n ~ 5 as well as with 2 ~ m ~ 7 and n = 6. Esper used the
character formula 3.22, other people used various tricks like results concerning the decomposition of the corresponding symmetric functions into S-functions.
101 Concerning this, we should ~ least notice what follows like 3.57 for the corresponding S-functions. 3.70
The symmetric function corresponding to the representation ([~][~]> = idGL(m,C)UI ([8][~]) is the product of the S-functions ~BI and ~I"
Proof: The symmetric function which corresponds to ([B][~]) is (nI+n2 ) :
1
(nl+n2)!
~(p) ~(o)
~Sn1+n 2 nl !n2 ! IcSn1+n2(w) I ~oEcSnl+n2 (~) nSnl• 2 n I+n 2
--~--ak(~) k,m k=1
~(p) I
C~SnlX Sn 2
P~Snl
~(o)
n I+n2 -[]-- oak(~)+ak(~ k,m k=1
nl oak(0))( I__ ~ t P) ~ k,m n2! k=l
n2 ak(a) ~(c) V ~k,m ) k=1
a~Sn2
q.e.d.
102
This yields a result about S-functions
which corresponds
to
1 4.41:
3.71
{a} = I Iai + j -
(Where we put
I0~ := I and
i}l.
Im~ := O, if m < 0.)
I should not forget to mention
that mostly
3.71
is proved first and
then serves to obtain I 4.41 via the isomorphism "Fundamental 3.53.
Theorem" which was mentioned
in the remark
(In fact 3.70 shows that the mapping
homomorphism following Theorem".)
3.70 and the bijectivity
in the
following
is a multiplicative
so that together with the additivity
3.45)
described
(see the remark
yield the "Fundamental
Chapter
III
An application
to combinatorics:
The theory of enumeration
It is shown how characters the Redfield-P~lya
of wreath products
enumeration
theory.
characters of several permutation to evaluate
action
can be applied to
The evaluation
representations
Schur-functions
Furthermore
groups
it is shown how
come in and how these can be used in
theory.
of the
of G~H allows
the number of orbits of these permutation
as well as their cycle-indices.
enumeration
under group
104
4. E n u m e r a t i o n
under g r o u p a c t i o n
In m a t h e m a t i c s
as w e l l as its a p p l i c a t i o n s
faced w i t h the p r o b l e m classes
of a g i v e n
Illustrations problems.
A concrete
colours. of w a y s
example
reads as follows. necklaces
To make it more precise:
in this way. different
the v e r t i c e s
Two c o l o u r i n g s
We are a s k e d for the n u m b e r
of the p e n t a g o n is no e l e m e n t
group D 5 of the p e n t a g o n w h i c h y i e l d s The s o l u t i o n
as it is i n d i c a t e d
are c a l l e d of the s y m m e t r y two c o l o u r i n g s
is 8, for if we
we o b t a i n 8 d i f f e r e n t
in Fig.
Fig.
can be c o l o u r e d
one of these
of this p r o b l e m
denote by 9 and o the two colours, colourings
We are a s k e d
w i t h 5 beads in two
of a p e n t a g o n
if and only if there
from the other.
relation.
s i t u a t i o n are the s o - c a l l e d n e c k l a c e -
of d i f f e r e n t
in w h i c h
often
of e q u i v a l e n c e
set u n d e r a g i v e n e q u i v a l e n c e
of this
for the n u m b e r
to count the n u m b e r
we are quite
I
I:
105
Further problems
of this kind and related problems
What is the number
of colourings
of the edges
are e.g.:
(vertices,faces)
of the cube in 2 colours? What is the number
of labeled graphs
What is the number
of chemical
say of C n H2n+l
isomeres
of automata with n states,
are primitive
of elements
of a given finite
of this kind of problems
J.H. Redfield,
G. P~lya and N.G.
especially
F. Harary)
symbols,
field K which
with respect to a given subfield E of K?
The theory which has been developed
impetus
k input
symbols?
What is the number
treatment
of a certain molecule,
OH?
What is the number and m output
of order n?
in order to allow a unified is mainly
de Bruijn.
when it was applied
to enumeration
problems
the work of It received an
(in particular
by
in graph theory during the
last twenty years. This theory originated
in papers written by Cayley and several
chemists,
which contain results
molecules
(cf. Cayley
[1],
on the enumeration
[2] and the references
J.H. Redfield was the first mathematician this subject where a certain polynomial cycle-index
of the permutation
in P61ya
[1]).
who wrote a paper on
is used,
group considered,
main tool of this theory in its present
of organic
form.
the so-called which is the
This paper,
106
published in 1927 and entitled "The theory of group reduced distributions",contains
deep results and is not easy to read.
This may be the reason for that it was overlooked for nearly thirty years.
It was mentioned first by D.E. Littlewood in
his book on group characters
(Littlewood [2]), in the chapter
where Littlewood discusses multiple transitivity of permutation groups. Redfield discusses in his paper the use of symmetric functions for enumeration problems and my goal is now, to point to these results and to shed light upon the connection with the preceding results on representation theory of symmetric groups and wreath products. Hitherto several papers were published where Redfield's results and methods are discussed
(Foulkes [I], [2], Harary/Palmer
[3]). H.0. Foulkes in particular pointed to the close relationship with the representation theory of finite groups. The second paper on that subject was published in 1937 by G. P61ya. Its title is "Kombinatorische Anzahlbestimmungen Graphen und chemische Verbindungen"
fur Gruppen,
(P61ya [1]) and it contains
both a systematic introduction and an excellent and extensive treatment of these enumeration techniques which were initiated by chemists and influenced by Cayley during the nineteenth century. During the fifties of this century more and more of such enumeration problems were attacked. N.G. de Bruijn was the first to write a survey article on that subject (de Bruijn [1]), it was published in 1964 and is still a standard reference. N.G. de Bruijn obtained several generalizatiorsof P61ya's results
[4]).
(cf. de Bruijn
107
This theory has meanwhile been applied by many authors to enumeration problems,
especially in graph theory. A comprehensive
treatment of these methods and results has recently been published by F. Harary and E. Palmer
(Harary/Palmer [I]). Short and
stimulating introductions to this theory were given by Marary as well as by de Bruijn
(Harary/Beineke
[I], de Bruijn [5]). The group
theoretical aspects were pointed out in very interesting but hitherto unpublished notes by A. Rudvalis and E. Snapper (Rudvalis/Snapper [I]). Besides this, various textbooks on combinatorics contain chapters on enumeration under group action (cf. Berge
[I], Liu [I]).
After this short sketch of the history of this theory let us return to our introductory example where we asked for the number of colourings of the regular pentagon by the colours o and Such a colouring of the pentagon may be regarded as a mapping from the set
~I,...,5} of labeled vertices of the pentagon into the set
11,2} of the two colours. Since we shall denote the set of all the mappings from a given domain D into the given range R by R D as usual, the set
~1,2}
11''
" ~ ''51
:=
t,~l~o'b
.....
51
~
I1,2}}
may be regarded as the set of all the 25 = 32 colourings of the pentagon by two colours. It is clear that the solution 8 of the necklace-problem is the number of orbits of the permutation group induced by the symmetry group D 5 of the regular pentagon on the set
11,2} ~I
95~ I
108
The p e r m u t a t i o n
group
of the following
permutation
D5 ~ Hence
~
Burnside's
mutation
group
induced
(~-
~1,2~ ~1 ..... 5} is the image
representation
r
lemma, w h i c h (in terms
by D 5 on
:= ~ o
gives
of D5:
)
the number
of its character)
of orbits
provides
of a per-
a solution
of
our problem. Since sequel
all the e n u m e r a t i o n
w h i c h will be m e n t i o n e d
are in a sense g e n e r a l i z a t i o n s
shall now argument
4.1
theorems
state which
("Lemma
this w e l l - k n o w n allows
and we
generalizations
a permutation
denotes
the number
p ~ P, then the n u m b e r 1 ~T
D peP
shall prove later
we
it by an
on:
of
of P we consider
- -
on the
IX I - d i m e n s i o n a l
points w h i c h
set X and if
remain
fixed under
of P" on X is equal
its natural vector
le I ..... elxll
: p .Autc
on a finite
to
a I (P)"
Instead
w i t h basis
group
of orbits
Proof:
If Wl,...,
result
of Burnside,
of Burnside")
If P denotes al(p)
suitable
of this lemma
in the
(clxl)
~t are the orbits
representation
space V := C Ixl over C
:
:
p
~ (e i
~ ep(&)).
of P on X, then
V i := <ejlj E wi>
109
is an invariant subspace. The salient point is, that V i contains exactly one subspace which affords the identity representation of P. (This holds since P acts transitively on
D
lejlj ~ ~i ~ and the subspace is generated by
ej~
J~i Hence the number of orbits of P on X is equal to the multiplicity of the identity representation in ~. But this multiplicity is equal to the dimension of the direct sum of subspaces of V which afford the identity representation.
Furthermore the vector space V is
projected onto this direct sum by left multiplication with the centrally primitive idempotent
The dimension of a projection is just its trace. Hence we have obtained 4.2
no.of orbits of P = (~P,IP) ~
t~(~(p)) =
at(p). P q.e.d.
The symmetry group D 5 of the regular pentagon consists of 5 cyclic permutations:
I permutation of type (5,0,0,0,0), 4 permutations of type (0,0,0,0,1), and 5 reflections: 5 permutations of type (1,2,0,0,0).
II0 It is clear that a c o l o u r i n g if ~ is c o n s t a n t points where
on each cyclic
of the p e r m u t a t i o n c(w)
a g a i n denotes
Thus @.1 y i e l d s
~ is fixed under
induced
(1 .2 5 + 4"21
W i t h this we have
of d i f f e r e n t
describe Without
generalizations. loss of g e n e r a l i t y
M
We c o n s i d e r
now in a more g e n e r a l
We are g i v e n two finite we may assume
on N, a p e r m u t a t i o n
These
of this
special
lemma. form in order to sets M and N,
say.
that
I ..... nl.
~ from N to M:
{~I~o : N -* M}.
W i t h the aid of two p e r m u t a t i o n
for a d e s c r i p t i o n
of w.
necklaces:
problems
the set M N of all the m a p p i n g s
:=
factors
of B u r n s i d e ' s
:= 11 ..... m I c ~ ~ N ::
MN
of fixed
+ 5 . 2 ~) = 8.
shown how e n u m e r a t i o n
this p r o b l e m
the n u m b e r
11,2111' .... 51 is 2 c(w),
of cyclic
kind can be solved by an a p p l i c a t i o n Let us r e s t a t e
Hence
by w on
the n u m b e r
for the n u m b e r
1
factor.
w ~ D 5 if and only
groups,
group acting
G acting
on M N is d e f i n e d
on M and H a c t i n g and we are a s k e d
of its orbits.
orbits are c a l l e d patterns.
There are three main types
of p a t t e r n s
permutation
is defined:
representation
depending
on the w a y how the
111
4.3 Def.: (i)
G ~ S m, H ~ Sn, define the following permutation representation of G~H: p: G%H * SMN: (f,~) w (r ~ ~, ~(i) := f(i)e(n-1(i))~ p[G%H], the image of this representation is called the exponentiation o~f G and H and denoted by [e] H, i.e. [G] H
(ii)
:= p[g~H].
The subgroup p[diag G*'H'] (recall that diag G*'H' =
~(g,...,g;~)Ig~G ^ ~ H I )
is called the power
o_~f G and H and denoted by GH: G H := p[diag G*'H'] ~ p[G~H] = [G] H (iii)
The subgroup p[H'] = p[l(e;n) I~ ~ HI] is denoted by EH:
E H := p[H'] = IISm [~ = [IISml]HNotice that 4.4
E H < GH < [G] H,
so that enumeration problems concerning G H and E H can be solved by restriction to suitable subgroups once we have solved the corresponding enumeration problem concerning [G] H. Hence we would like to evaluate the number of orbits of [G] H first. In order to do that we first characterize [G] H in terms of representation theory of wreath products (Kerber [8]):
112
If G <_ S m, H <_ Sn, and ~ denotes the natural re-
4.9
presentation
of G, then ~ ~ is a representation
of
G%H with n
H acts on the vector
Proof: n @
6m, on which ~# 9 acts.
space ~ mn which is isomorphic to If I%, .... em} is a basis Of C m,
then {e
:= e~(I) @...| e (n)
is a basis of ~ C m, on which
(f,~) e
= f(1) e ( - 1 ( 1 ) )
|174
= ef(1)q~(.rCl(1))| =
(f;w) e G%H acts as follows:
f(n)e (~-lgn))
ef(n)~(n-l(n))
e
if ~(i) := f(i) r
I <_ i _< n. q.e.d.
This shows that the permutation character of [G] H is equal to the character of 4.6
~:
u (f~w)6 G~H (a1(P(f;~))
= X # ~(f;~)).
We need only apply 2.6 in order to get this character value explicitly:
s 4.7
al(P(f;~))
= X ~ ~(f;~)
c(~) =
~ i=1
a1(gi(f;~)).
113 (This expression can be simplified a bit by using the type notation for the conjugacy classes.) Burnside's lemma together with 4.7 yields now (Kerber [8]): 4.8
("Exponentiation group enumeration theorem, constant form") The number of orbits of the exponentiation
[G] H of G ~ S m and
H } S n on M N is equal to c (.,-,)
1GInlHI ( f~ N)eG-H
a I (gi(f;~)).
i=1
By restricting to the subgroup G H we obtain the following theorem (de Bruijn [2], Harary/Palmer [2]):
4.9
(" Power group enumeration theorem, constant form"): The number of orbits of the power GH of G ~ S m and H ~ S n is equal to n
T ~
Proof:
(g,~)~axH
"=
If the j-th cyclic factor of w is of length r, then
gj(f;.) = g~ if the value of the constant mapping f is g. In this case we have
114 a l ( g j ( f , w ) ) = al(g r) = ~ S'as(g)sir
This together with 4.8 yields
the statement. q.e.d.
Since
al(o(e;~)) = mC(w) we obtain for the subgroup
4.10
The number
of orbits of E H is equal to
1 ~
THT I~H
If we are interested permutation
character
decomposition
mC(W)
in a numerical
of [G] H so that the question arises
depends heavily
tables of wreath products groups mostly
that we notice
(S~nger
of G%H
of Sm~S n
for the
constituents.
of wreath products
[1], Gretschel
first that for the natural
n n # ~ G = @ ~ S m ~ G~H
the
on G but the known character
are tables
G ~ S m and ~S m of S m the following
4.11
example we need evaluate
of # ~ into its irreducible
This decomposition
symmetric
E H (P61ya[1]):
holds:
of
[1], Hilge
[1]),
representations
~G of
so
115 n
Hence we may ask for the decomposition of ~ ~ S m, for which the
following holds: n
4.12
n
# ~ S m = ~ ([m] + [m-1,1]).
We apply 2.59 in order to derive that
4.13
n
n
~([m] + [m-1,1]) =
~
(m;r)~(m-1,1; n-r) ~ Sm~S n
r=o
Hence e.g.
.@.( [ 2. ] +, [ 1 2 ] )
= 3@ [1 2] + ( [ 2 ] ~ [ 1 2 ]@[1 2] | ( [ 1 ] @ [ 2 ] ) ,)
+ ([2]@[2]@[12]
| ([2]@[1])')
f
~. S2 ",..S3
$2',,S 3 + @ [ 2 ]
The characters of these 4 ordinary irreducible representations of $2~S 3 can be found in the character table in S~nger [I]. An addition of their values shows that the resulting permutation S3 character of [$2] has only 3 values ~ 0, these values and the orders and representatives of the corresponding classes in $2~S 3 are:
value
4.14
class order
representative
of class
1
(e; 1)
6
[e; (12))
8
(e; (123))
116
An inspection
for the
rectriction
since (e; I), S3 in S 2 . After
(e;
of the conjugacy classes yields S to $23 the same character values,
(12)) as well as (e; (123)) are contained
a suitable permutation
change
of class orders we obtain for the S3 character of S 2 :
value
representative
class order
(e; 1) 4.15
The restriction
(e;
(12))
(e;
(123))
to the subgroup E $3 yields the table 4.15
again.
The tables transitive,
S3 that [$2] has one orbit only, it is S3 while S 2 has two orbits and E $3 has four of them.
show
Other groups which are of special enumeration
problems
by permutation
groups
are certain on certain
interest
in graphical
subgroups which are induced subsets
of the power
set of
the set of symbols.
4.16 Def. :
Let P denote a finite
a permutation
group which acts on
set X.
(i) If for I ~ k ~
IXI, X (k) denotes the set of
117
subsets of order k of X, then
S(k ) : P--~Sx(k)
: ~(fXl,...,Xk~-~l~(xl),
i s a permutation r e p r e s e n t a t i o n
. . . . ~(Xk)~)
of P (and f a i t h f u l ,
i~
k < IX{). Its image
p(k) :: 6 p] is called
(ii)
the k-sets group o_~fP.
The image of
i--'~ ( (x I .....
8 k : P--@Sxk
x k) ~
( W(X I ), ....
~'(X k) ) ),
i.e. the group
pk :=
k[p ]
is called the k - s e q u e n c ~ group o_~f P. 6 k is faithful.
(iii) The diagonal diagX k of X k is an orbit of 8 k, so that the restriction of 6k[P] to xk\diagX k is a permutation representation if k > I and
IXl > I, it is denoted by
S[k]= S[k ]
:
~-r
8k(W) lxk\diagXk
118 Its image
p[k] :: S[k][p ] is called
In particular
the reduced k-sequences
the k-groups
In terms of representation
of symmetric
group of P.
groups are of interest.
theory they can be described
as
follows:
4.17
(k) is similar Sn
to the permutation
S n on the left cosets
group induced by
of S k • Sn_ k via left multiplication.
(If P ~ S n, then p(k) is similar to the restriction this permutation
representation
of
of S n to P.)
The check is easy. 4.17 implies
4.18
for the permutation
~(k) ~ s representation ~n
character
of these groups:
of S n) has [k][n-k]
as permutation
character.
The irreducible an application which yields
constituents
of [k][n-k]
can be obtained by
of the Littlewood-Richardson-rule
(I. 4.51)
119
4.19
~(k) (as representation of Sn) has the character
~n
k D [n-r,r], if k ~ n/2. r--o
An important example is that
4.20
[n-2,2] + [n-1,1] + [ n ]
is the character of S~2J . t~ n
A typical problem of graphical enumeration is the following one: What is the number of types of graphs
(without multiple
edges and without loops, i.e. "Michigan-graphs")
with n
points?
The n points of such a graph form (~) pairs of points so that the graph itself may be considered as a mapping ~ from this set of (2) pairs of points into a two element set, say
I0,1~,
where ~(i) = 0 means that the pair with number i is disconnected while ~(j) = 1 indicates that the pair with the number j is connected.
Two such graphs ~ and ~ are said to be of the same t ~ e
if
and only if there is a w ~ S n which by application to the n points yields $ from ~, i.e. if and only if there is a E S n such that
120
v I <_ i _< (~) (~(s~)(~-1)(i)) : ~(i)).
Hence the number of types of graphs with n points is equal to the number of orbits of S (2) 11S2 ~ n
4.21
Let us consider the case when n := 3 for an example. Since S~ 2) is similar to S 3 we can use the table 4.15 which shows that E $3 has four orbits so that there are 4 types of graphs with three points:
/ /\
9
9
9
9
9
a
9
Fig. 2
If we consider the types of graphs up to complementation only (i.e. if we allow to substitute disconnected for connected and connected for disconnected),
then we obtain the number of
S~ 2) orbits of S 2 as solution. Hence 4.15 shows that there are just two types of graphs up to complementation which contain exactly four points. It is obvious that these are the following two:
121
(its complementary
(its complementary
graph is
graph is
/
.)
Fig. 3 In the next section we shall return to this problem since then we shall have additional
tools.
Now one may ask for the number of orbits of p(k), where P _< Sn, so that p(k) ~ o~(k) n 9 Livingstone and Wagner gave a theorem on this which turned out to be very useful
(cf. Livingstone/Wagner
the proof (Robinson
[I]), Robinson simplified
[7]). The theorem reads as follows:
4.23 If P ~ S n and 2 ~ k ~ n/2, then p(k) has at least as many orbits as p(k-1) Proof: 4.17 yields that [n-k][k] tation representation
~ P is the character of the permu-
p(k) of P. Hence 4.19 yields for the number of
orbits of p(k):
4.22
IPI-1D_,
"x[n-k][k](")
= IPI-Ic
~P
k ~(n-r,r) E, (~)
~ep r=O
= IpI_ID ~P
k-1 ,S ~(n-r,r) r=O
( " ) + IPI -ID
f(n-k,k)(w)
w~P J
> I P I-1Z ~P
k-1
_>o
r.;(n-r ,r) t~j~ = no. of orbits of p(k-1 )
r=0 q.e.d.
122 While
the 2-sets
group S ( 2 )
the 2 - s e q u e n c e s The f o l l o w i n g k-sequences
4.2a
o f Sn c o u n t s
group $2n of S n counts
obvious
lemma yields
(as r e p r e s e n t a t i o n
k
relations
on n ~ ~oints.
the c h a r a c t e r
of the
of Sn) the c h a r a c t e r
k ~s
of n points,
group:
Skn has
|
graphs
n
=
~
k
(In]
*
[n-1,1])
~; (k) r--o
=
r | [n-1,1]
o
(we p u t
(ko) |
[n-1,1]
:= [ n ] ) .
For k := 2 we o b t a i n the c h a r a c t e r
of S 2 : n
2 |
2 ([n]
+ [n-1,1])
= [n]
+ 2[m-1,1]
+ |
[n-1,1].
2
Since
(0 In-l,1],
4.25
S 2 has e x a c t l y n
This
follows
In])
= ([n-l,1],
eTso d i r e c t l y
~rom the d e f i n i t i o n ._
of S 2 c o n s i s t s n
Thus S~2]~ p o s s e s s e s
4.26
we o b t a i n
n
of the pairs
exactly
of S 2
(i,i),
One of these 9
I < i ~ n. -
one orbit:
S [2] is transitive. n
a.24 yields
a.27
= I,
two orbits
-
two orbits
[n-1,1])
for the c h a r a c t e r
S n[k] has
of s~k]:
(as r e p r e s e n t a t i o n
of S n) the c h a r a c t e r
123 k
(~
Sn ) _ ~s n :
k r ~ (k) | r=2
].
This yiel_ds
4.28
(i) The number of orbits of S k is equal to n
k
r
(rk) (| [n-1,1],
In]).
r--o
(ii) The number of orbits of S [k] is equal to n
k
r
r=2
(k) (~ [n-1,1], In]).
For the number of orbits of pk and pt. kj[I, where P <_ S n, we obtain
k k-1 (@ ~Sn{ P, IP) = ( | ~Sn~ P, ~Sn [ P) k-1
= ( |
~Sn@ P, [n] [ P + [ n - l , 1 ]
k-1 = ( | ~Sn~P
$ P)
k-1 [ n ] ~P) + ( | ~Sn{ P, [ n - l , 1 ] $ P ) i
I
>0 k-1 ( @ ~Sn$ P , I P ) .
~
Hence we h a v e
4.29
(i)
pk h a s a t l e a s t
a s many o r b i t s
as has pk-1
(ii) p[k] has at least as many orbits as has p[k-1]
124 Let us consider
the case n
:= 2 for an example.
We have
2 4.30
2
@ (In]
+ [n-1,1])
= 2,In]
= [n]
+ 3.[n-1,1]
for the last equation
+ 2.[n-1,1]
+ [n-2,2]
+ @ [n-1,1]
+ [n-2,12],
confer M u r n a g h a n
[3] or use a c h a r a c t e r
table.
This yields
e.g. that S 2 has 2 orbits while S~2]F is transitive. n
If k := 3 we obtain
for the number
o = (3)(@ [ n - 1 , 1 ] , [ n ] )
(@ ,~Sn,[n])
of orbits:
I + (3)(@ [ n - 1 , 1 ] ,
In])
3 = I
Since
I + 3
(cf. M u r n a g h a n 3 (| [n-1,1],
we obtain
o + 3
(@
[n-1,1],
[n]).
[3]): 2 [n]) = (@ [n-1,1],
that S 3 has exactly n
has exactly
- I + I
4 of them).
5 orbits
[hi),
(and hence
that S [3] n
125 The orbit of the k-tup!e
(i 1,...,ik) under S kn is obviously
characterized by the indices ~ of the coordinates i
U
of
(il,...,ik) which are equal. Hence there are as many orbits of S k ss there are partitions
of the set
~I
n
,kl ~~176176
I.e "
the "
number of orbits of ~n~k is Just the Bell number B K (cf. Comtet [1]). This yields a characterization of Bell numbers in terms of representation theory:
4.31
k V n >_ k (Bk = (| ~Sn,[n]) =
k r D (k)(| [n-1,1],
In])).
r=o
k (Notice that B k and hence also (| ~Sn,
[n]) is independent of n!)
The first values of B k are (cf. Comtet [I], p. 212):
4.32
kj1 Bk
I
2
3
4
5
2
5
15
52
6
7
203
877
There are many results known about Bell numbers, recursion formulae, generating functions etc.
(cf. Comtet [I]), 4.31 shows how these
may be applied to representation theory and conversely.
We have discussed ~(k) S k and ~[k] a little bit. Besides these ~ ' n ~ three there is a fourth representation of S n which is of some importance in combinatorics.
It is the representation,
the image
of which is induced by S n on the k-tuples of distinct elements of ll,...,n~. Let us denote this group by
126
s n( k )
The following is obvious:
4.33
s(k}n has (as representation of S n) the character
ISn~ k f
Sn = [ n - k ]
t Sn.
Since
4.34
[n-k]
t Sn = I n - k ]
[1]
...
[1],
I
I
k factors
the Littlewood-Richardson
rule yields that each [~], where
a ~ n and a I ~ n-k, oceurs under the irreducible constituents
of
[n-k]
t s n.
Since [n-k]
[1]
...
[1]
=
([n-k][1])
[1]
k = ([n-k+1]
k-1
+ [n-~,l])
[1]
...
L
k-1 = In-k+1]
[1]
...
...
L,
[1]
+ I
k-1
....
[1]
[1]
127
the following holds for p ( k )
the subgroup of S (k) corres-
ponding to P ~ S n :
4.35
p(k) has at least as many orbits as has p(k-1)
There is a famous theorem of Livingstone and Wagner (Livingstone/Wagner
[I], theorem 2) which is very difficult to
prove and which reads as follows:
4.36
If 2 <_ k <_ ~n and P <_ S n and p(k) transitive, a) p(k-1) is transitive
then
(so that P is (k-1)-fold transitive)~
b) if also k > 5, then pfk) is transitive
(so that P is
even k-fold transitive).
In the light of 4.19 we derive from 4.36 the surprising result which is equivalent to theorem 4.36 a) : 4.3?
If P <_ Sn, 2 _< k < ~ , and if
u 0 < r <_ k
(([n-r,r]
$ P,
IP)
= 0),
then even v
a~n
(n-k+1
< aI < n
~
([~3 S P ,
IP)
= o).
4.36 b) can be formulated similarly. We leave it here and shall return to multiply transitivity at the end of the last section.
128
5. E n u m e r a t i o n
of f u n c t i o n s by w e i g h t
In the p r e c e d i n g
s e c t i o n we d i s c u s s e d the e n u m e r a t i o n of the o r b i t s
of E H, G H and [G] H. This done we are in a p o s i t i o n to solve p r o b l e m s like the i n t r o d u c t o r y one w h e r e w e a s k e d for the n u m b e r of n e c k l a c e s with
five b e a d s in two colours.
The s o l u t i o n of this i n t r o d u c t o r y
e x a m p l e is 8 as it is i n d i c a t e d in Fig.
But Fig. Fig.
I y i e l d s m u c h more than this
I shows a r e p r e s e n t a t i v e
1.
s o l u t i o n 8 only.
In fact
for e a c h one of these 8 c l a s s e s of
n e c k l a c e s w h i c h are the o r b i t s of E D5 on the set P a r t of this i n f o r m a t i o n g i v e n by Fig.
~1,2J 11'2'3'4'5j'"
I is how m a n y n e c k l a c e s
there are w h i c h c o n t a i n e x a c t l y 3 b e a d s os c o l o u r o. In fact a somewhat incomplete
d e s c r i p t i o n of Fig.
I c o u l d r e a d as
follows: I necklace with
5 b e a d s of c o l o u r
9 ,
I n e c k l a c e w i t h 4 b e a d s of c o l o u r
,
and I b e a d of c o l o u r
o ,
2 necklace~with
3 b e a d s of c o l o u r
9
and 2 b e a d s of c o l o u r
o
,
2 necklace~with
2 b e a d s of c o l o u r
9
and 3 b e a d s of c o l o u r
o
,
of c o l o u r
9
and 4 b e a d s of c o l o u r
o
,
I n e c k l a c e w i t h 5 b e a d s of c o l o u r
o
I necklace with 1 bead
In o r d e r to e x p r e s s this d e s c r i p t i o n of Fig. of a m a t h e m a t i c a l
I in t e r m s of elemellts
s t r u c t u r e we e x p r e s s the c o l o u r
~ by the v a r i a b l e x
a n d the c o l o u r o by the v a r i a b l e y of the p o l y n o m i a l
r i n g G[x,y].
By i n d i c a t i n g the n u m b e r of b e a d s in a c e r t a i n c o l o u r by the exponent
of the c o r r e s p o n d i n g v a r i a b l e ,
d e s c r i p t i o n of Fig.
1 by the
we may e x p r e s s the p r e c e d i n g
f o l l o w i n g e l e m e n t of ~[x,y]:
129
5.1
x 5 + x4y + 2x3y2 + 2x2y3 + xy4 +
This element problem.
of ~[x,y]
"generates"
the solution of the necklace
The sum of all its coefficients
and the coefficient
is the number
of orbits
of xay b yields the number of orbits which
consist of a beads in colour This shows that it makes generating
y5
9 and b beads in colour
o .
sense to ask for a method to produce
functions,which
describe
the orbits
of [G] H, G H and E H
similarly. It will turn out that a method can be described which yields polynomial,which inventories a polynomial functions
is called the pattern
the patterns)
is again
for M, the range of the describes
the store
I9 ,o }
in fact we shall derive a result which
how 5.1 can be obtained within G[x,y].
function
(x+y for example
of the above example),
(since it
from the store enumerator,which
but a generating
considered
inventory
such a
shows
from x+y by a well defined manipulation
Again we discuss
first in order to get results
the problem with symmetry group
corresponding
[G] H
to G H and E H as
corollaries. Given permutation
groups G ~ Sm, H ~ S n define
We assume that w: M ~ K, the store enumerator, from M into a field K of characteristic orbits of G (w can be suitably w defines a weight w*(~)
5.2
[G] H on M N. is a given function
0 which is constant
chosen for the problem considered).
for each ~ ~ M N by
w*(r := H
w(~(i)).
i~N Since w is constant
on the
on the orbits of G, the following
holds:
130
5.3
w* is constant on the orbits of [G] H, G H, E H.
The check is very easy. Denoting by ~1,...,~r the orbits of [G] H and by
Wi
:= W*(~0),
~ ~
Wi,
the value of w* on ~i' I ~ i ~ r, the expression r
5.4 i~1
w
EK
gives us some information on the orbits of [G] H on M. In fact if for a given enumeration problem concerning w was carefully chosen,
[G] H the store enumerator
5.4 is sometimes the desired solution of the
enumeration problem in question. Hence we would like to evaluate 5.4. In order to do this we consider the representation # 9 of G%H n acts on V := | K m. If K m = <<e I ..... em>>, then (cf. the proof of ~.5 )
{e
:= ee(1) @...|
e (n) I ~ ~ MNI
is a basis of V. # 9 induces a permutation group acting on this basis which is similar to [G] H since
(f,~) e
....
| ef(i)~(~-1(i))
@...
which
131
The subspaces V i which are generated by the orbits, i.e.
V i := <e
I~ e
wi>,
are obviously invariant so that r V =i=elV i
5.5
is a decomposition of V into invariant subspaces.
The representation
afforded by V i contains the identity representation exactly once since [G] H acts transitively on its orbits. Let us denote this subspace of V i which affords the identity representation by Vil. We have (recall the proof of Burnside's lemma): Vil = <
~
e :>,
1
<_ i _< r.
The centrally primitive idempotent =
5.6
(f;~)
1
e :
IGInlH I
J (f;~)~G~H
projects V onto the subspace r
9 vil <_ v i=1 so that its trace is r, the number of orbits. In order to evaluate 5.4 we need a slight change only. Instead of the operator 5.6 we consider the weighted operator
5.7
1 ew := IGInlHI
W o (f;n) = Woe. l (f;~)~G~H
where W is the linear transformation which multiplies each basis vector by the weight of its orbit:
132
5.8
W
: V ~ V : e~ ~ w*(r
G a t h e r i n g up we o b t a i n
5.9
(Lehmann
[I]):
If w: M * K is a f u n c t i o n characteristic and if w* function
0 which
9 e
from M into a field K of
is c o n s t a n t
: M N * K denotes
on the orbits
the c o r r e s p o n d i n g
of G
weight
d e f i n e d by
w*(~) := ~ w(~(i)), ieN then w* is c o n s t a n t the value
on each orbit of [G] H.
of w* on the i-th orbit
the p a t t e r n
inventory satisfies
r wi =
I IGInlHI
where W denotes
the linear t r a n s f o r m a t i o n
We are left w i t h the q u e s t i o n
e
d e f i n e d by
~ w*(~)e
for the trace
of W o (f;~).
an a n s w e r we p r o c e e d as in the p r o o f s
the trace
Putting
U
the following equation:
, t r a c e ( W o(f;~)), (f,~)~G~H
W: n@ K m ~ n| Km:
2.6 w h e r e
I < i < r, t h e n
y
i=I
order to p r o v i d e
of [G] H
If w i denotes
of (f;w) w a s derived.
<w1 Io
:= w(m)
In
of 2.5 and
133
and denoting by ~(g) = (dik(g)) the matrix representation associated with the natural representation ~G on K m with respect to the basis ~eI ..... em~ we have: Wo(f;(1...n))
@ ei~ = W(
~ i diin (f(1))'''dkin_ 1 (f(n))e i @...@ ek) i,...,k
= i ...,k'w(i)dii n(f(1))...w(k)dkin_l(f(n))
el|
e k.
This yields for the trace: tr(W o (f, (1...n))): ]
w(i I )diq in(f(q )).- .w(i2)di2il (f(2))
= tr(U'~(f(1))
9 U'~(f(n))...U'D(f(2)))
= tr(un.~(f(1).f(n)
Since f(1).f(n)...f(2)
... f(2))).
is the cycle product associated with (1...n)
with respect to f~ for a general element we obtain similarly:
5.10
c(w) tr(W o (f,~)) = ~ t r ( U ~=I
k
c(w) ~ -~(g~(f;w)))=~
, k~ ~ , w(j) jEFixg ~( f, w) "
where k~ denotes the length of th~ ~-th cyclic factor w~ of w:
~ = (j~ ~(J~).-.~
k~-1
(j~)),
and where g~(f~w) is the corresponding cycle product:
134
-k~+1
g~(f;~) : f(j~)f(~-1(j~))..,
f(~
(j~)).
This altogether yields the desired enumeration theorem (Lehmann [I]):
5.11
("Exponentiation group enumeration theorem, weighted form") Under the assumption of 5.9 we have for the pattern in~ventory of [G]H:
c(~)
r
w(j) k~
wi :
i=1
iOinlHi
Restricting to GH 5.12
j~Fix(g~(f;~))
(f; ~)~G~H
yields (de Bruijn
L2],
Harary/Palmer [2]):
("Power group enumeration theorem, weighted form") Under the assumption of 5.9 we have for the pattern inventmry of GH: S
n
I wi
:
>
IOl IHI
i=1
#
j
--~vl
/
(g, ~)~OxH k=l jcFix(gk)
A further restriction down to E H yields (P61ya [I]): 5.13
The pattern inventory of E H is t
i=I
n
~H
m
k=1 j=1
(In this case every function w: M * K satisfies the assumption of 5.9 since G :=
llsm~.)
135
The following definition enables us to systematize our approach:
5.14 Def. :
If P is a permutation group acting on a finite set X, i.e. P ~ SX, then the following element of the polynomial ring O[x I ..... Xlxl] is called the cycle-index o f P:
cyccp
:: 147-).' 77-JxtXkakCp .,I.
"'
"
I~P ~:I=
We shall sometimes display the variables, writing
Cyc (P;x 1 . . . . . Xlx I ) instead of
Cyc(P).
Using this notation we define what we understand by P61ya-insertion of the polynomial f(x,y,z,...) Cyc (Plf(x,y,z .... )) :=
= -[~
Cyc (P;f(x,y,z .... ) .....
f(xlXl,ylXl,~Ix!...))
J~i f(xk,yk,zk .... ) ak(p). p~P k=l
This yields a simple expression for 5.13: 5.15
The pattern inventory of E H is obtained from Cyc(H~ by P61ya-insertion of the store enumerator t i.e. it is equal to m
cyc(Hl~,w(j~) r Let us list some examples of cycle-indices of special groups:
136
5.16
(i)
cyo(~1 s I) = x~, m
(ii)
Cyc(C n := ((1...n))) = I ~ . r iln
n/i
(where r denotes the Euler function, i.e. r
:= Ilk ~ ~Ik ~ i ^ (k,i) = I}I) , n
(iii) Cyc(Sn) =
xkak
I._/._(~__) k=1 ak!
ann
n
(I+(-I) a2+a4+''" ) k~_l~ I
(iv) Cyc(An) =
Xk ak , (~-)
a~--~ n
I (n-I)12 ~XlX 2
if n is odd
(V) Cyc(Dn) = 89 Cyc(Cn) + ~(x2n/2
+
2 ~n-2)/2), if n is eve~ x1"x
(vi) The cycle-index of the regular representation RG of a finite group G:
Cyc(Ra)
1 ~----' ~(k)x/GI/k
= ]ZT
, k l ICl
= ~
1 ~xlCl/ll Il
g~G
where ~(k) denotes the number of elements in G with order k.
'
137
5.16
(iv),
(v) yield e.g.
=
so that we obtain for the cycle inventory of E D5 (apply 5.15 to w:
Cyc(D51x+Y)
11,2 l * Q[x,y]
I ~x . : 2 ~ y)"
= ~0 ((x+y)5 + 5(x+y)(x2+y2)2
+ 4(x5 + yS))
= x 5 + x4y + 2x3y2 + 2x2y3 + xy4 + yS.
This is the polynomial the introductory
5.1 which describes Fig. I, the solution of
enumeration problem.
Notice that it is obtained by
P61ya-inserting x+y into the cycle-index of D 5. This process can be generalized by inserting
oi, t = x I +...+ x t ( t an arbitrary number
which may be suitably chosen for the enumeration in question). resulting polynomial
5.17
Grf(P,t)
::
is Redfield's
~roup reduction function:
Cyc(Plol,t). 1 ~ J - ~ ak(p)
= ~
~ PeP k=1
Comparing this with the definition of Schur-function we obtain that 5.18
Grf(Sn, rn) = Inl,
a basic result which shows how S-function enter the theory of enumeration under group action.
The
138
In 5.18 only the special S-functions
In~ occur. The general case
~g} occurs if we consider a generalization
of cycle-index which
is defined as follows:
If P denotes a permutation group acting on a finite set X
5.19 Def.:
and if F is an ordinary representation character
of P with
MF,then the polynomial
Cyc(P,F)
:= p~P
k=1
is called the ~eneralized cycle-index of P with respect t_~o F.
Hence 5.20
Cyc(P) = Cyc(P,I),
if I denotes the identity representation
5.21
~
n ~
la~ = Cyc(Sn,[a]lal, n)
We prove now an interesting
5.22
P ~ Sn
of P. We obtain furthermore:
~
lemma of Foulkes
Cyc(P,F)
= Cyc(Sn,
F } Sn)
Proof: n Cyc(Sn,FP~ S n ) =
I ~, ~,
~Sn(w)~xkk(W)
~S n I
n! ~S n
k=1 ,
IPI IcSn(~)l ~cSn(~)n P
(Foulkes [I]):
k--1
139
n
]-~ > =I
V ;x
xF(p)
p~P'
k(p) = Cyc (P,F).
k=1 q.e.d.
A corollary is
5.23
P ~ Sn
9
Cyc(P) = Cyc(Sn, IPt Sn).
This shows how S-functions can be used in enumeration theory. 5.23
implies for example that Cyc(A n)
=
Cyc(Sn, IAn~S n)
=
Cyc(Sn,[n]+[In])
=
Cyc(Sn,[n]) + Cyc(Sn,[ln]),
so that
Grf(An,n) = Inl + llnl.
140 Applzcations
of this fact will be discussed
return to 5.11,
5.12 and 5.13.
w the trivial weight
w
In the case when we take for
: M
>
K
: m
of orbits
i
>
1 K,
of [G] H, G H
the graph theoretical
this new aspect.
Let us
function
then we obtain the number Let us consider
later.
example
We saw that the number
E H respectively
of section 4 under
of types of graphs with
2) n points
is equal to the number of orbits
of E
, 5.13 yields
now :
5.24
The number
cyc(s
of types of graphs with n points
is equal to
2) 12) =
We saw furthermore
that the number of types of graphs with n
points up to complementation
is equal to the number
of orbits
S (2) of s2n . Hence by 5.12 it is equal to
I
@ ~Sn(2)
~- (Cyc(S
ak( )
k=l
ak( ) ~Sn(2)
; 2,2 .... ) + Cyc(S
Since this consideration
k=1 21k
0,2,0,2,...))
up to complementation
divides the types
into pairs of types and the self-complementary
graphs; we obtain
furthermore
the well-known
result:
141 5.25 The number of self-complementary
graphs with
n
points
is equal to
Cyc(S~2);
O, 2, O, 2 .... )
For twice the number of types of graphs up to complementation
is
equal to the number of types of graphs plus the number of types of self-complementary
graphs
(with
n
points in each case),
so that we need only apply 5.24.
If for example n := 3, then we obtain for the number of selfcomplementary
graphs with three points:
Cyc(S~2);
0,2,0) = Cyc(S3;
0,2,0)
= ~I (x ~ + 3XlX2 + 2X3)xl =x3=O , x 2 =2 = O.
Hence there is no such graph as Fig. 2 shows as well. A formula for the number of self-complementary can be found in Wille
[I].
m-placed relations
142
6. Some c y c l e - i n d i c e s
I w o u l d like first to discuss how c y c l e - i n d i c e s in some
special
can be e v a l u a t e d
cases.
Let us first c o n s i d e r
how the c y c l e - i n d e x
g r o u p s can be e v a l u a t e d
of c e r t a i n p r o d u c t s
once the c y c l e - i n d i c e s
of
of the factors
are known. 6.1 Def.:
If P, Q denote p e r m u t a t i o n
groups
on finite
sets X, Y,
w h e r e X R Y = ~, then we call (i) the direct
sum P 9 Q o_~f P and Q the p e r m u t a t i o n
with underlying
group
set P x Q w h i c h acts on X 0 Y as
follows:
p(i),
i~ i ~x
q(i),
if i e Y
(p,q)(•
I
(ii) the direct p r o d u c t
P | Q o f P and Q the p e r m u t a t i o n
group w i t h u n d e r l y i n g
set P x Q w h i c h
acts on
X • Y as follows: (p,q)(i,k)
It is obvious
6.2
:= (p(i),q(k)).
that the f o l l o w i n g
Cy c ( P e Q) = C y c ( P )
In order to d e t e r m i n e
holds:
9 Cyc(Q).
the c y c l e - i n d e x
we r e m a r k that an i-cycle
of the direct p r o d u c t
of p e P and a k - c y c l e
P | Q
of q e Q t o g e t h e r
143
yield just gcd(i,k) cycles of length lcm(i,k).
Ixl,IYl 6.3
Cyc(P e Q ) :
I
>p
gcd(i,k)ai(P)ak(q) x
; ,q
This yields
i,k=1
lcm(i,k)
9
Other products of permutation groups which are of importance combinatorics
are e.g. the groups E H, G H and [G] H which were
mentioned in the preceeding presentation
in
sections and the permutation re-
~[G~H S which was introduced in section 1, it is usually
denoted HKG] in combinatorics:
Let us call this group the composition of G and H. Combinatorists called it the wreath product since they considered wreath products of permutation groups mainly and ~[G~H] then is the natural representation
of G~H in a sense.
Besides these two permutation representations
HKG] and [G] H
of the wreath product G~H of G ~ S m and H ~ Sn there is a third one which occurs as symmetry group of combinatorial
structures.
In order to define this group we consider the set W of all the nxm-matrices
in which each row contains all the elements of
b,...,ml. We notice that
IWI = m! n, so that the
permutation representation
144
6.5
c :Sm~S n * Sw
:
(f,~) ((aik) ~* (f(i)a _1(i),k))
is of dimension m! n. An easy check shows that O[Sm~Sn] is similar to the permutation group induced by @ RS m (RS m the regular representation of sm) on the natural basis of the representation space. Let us denote the subgroup o[G%H] by [G]H: 6.6
[G]H := g[G%H].
An application of 2.6 yields for its permutation character:
c(~)
=I ! c(~), if 2.1;kalk(f;~) = c(w)
6.7 a1(o(f;~)):--~i xR(g~(f;~)) =
, elsewhere
In order to define a further interesting permutation representation we introduce an equivalence relation on W. Two elements (aik), (bik) of W are called equivalent if an only if (bik) arises from (aik) by a permutation of the columns of (aik). In other words: (aik)~bik)
:<=~ ~p e S m ((aik) = (bip(k))).
Denoting the class of (aik) under "~" by [(aik)]~ , we obtain from a a further permutation representation of Sm~Sn:
6.8
T:Sm~S n ~ Sw/~ : (f,~) ~([(aik)]
~ [~(f,~)(aik)]
).
In terms of representation theory it can be characterized as follows:
145
6.9
~[Sm~Sn] is similar to the permutation group induced by (the left multiplications of elements of) Sm~S n on the left cosets of diagSm*S n'
The check is easy. It implies for the character:
6.10
The permutation character of m[Sm~Sn] is the character of I diagSm*Sn' t Sm~Sn, i.e. Sm~S n al(~(f;w)) = mvn_ I IC (f,~) D diagSm*S n'l 9
S
IC m
The subgroup
~S
n(f;~)i
w[G~H] (for G ~ S m, H ~ Sn) of
=[Sm~Sn] was intro-
duced by E.M. Palmer and R.W. Robinson (Palmer/Robinson [1],[2]), denoted by [H:G] and called the matrix ~roup of G and H: 6.11 6.10
[H;G] := ~[G%H] ~[Sm~Sn]. implies
6.12 The permutation character of the matrix group [H:G] is the character of I diagSm*S n' ~ Sm~S n ~ G~H.
Having obtained the permutation characters of H[G], [G] H, [G]H and [H;G] we ask for the
cycle-indices.
In order to evaluate these
we consider the equation which holds for the elements al(P) of the cycle type (al(P) ..... alXl(p)) of an element p e P acting on a finite set X:
146
6.13
Vr e N (a1(pr) =s~r Sas(P))"
An application of a Moebius-inversion to this equation yields
I ~,~( as(P) = ~ rls
6.14
s ~) a1(pr).
This together with the preceding results on the permutation characters yields the desired cycle-indices:
6.15
("Cycle indices of H[G], [a]H, [G]H, [H;G]") (i)
The cycle-index of the composition H[G] is equal to m.n
IGInlHI
(e;'rr)~C-~
~(f;~))
=
where ak(~(f;~)) = ~ E ~ ( k ) i Ik
al(f(j)) jeFix(~)
(ii) The cycle-index of the exponentiation [G]H is equal to mn
k (p(f;~))
1
lalnlHl (f;~)~a'.Hk=1 where
ilk
~-~=I
al(g~(f...f i_ I;~i))
147
(iii) The cycle-index of [G] H is equal to m ,n 9
iGInlHI(f,~)~G~H
ak(a(f,~))
=
where
6
c(~i),~alj(f...f
ilk
i_ 1;~i)
(iv) The cycle-index of [H;G] is equal to n-1 ~
lalnlHl
m~
ak(x(f'w))
Xk (f, ~)~G~H
where S ~S 1 ~-~ ,k, ,n-1 IC m n((f,~)i) n diagSm*S n' I ak(T(f,w)) : [ / , W < I ;m" i Ik icSm~Sn((f;~)i)l
There are other expressions obtainable for the cycle-indices of H[G] (see P61ya Eli ) and [G] H (see Palmer/Robinson
[1]) which are
easier to handle since they express the cycle-index of these products in terms of the cycle-indices
of the factors. But l
preferred to stress the fact that a knowledge of the permutation character is sufficient. The method how we obtained the cycle-index of [H;G] points to a more general situation,
if Q is a subgroup of a finite group P,
then P induces a finite permutation group on the left cosets
148
of Q in P. Denoting this permutation group by P/Q, we obtain
6.16
("Exterior cycle-index") Cyc (P/Q) =
P:Q
1
ak(P)
I~P
where I ak(P) := ~ ~ .
k ~(~)
IP:QI
IcP(p~) n QI
i Ik
ICP(pJ I
This generalizes a result of de Bruijn (de Bruijn [3]). It holds since the permutation character of P/Q is the character IQ ~ P of
P~and xIQtP(P) =
IP:QI IcP(~ n QI IcP(p) I
as it is well known. Further expressions for the cycle-indices
of H[G], [G] H, [G] H and
[H;G] can be obtained by an application of 5.22 which says that Cyc(P) = Cyc(Sn, IP t Sn). Hence e.g. 6.fl7 Cyc (Sn[Sm]) = Cyc(Smn,(m;n ) ~ Smn ) : Cyc(Smn,[m]Q[n]) m/q
x[m] (D [n] ( ~ ) V x k k ( ~ ) = ~
~Smn
k=fl
149
This implies that
Grf(Sn[Sm])
= Cyc(Sn[Sm]l~l,mn)
= ~
I
~---~x[m]|
-~- ak(~)
~)ll~k,mn
~Smn
,
k=1
In the same way we obtain:
6.18 Cyc (Sn[Sm] ,(~; 13)I~i" 1 ,mn ) =
i_!~_~, ~-~,X[ ~]0[ 13](~)H~rk,mn ~ ak(~) ~S
k=1
mn
Hence tables concerning decompositions of symmetrized o~ter products [a]G[~]
can be used in order ot obtain generalized
cycle indices of compositions Sn[Sm]. E.g.
[2]0[/*]
= [8] + [ 6 , 2 ]
+ [42 ] + [422 ] + [24 ]
This implies
Grf(S4[S2])
= i8t + 16,2t + 142t + 1422t + 124t
For the more general case 6.18 a (given) decomposition of [~]Q[6] yields Cyc(Sn[Sm] , (a;~)l@1,mn) as a sum of s-functions
[u
~mn.
This sum is uniquely determined since the function Cyc(Sn[Sm],
(~;p)I@1,mn) is symmetric and homogeneous os weight mn
and the S-functions
~},~mn,
form a basis for the vector space
of all these symmetric functions over @. This sum of symmetric functions is denoted by la~O 16~:
6.19
V aNm,B~n
(~lel~
:= Cyc(SnESm],(a;6)l@1,mn)).
In fact it is the so-called oster plethysm
which Littlewood
introduced (Littlewood [5]) in terms of the theory oi invariant
150
matrices. o_~f ~al
Let us call
and
I~I (~)18} the symmetrized outer product
I~.
A special case of 6.19 is
6.20
~m~ C)~n~ = Cyc(Sn[Sm]IC l~mn) = Grf(Sn[Sm],mn ).
This equation is basic for the application enumeration theory
of S-functions
(cf. Read [I]-[5] Foulkes
[I]-[3]).
in
(It should
be mentioned that conversely the enumeration theory can be applied in order to obtain results on S-functions
Important applications superposition
of graphs
(cf. P.A. Morris
[I]).)
are the enumeration problems concerning the (already Redfield's paper deals with this
subject). An example will be given in the next section.
Having evaluated the cycle-index polynomial
for several permutation
groups one may ask how much of the group structure by the cycle-index polynomial. structure,
for nonisomorphic
same cycle-index. number
p
is reflected
It does not reflect the complete
permutation groups may have the
P~lya mentions,
that for each odd prime
and each natural number m > 2 there is a non-
abelian group of order pm, each nonidentity of order p. Hence its regular representation index
I V
(x~ m +
pm-1 (pm-1 )Xp
.
element of which is has the cycle-
151 The regular representation of the abelian group Cp~...• (m factors, Cp := ((1...p))) obviously has the same cycleindex. Hence the cycle-index reflects only a part of the group structure. In a sense it reflects only the average cycle-structure of the elements. In this connection one may put the question what can be said about the behaviour of the cycle-structure
of a group element
g E G under various permutation representations and Hales have considered this problem us consider their results.
of G. Golomb
(Golomb/Hales [1]), let
They started with the following
definitions:
6.21 Def.: Let
G
Then gl
be a group and gl,g 2 be two elements of G. and g2
are called strongly enumeratively
equivalent if an only if their images have the same set of fixed points for every permutation representation of G, and they are called enumeratively equivalent if the numbers of fixed points of ~(gl)
and
~(g2) are equal for all permutation
representations
T
of
G.
Their first results show how these two concepts can be characterized from the group-theoretical point of view:
6.22
If
G
(i) gl
is a f i n i t e group and gl,g2~G, and
g2
then
are strongly enumeratively equivalent
152
if and only if they generate the same cyclic subgroup of
G, and they are
(ii) enumeratively equivalent if and only if they generate conjugate cyclic subgroups of
G.
Proof: (i)
a) Suppose
(gl)
= (g2>
permutation representation of
and that
T: O ~ S X is a
G. If x e X is fixed under
~(gl ), then x = r(gl)-l(~(gl)(x))
= r(gl-1)(x). Hence
x
is fixed under each power of T(gl) , in particular under T(g2). b) If , say
g2 4
tation representation r := l(gl)# G, i.e. the permuation representation of symbol
G
on the left cosets of (gl).
The
(gl > is fixed under ~(gl ) , but it is not fixed under
~(g2)" (ii)
a) If (g2 > = h(gl)h -1, say
is fixed under gl' then h(x) al(g 1) ~ al(g2), since tain
h
g2 = hg~ h-l' and if x e X
is fixed under g2' so that
is a bijection. We similarly ob-
al(g2) ~ al(gl)-
b) If al(~(gl)) = al(T(g2)), for each permutation representation of G, we consider T := I ~ G. Since (g1> is fixed under ~(g!), there is (since a1(T(gl))=al(w(g2))
a point fixed under T(g2) , say
the point h(gl>. It satisfies h-lg2 h ~ (gl), so that (g2 > ~ h(gl)h -I. Analogously we obtain (gl) ~ h'(g2>h '-1, so that the finiteness of the order IGI of G yields the contradiction:
153
-_~g2 ~ = h(gl)h -I for a suitable h ~ G.
q.e.d. Since two group elements
generate
the same cyclic
if and only if each is a power of the other, call such elements questions
relatives,
in which cases any two conjugates
(i)
If
G
is a group,
are relatives
are conjugates
are relatives.
(i) of the following
tioned that B. Fein had pointed
6.23
Golomb and Hales
the results 6.22 raise the
in which cases any two relatives
Hales obtained part
subgroup
out,
that
and
Golomb and
theorem and men(ii) holds:
then any two conjugates
in
G
if and only if each subgroup
of
G
is normal. (ii)
If
G
is a finite group,
then any two relatives
are conjugates
if and only if all the ordinary
characters
G
of
are rational-valued.
Proof: (i)
a) Assume
that conjugates
h ~ H ~ G ~ g. Then relative
of
ghg -I
h, so that
are relatives
is conjugate
to
and that h
and hence a
(ghg -I) = (h) ~_ H. This implies
H~G.
b) If
G is a group and g1'g2 ~ G are conjugates,
(gl) is normal:
(gl) = (g2)
so that
gl
and
g2
then since are relatives.
154
(ii) There
is a t h e o r e m
the c h a r a c t e r s
of a finite
and only if for each g
and
g
t
(cf. Serre
g ~ G and
are conjugates.
each o r d i n a r y
character
G o l o m b and H a l e s
the n e c e s s a r y
If G ~
(i)
G
12.5)
t ~ 2
Hence
saying that all
are r a t i o n a l - v a l u e d with
(t,
if r e l a t i v e s
has r a t i o n a l
values
if
l(g) l) = I
are conjugates,
only and vice versa.
called a group G to be of class
only if any two r e l a t i v e s
6.24
group
[I],
in G are conjugates.
~
if and
They o b t a i n e d
conditions
~
1 <
, then
IGI < - = 2 I I G I , a n d
(ii) Z(G)
has e x p o n e n t
G/[G,G]
h a s exponent 2,
2.
Proof:
(i) If g ~ G\IqGI,
then g and g-1 are r e l a t i v e s
jugates
since G ~ 3.
lIG,gl,
so that,
Then, if g = g-1 , G c o n t a i n s
since
there is an h E G w h i c h
IG I is finite, satisfies
IGI
a n d hence conthe
subgroup
is even. And if g @ g-1
g = h g - l h -1 , so that
tion by h" is an inner a u t o m o r p h i s m
of e v e n order.
"conjuga-
Hence
IG I is
even in both cases. Furthermore g
2
[g,h]
= g h g - l h -I = g2 so that for each g ~ G
is in [G,G].
(ii) If g ~ Z(G) relatives
then all its c o n j u g a t e s
are equal to
g
itself.
Hence
and hence g = g
-1
all its
, i.e.
g
q.e.d.
2
= 1.
155
G o l o m b and H a l e s m e n t i o n quaternion
that the symmetric
g r o u p of order 8 are of class ~
g e t h e r w i t h G and H, the g r o u p s In the light of the p r e c e d i n g products
and t h e o r e m
by the f o l l o w i n g
6.25
on c h a r a c t e r s these
JR .
of w r e a t h
results
theorem:
conditions
G is a symmetric
(ii)
G is the q u a t e r n i o n
6.23
GxH and G~S 2 are of class
results
(i)
(iv)
, and that to-
(ii) we c o m p l e t e
Each of the f o l l o w i n g
(iii)
Since
6.23
group and the
is s u f f i c i e n t
for G e ~ :
group, group of order
8,
G ~ H x I, H and I E ~ , G ~ H~I, w h e r e H,
I E
is a p e r m u t a t i o n
group
such that I n S
e
G
~
~
and I
of finite , for all
(ii) h o l d s we n e e d only m e n t i o n
group of order
8
has the same c h a r a c t e r
degree
n
~ ~ n.
that the q u a t e r n i o n table as
$2~S 2.
156
7. The construction of patterns
The preceding sections on enumeration under group action were devoted to (i)
the enumeration of the number of patterns which is given by an application of Burnside's lemma to the permutation character
(enumeration theorem,
constant form),
(ii) the evaluation of cycle-indices which yield (by P61ya-insertion of store enumerator) a generating function for the problem with symmetry group E H, (ii~ the enumeration theorem in weighted form which enumerates functions
~ E M N by weight with respect to E H, GH, [G] H.
We should not leave this subject without saying a word on how these patterns of functions are not only enumerated but even constructed. I.e. we would like to know how to construct a representative
for
each orbit of the symmetry group in question. There is in fact a method available which can be used at least for the enumeration Rroblems concerning patterns of functions with respect to E H,
H <_ S N
~ ~ MN
(see Ruch/Hasselbarth/Richter
[1], Brown/Hjelmeland/Masinter
[1], Brown/Masinter
[1]).
In the case when E H, for a given H ~ Sn, is the symmetry group we can use first the enumeration theorem
5.15 in the weighted
form, in order to obtain the number of patterns of functions ~ M N, the elements of which have say r I values equal to 1 (e M), r 2 values equal to 2 (e M),
...,r m values equal to m (e M), so that
157
r i ~ Z> o' ~. ri = n. -
Let us assume
1
that there are in fact p a t t e r n s
We ask for the exact n u m b e r we w o u l d
Instead
like to c o n s t r u c t
of the m a p p i n g s
corresponding follows. which
elements
If i I ~
are m a p p e d
~
of this
a representative
~ ~ MN
~ E SN = S n
onto I ~ M =
:=
~
,
special
type and
of each of them.
of this type we may c o n s i d e r , where
... ~ it1 are the e l e m e n t s
(i~)
If J l
of p a t t e r n s
of this type.
~ is d e f i n e d as of N =
ll,...,n 1
~I .... ,m}, then we put
1 <
~ < r 1.
"'" ~ Jr~ are the e l e m e n t s
of N w h i c h are m a p p e d onto
2 ~ M, then we put
(j~)
In terms
of these
as follows
type
elements
(recall
I ~ i ~ m):
:= r I + ~, 1 _< ~ _< r 2.
that b o t h
~ ~ S n the e q u i v a l e n c e ~ and ~ have r i v a l u e s
~ and ~ ~ M N are e l e m e n t s
( r l , . . . , r m) if and only if there
( or S r 1 @
... ~ S r
if you like) m = ~o
~ o
of ~, ~ reads
-1
and a
equal
of the same p a t t e r n is a
c ~ St1
w E H
which
i, (with
x ... x Srm satisfy
158
In other words:
7.1
~ and ~ e M N of type of the same pattern the same double
SrlX
of Srl
(rl,...,r m) are r e p r e s e n t a t i v e s if and only if ~ and ~ belong
to
coset
...
Sr
•
x ... x S r
~ H
m
and H in S nm
Let us illustrate
that by the i n t r o d u c t o r y
We ask for a c o n s t r u c t i o n three
of w h i c h
Evaluation
exist.
Srl
5.1, where
in fact two n e c k l a c e s
~s of r e p r e s e n t a t i v e s
9 ..
patterns we have cosets
considerations
x
have
of type
(rl,...,rm).
to evaluate
a system
problem. five beads
two of colour
of type
(rl,r 2) = (3,2)
shown that a complete
of the double system
In case
system
cosets ~'
"'''
of the necklace
of r e p r e s e n t a t i o n s
Cs
~1,(12),(13),(23),(123),(132)}
of the double
x
I1,(45)}
and D5 =
11, ( 1 2 3 4 5 ) , ( 1 3 5 2 4 ) , ( 1 4 2 5 3 ) , ( 1 5 4 3 2 ) , (25)(34),
in S 5 .
(13)(45),
(24)(15),
of the
problem
of
S 3 x S2 =
o~
2 is the c o e f f i c i e n t
how they can be constructed.
~ H yield a complete
Srn
with
and the remaining
We are left w i t h the question
..... x
9
of C y c ( D 5 1 x + y ) yields
The p r e c e d i n g ~I
of all the necklace
are of colour
of x3y 2 so that there
necklace
(12)(35),
(14)(23)I
159
It t u r n s out t h a t [ 1 , ( 1 4 5 3 2 ) ] i s
~1
=
such a system,
1 2
3 4
5)
1 2
3 4
5
1 2
3 4
5)
4
2
3
i.e. we
and
$2
=
1
5
The c o r r e s p o n d i n g m a p p i n g s are
1
el:
~
O
2~
9
2 '-"';~ 9
3~e
9
3~-~
9
4~
o
4 ~-~
O
5 ~
O
The n e c k l a c e s are
and
obtain
160
It is not very difficult complete
to make a computer produce
system of representatives
of double
degree n of the group H is relatively this a Todd-Coxeter of representatives
small.
algorithm which produces of left cosets
cosets
One may use for a complete
for higher degrees
found in the papers Brown/Hjelmeland/Masinter[1]
elucidation
applications
Srl • ... • S r
can be
and Brown/Masinter structure
of double
of pattern
cosets
~ H in S . The preceding sections have shown that n of patterns is also equal to the coefficient of m
in Cyc(Hlel,m)
The number
= Grf(H,m).
of double cosets
coefficient
This yields:
SrlX'''XSrm ~ H
in S n is the
rI rm of x I ... x m in Grf(H,m).
In other words:
Grf(H,m)
= Cyc(Hle I m ) is the generating t
function
Recall
7.3
of type
rm
x I ... x m
7.2
to chemical
It shows that the number
(rl,...,r m) is equal to the number
rI
system
are discussed.
Let us return to 7.1.
this number
if the
of H ~ r s t .
Tricky methods which can be applied
[I] where also special
such a
for double cosets Srl • ... • Srm ~ H i n S
that 5.22 yields
.Cyc(HI~It m ) ,
:
~; ~Pn
= Cyc(S n, IH I' S n
(IH ~ S n, [~]). I~I.
l(~1~m)
n"
161
A generalization
of this is discussed
on group characters
in chapter
in Littlewood's
IX ("Structure
book
of groups"),
section 5 ( T r a n s l t i v i t y " ) .
If both G and H denote
subgroups
of S n, then the number
cosets G w H in S n is called the transitive
of double
factor of G and H
and denoted by
N(G,H).
Littlewood
7.4
then shows that the following
N(G,H)
=
)-~
(IG~ S
n'
This result has interesting functions
of multiply
corollaries
holds:
[ a ] ) ' ( I H t S n, [a]).
corollaries
transitive
let us consider,
concerning
groups.
how N(G,H)
group reduction
Before we discuss
these
fits into the preceding
considerations.
This consideration cf. also Foulkes
leads to Redfield's
paper
[I], [2], Harary/Palmer
to the enumeration
(Redfield
[3]).
[I],
It is devoted
of what he calls group-reduced
distributions.
The starting point is again the set W of n x m - matrices A = (aik),
each row of which contains
M := ~I ..... m I (cf. section 6). Hence
IWI = m! n.
the m elements
of
162
Redfield
Every
calls
column
the set M the range.
of an element
between
n elements
we call
two elements
n-tuple
of correspondence4
7.5
I.e.
each of which
(aik) ~ (bik)
belongs
of W equivalent
:r
We notice
lwl~l Such a class,
equivalence
[A]~
=
= (b
i,~
-I
(k)
))"
from B by a p e r m u t a t i o n
of
m! n-1
of W/~,
subgroups
[B]~
the same
that
relation
~
to the range M. Hence
if they constitute
~ ~ E S m ((aik)
an element
If we are given
a correspondence
we define
A ~ B if and only if A arises
columns.
7.6
A e W constitutes
is called
G I , . . . , G n of Sm,
a range-correspondence.
then they
define
an
on W/~ as follows:
:<~W
I ~ i ~ n ~Oi ~ Gi
([(aik)]~
= [(Pibik )]~)"
We ask for the n u m b e r
I(w/~)/~l of equivalence
classes.
The classes
distributions.
Considering
the
are c a l l e d g r o u p - r e d u c e d
special
element
163
1 ... m 1
A O :=
....... I ... m
~ W,
we see that for each A ~ W there are uniquely ~i ~ S n w h i c h
determined
satisfy
A = (A
(k))=:
( A ....
, ~n A) Ao "
7.6 then reads as follows:
7.7
[A]~~
[BI~(==~ ~ Pi ~ G i '
~
Sm,((~ . . . .
'~nA) = ( P i ~
a . . . . . . P n ~ a))
Hence
[A]~ ~ [B]~~zz~ ~ ~
S m ( ( ~ ..... ~n) g
This shows that Redfields
group-reduced
a sense the orbits of the p e r m u t a t i o n
G1~...XGn~ne
distributions
).
are in
group induced by S m on
the cartesian product
x Sm/G i i of the sets Sm/G i of right cosets of the G i in S m. The permutation
group has in fact the character n
@ (I G i ~ Sm} ~
i=I
so that in terms of r e p r e s e n t a t i o n
theory we obtain forthe
164 desired
number
of g r o u p - r e d u c e d
distributions:
l(wl~~_- i = (~(1oir Sm)~ [rn]).
7.8
1
If in p a r t i c u l a r
n
:= 2, then this number
2 (~(IGi~ Sm)~ [m]) I which
is the number
theorem
or prove
a sense
just a p e r m u t a t i o n
groups
G I w G 2 in S m (use M a c k e y ' s
that an element
of W/~ is in
of S m if n = 2). Hence
This can be applied
7.8 yields
to m u l t i p l y
transitive
GI .
lemma:
G ~ S m is k - f o l d i.e.
7.8 yields
7.10
co~s
it by noticing
We use the following
7.9
to
= (IG I ~ S m, IG 2 ~ Sm),
of double
7.4 as a corollary.
is equal
transitive
if and only if Sm_ k. G = S m,
if and only if N(Sm_k,G)
= I.
now as a corollary:
G ~ S m is k - f o l d
transitive
if and only if
I = ([m-k]t S m, IG ~ Sn~).
The b r a n c h i n g the identity
rule
(I 4.52)
representation
and 7.10
imply
[m] exactly
(since
I G ~ S m contains
once as does
[m-k] ~ Sm):
165
7.11
G < S -
is k - f o l d
V ~m
In terms
7.12
Since
(Littlewood
reduction
u a~m
< m~
(m-k ~ ~
=
but not
order to evaluate
Let us r e t u r n
5.17,
5.23)
(Am) :
~m I +
transitive
(m-1)-fold group
if and only if
lal is not a summand
Im I and Grf
that S m is m - f o l d
transitive
(apply
[2]):
transitive
Grf(Sm)
[a]) = 0).
function
G ~ S m is k - f o l d
yields
if and only if
(m-k ~ c I < m , @ ( I G ~ S m,
of the group
we obtain
transitive
m
to 7.8 and discuss
~Iml, this also
and that A m is (m-2)-fold
transitive.
reduction
of Grf(G)).
7.12 can be used in
functions.
some of its applications.
7.8 can be used in order to solve p r o b l e m s
concerning
superposition
that the number
different
of graphs.
superpositions
with automorphismen
Redfield
observed
of graphs F I .... ,Fn
groups
G1,
71
:" T2
with m points
..., G n is just 7.8.
If for example
:= T3
:=
so that
G I = G 2 = G 3 = $2[S 2]
the of and
166
we obtain for the number
of superpositions:
3 3 ( ~ ( l G i f $4) , [ 4 ] ) = ( ~ ( ( 2 ; 2 ) t $ 4 } , [ 4 ] ) 1
= (~([2]e[2]),[4])
= (~ ([4] + [22]),
[4])
= 5
(use the character table of S 4 and the decomposition of [ 2 ] Q [ 2 ] ) . These 5 superpositions are indicated below:
i',
l~.
\
.i ".L
........"
f~1
i'j
!.j
<~_>
Fig. 4
References (in addition to the references in Vol. I) Benson,C.T./ Curtis, C.~:
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[1] Principles of combinatorics. Academic Press, New York, 1971.
Brown,H./ [I] Constructive graph labeling using double cosets Hjelmeland,L./ Discrete Math. ~ (1974), 1-30. Masinter,L.: Brown,H./ Masinter,L.:
[I] An Algorithm for the construction of the graphs of organic molecules. Discrete Math. ~ (1974), 227-244.
deBruijn,N.~: [I] P61ya's theory of counting. Applied Combinatorial Mathematics (E.F. Beckenbach, ed.), 144-184. Wiley, New York, 1964. [2] Generalization of P61ya's fundamental theorem in enumerative combinatorial analysis. Nederl. Akad. Wetensch. Proc. Ser. A 62 (1959), 59-69. [3] The exterior cycle index of a permutation group. Studies in Pure Math. (Presented to Richard Rado). 31-37. Academic Press 1971. [4] A survey of generalizations of P61ya's enumeration theorem. Nieuw Arch. Wiskunde 19 (1970), 89-112. [5] P~lya's Abz~hl-Theorie: Muster fGr Graphen und chemische Verbindungen. Selecta Math., ed. K. Jacobs, vol. III, 1-26. Springer-Verlag 1971. Butler, P.H.: [I] Coupling coefficients and tensor operators for chains of groups. Trans. Roy. Soc. London (to appear) Butler, P.H./King, R.C.: [I] Branching rules for U(N)oU(M) and the evaluation of outer plethysms. J. Math. Phys. 1~4 (1973), 741-745. [2] The symmetric group: Characters, products and plethysms. J. Math. Phys. 1__4 (1973), 1176-1183.
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[3] Symmetrized Kronecker products of representations. Can. J. Math. 26 (1974), 328-339.
Carter,R.K:
[I] Conjugacy classes in the Weyl group. Seminar on Algebraic Groups and Related Finite Groups. The Institute for Advanced Study, Princeton, N.J., 1968/69, pp. 297-318. Lecture Notes in Math. vol. 131, Springer, Berlin 1970. [2] Conjugacy classes in the Weyl group. Compositio Math. 25 (1972), 1-59.
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[I] On the mathematical theory of isomers. Philos. Magazine 47 (1874), 444-446. [2] On the analytical forms called trees, with application to the theory of chemical combination~ Report of British Association for the Advancement of Science, 1,,875,, 257-305
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[1] Zur Darstellungstheorie gewisser Verallgemeinerung e n d e r Serien yon Weyl-Gruppen (in preparation)
Clausen,M.
[I] Zentralisatorenverb~nde von Moduln ~ber halbeinfachen Gruppenalgebren; zur Theorie der Symmetrieklassen in Tensorr~umen. Diplomarbeit, Gie~en I~74.
Comtet, L.: [I] Advanced Combinatorics. D. Reidel Publishing Compan~ 1974. Derome, J.-R.: [I] Symmetry properties of the 30-symbols for an arbitrary group. J. Math. Phys. Z (1966), 612-61
Dieudonn@,J.A./ [I] Invariant theory, old and new. Academic Carrell, J.~: Press, New York, 1971.. Esper,~:
[I] Ein interaktives Programmsystem zur Erzeugung der rationalisierten Charakterentafel einer endlichen Gruppe. Staatsexamensarbeit, Aachen 1974.
169
[2] Tables of reductions of symmetrized inner products ("inner plethysms") of ordinary irreducible representations of symmetric groups. (to appear) Felt, W.:
[I] Characters of finite groups. W.A. Benjamin Inc., 1967.
Foulkes,H.~:
[I] On Redfield's group reduction functions. Canad. Math. 15 (1963), 272-284. [2] On Redfield's range-correspondences. Canad. J. Math. 18 (1966), 1060-1071. [3] Linear graphs and Schur-functions. Conference on Comb. Math., Oxford 1969.
Frobenius,F.~:[1] Ober die Charaktere der symmetrischen Gruppe. Sitzgsber. Preu~. Akad. Wiss. 1900, 516-534. Golomb,S.W./ Hales,A.W.:
[I] On Enumerative Equivalence of Group Elements. J. Comb. Theory ~ (1968), 308-312
Gretschel,B.:
[I] Berechnung der Charakt~rentafeln yon Symmetrien symmetrischer Gruppen. Diplomarbeit Gie~en, 1973.
Harary,F./ Beineke,L.:
[I] A seminar on graph theory. Holt, Rinehart and Winston, New York, 1967
Harary,F./
[I] Graphical enumeration. Academic Press,
Palmer,E.:
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Hilge,A.:
[I] Berechnung der Charakterentafeln von Symmetrien symmetrischer Gruppen. Diplomarbeit Gie~en, 1973 .
Humphreys,J.E.:
[I] Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics ~, Springer-Verlag, Berlin 1972, xii + 169 pp.
170
Kerber,A.:
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Klaiber,B.:
[I] Fortsetzbarkeit und Korrespondenz yon Darstellungen. Dissertation, Mainz 1969 .
King,R.C.:
[I] Branching rules for GL(N) O ~ m and the evaluation of inner plethysms. J. Math. Phys. I_~5 (1974), 258-267.
Knutson, D.:
[I] k-Rings and the representation theory of the symmetric group. Lecture Notes in Math., vol. 308 Springer-Verlag 1973.
Kostant,B.:
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Lehmann,W.:
Littlewood,D.E.: [4] Plethysm and the inner product of S-functions. J. London Math. Soc. 32 (1957), 18-22. [5] Polynomial concomitants and invariant matrices. J.London Math. Soc. 11 (1936), 49-55 Livingstone, D./Wagner, A.: [I] Transitivity of finite permutation groups on unordered sets. Math. Z. 90 (1965), 393-405. Liu,C.L.:
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Mayer,S.J.:
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Morris,P.A.:
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173
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Subject
basis
Index
5
group , diagonal
Bell
of
lemma
of
of
groups
contragredient
generalized
cycle-index
138
108
graph,
of
119
type
cycle-index
basis
-
5
154
-s,
composition
143
-s,
direct
product
-s,
direct
sum
96
-s,
exponentiation
represen-
tation
135
-
, k-sequences
- -
, exterior
148
-
, k-sets
- -
, generalized
138
-s,
cycle
product
9
-
direct -
of
basis
product
sum
of
of
group groups
groups
theorem
matrix
of
exponentiation constant
constant
-
, reduced
-
, simple
142
-s,
-
Weyl
- of -
form
equivalent
, strongly group
form)
-
(weighted
form)
of
the
of
-
145 -
99 154
k-sequences phase
-
-
118 99 8
distribution
162
function
137
reduction
inner
symmetrized
product
69
k-sets
group
i17
group
117
134 151
111
113
lemma
of
Burnside
matrix
i08
group
145
cycle-index
104
necklace-problem non-simple-phase
134
kind,
117 117
group
group
representation
99 99
148 operator,
first
111
theorem
(constant
exterior
142
151
, enumeration
, - - -
-
-
-
k-sequences
, weighted
exponentiation
of
142
134
113
enumeratively -
form
group,
form
. . . . .
-
class
6
113
, weighted
power
of
group,
form
. . . . .
143
-
group
142
of
of
-
-
group-reduced enumeration
of
, non-simple-phase
-s, diagonal
9O
125
group, class composition
formula
6
number
Burnside,
Frobenius'
outer
representation 96
weighted
-
plethysm
symmetrized
131 149
product
65
175
pattern
110
- inventory plethysm, Pblya
power -
of -
129
outer
149
insertion
135
polynomial
identity 9
91
group
111
- 9 enumeration
theorem
(constant
form)
- ,
(weighted
problem 9
necklace
product,
cycle
-)
-
(of S - f u n c t i o n s )
range
group
relatives
-
- - second - - third ,
-
kind
-
kind,
118
99
phase
99
89
representation 96
S-functions
89 group
99
- - representation standard store
wreath
96
of the
simple-phase
polynomial
99 identity
enumerator
strongly
91 129
enumeratively
equivalent
factor
(f;~)
150
of
96 161 i0 27
weighted
65 150
96
-
Schur-functions second
of
Weyl
96
non-simple-phase
, simple
transitive
69
96 first
65
representation
- - a graph
contragre-
dient - of the
kind 9
- - F*
153
representation 9
third
134
162
k-sequences
69
, - (of S - f u n c t i o n s )
113
162
correspondence
reduced
inner
9 outer - -
type
9 inner
9 - outer
-
product 9
119
104
-
, symmetrized
, - -
symmetrized
151
operator
groups product
8
5
Vol. 342: Algebraic K-Theory II, "Classical" Algebraic K-Theory, and Connections with Arithmetic. Edited by H. Bass. XV, 527 pages. 1973. Vol. 343: Algebraic K-Theory III, Hermitian K-Theory and Geometric Applications. Edited by H. Bass. XV, 572 pages. 1973. Vol. 344: A. S. Troelstra (Editor), Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. XVlI, 465 pages. 1973. Vol. 345: Proceedings of a Conference on Operator Theory. Edited by P. A. Fillmore. Vl, 228 pages. 1973.
Vol. 371 : V. Poenaru, Analyse Differentielle. V, 228 pages. 1974. Vol. 372: Proceedings of the Second International Conference on the Theory of Groups 1973. Edited by M. F. Newman. VII, 740 pages. 1974. Vol 373: A. E. R. Woodcock and T Poston, A Geometrical Study of the Elementary Catastrophes V, 257 pages. 1974. Vol. 374: S. Yamamuro, Differential Calculus in Topological Linear Spaces. IV, 179 pages. 1974.
Vol. 346: Fu~ik et al., Spectral Analysis of Nonlinear OperatorS. II, 287 pages. 1973.
Vol. 375: Topology Conference. Edited by R. F. Dickman Jr. and P. Fletcher. X, 283 pages 1974.
Vol. 347: J. M. Boardman and R. M. Vogt, Homotopy Invariant Algebraic Structures on Topological Spaces. X, 257 pages. 1973.
Vol. 376: I. J. Good and D. B. Osteyee, Information, Weight of Evidence. The Singularity between Probability Measures and Signal Detection. XI, 156 pages. 1974.
Vol. 348: A. M. Mathai and R. K. Saxena, Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. VII, 314 pages. 1973.
Vol. 377: A. M. Fink, Almost Periodic Differential Equations. VIII, 336 pages. 1974.
Vol. 349: Modular Functions of One Variable II. Edited by W. Kuyk and P. Deligne. V, 598 pages. 1973. Vol. 350: Modular Functions of One Variable III. Edited by W. Kuyk and J.-P. Serre. V, 350 pages. 1973. Vol. 351 : H. Tachikawa, Quasi-Frobenius Rings and Generalizations. Xl, 172 pages. 1973. Vol. 352: J. D. Fay, Theta Functions on Riemann Surfaces. V, 137 pages. 1973. Voi. 353: Proceedings of the Conference.on Orders, Group Rings and Related Topics. Organized by J. S. Hsia, M. L. Madan and T. G. Ralley. X, 224 pages. 1973. Vol. 354: K. J. Devlin, Aspects of Constructibility. XlI, 240 pages. 1973. Vol. 355: M. Sion, A Theory of Semigroup Valued Measures. V, 140 pages. 1973. Vol. 356: W. L. J. van der Kallen, Infinitesimally Central Extensions of Chevalley Groups. VII, 147 pages. 1973. Vol. 357: W. Borho, P. Gabriel und R. Rentschler, Primideale in EinhtJllenden auflesbarer Lie-Algebren. V, 182 Seiten. 1973. Vol. 358: F. L. Williams, Tensor Products of Principal Series Representations. Vl, 132 pages. 1973. Vol. 359: U. Stammbach, Homology in Group Theory. VIII, 183 pages. 1973. Vol. 360: W. J. Padgetl and R. L. Taylor, Laws of Large Numbers for Normed Linear Spaces and Certain Frechet Spaces. VI, 111 pages. 1973.
Vol. 3 7 8 TOPO 72 - General Topology and its Apphcations. Proceedings 1972. Edited by R. A. AIo, R. W. Heath and J. Nagata. XIV, 651 pages. 1974. Vol. 379: A. Badrikian et S. Chevet, Mesures Cylindriques, Espaces de Wiener et Fonctions Aleatoires Gaussiennes. X, 383 pages. 1974. Vol. 380: M. Petrich, Rings and Semigroups. VIII, 182 pages. 1974. Vol. 381 : Seminaire de Probabdites Vllt. Edite par P. A. Meyer. IX, 354 pages. 1974. Vol. 382: J. H. van Lint, Combinatorial Theory Seminar Eindhoven University of Technology. Vl, 131 pages. 1974. Vol. 383: Seminaire Bourbaki - vol. 1972/73. Exposes 418-435. IV. 334 pages. 1974. Vol. 384: Functional Analysis and Applications, Proceedings 1972. Edited by L. Nachbin. V, 2?0 pages. 1974. Vol. 385: J. Douglas Jr. and T. Dupont, Collocation Methods for Parabolic Equations in a Single Space Variable (Based on C L Piecewise-Polynomial Spaces). V, 147 pages. 1974. Vol. 386: J. Tits, Buildings of Spherical Type and Finite BNPairs. X, 299 pages. 1974. Vol. 387: C. P. Bruter, Elements de la Theorle des Matroldes. V, 138 pages. 1974. Vol. 388: R. L. Lipsman, Group Representations. X, 166 pages. 1974. Vol. 389: M.-A. Knus et M. Ojanguren, Theorie de la Descente et Algebres d' Azumaya. IV, 163 pages. 1974.
Vol. 361 : J. W. Schutz, Foundations of Special Relativity: Kinematic Axioms for Minkowski Space-Time. XX, 314 pages. 1973.
Vol. 390: P. A. Meyer, P. Pnouret et F. Spitzer, Ecole d'Ete de Probabddes de Saint-Flour III - 1973. Edite par A. Badrikian et P.-L Hennequin. VIII. 189 pages.'1974.
Vol. 362: Proceedings of the Conference on Numerical Solution of Ordinary Differential Equations. Edited by D.G. Bettis. VIII, 490 pages. 1974.
Vol. 391 : J. W. Gray, Formal Category Theory: Adjointness for 2Categones. Xlt, 282pages 1974.
Vol. 363: Conference on the Numerical Solution of Differentiat Equations. Edited by G. A. Watson. IX, 221 pages. 1974. Vol. 364: Proceedings on Infinite Dimensional Holomorphy. Edited by T. L. Hayden and T. J. Suffridge. VII, 212 pages. 1974. Vol. 365: R. P. Gilbert, Constructive Methods for Elliptic Equations. VII, 397 pages. 1974. Vol. 366: R. Steinberg, Conjugacy Classes in Algebraic Groups (Notes by V. V. Deodhar). Vl, 159 pages. 1974. Vol. 367: K. Langmann und W. LiJtkebohmert, Cousinverteilungen und Fortsetzungss,~itze. Vl, 151 Seiten. 1974. Vol. 368: R. J. Milgram, Unstable Homotopy from the Stable Point of View. V, 109 pages. 1974. Vol. 369: Victoria Symposium on Nonstandard Analysis. Edited by A. Hurd and P. Loeb. XVlII, 339 pages. 1974. Vol. 370: B. Mazur and W. Messing, Universal Extensions and One Dimensional Crystalline Cohomology. VII, 134 pages. 1974.
Vol. 392: Geometne Differenhelle, Colloque, Santiago de Compostela, Espagne 1972. Edite par E. Vidal. Vl, 225 pages. 1974. Vol. 393: G Wassermann, Stability of Unfoldings. IX, 164 pages. 1974. Vol. 394: W. M. Patterson, 3rd, Iterative Methods for the Solution of a Linear Operator Equation in Hdbert Space - A Survey. III, 183 pages. 1974. Vol. 395: Numerische Behandlung nichtlinearer Integrodifferential- und Differentialgleichungen. Tagung 1973. Herausgegeben yon R. Ansorge und W. Ternig. VII, 313 Seiten. 1974. Vol. 396: K. H. Hofmann, M. Mislove and A. Stralka, The Pontryagin Duality of Compact O-Dimensional Semilattices and its Applications. XVl, 122 pages. 1974. Vol. 397: T. Yamada, The Schur Subgroup of the Brauer Group. V, 159 pages. 1974. Vol. 398: Theories de I'lnformation, Actes des Rencontres de MarseiIle-Luminy, 1973. Edite par J. Kampe de Feriet et C.-F. Picard. XlI, 201 pages. 1974.
Vol. 399: Functional Analysis and its Applications. Proceedings 1973, Edited by H. G. Garnir, K. R. Unni and J. H. Williamson. II, 584 pages. 1974.
Vol. 429: L. Cohn, Analytic Theory of the Harish-Chandra C-Function. ]11,154 pages. 19"74.
Vol. 400: A Crash Course on Kleinian Groups. Proceedings 1974. Edited by L. Bers and I. Kra. Wl, 130 pages. 1974.
Vol. 430: Constructive and Computational Methods for Differen tial and Integral Equations. Proceedings 1974. Edited by D. L. Colton and R. P. Gilbert. VII, 476 pages. 1974.
Vol, 401: M. F. Atiyah, Elliptic Operators and Compact Groups. V, 93 pages. 1974.
Vol. 431 : Semicaire Bourbaki - vol. 1973/14. Exposes 436-452. IV, 347 pages. 1975.
VoW. 402: M. Waldsct~midt, Nombres Traescendants. Viii, 271 pages, 1974.
VoI. 432: R. P. Pflu9, Holomorphiegebiete, pseudokonvexe Gebiete und das Levi-Problem. Vl, 210 Seiten. 1975.
Vol. 403: Combinatorial Mathematics. Proceedings 1972. Edited by D, A, Holton. VIII, 148 pages, 1974.
Vol. 433: W. G. Faris, Self-Adjoint Operators. VII, 115 pages. 1975. Vol. 434: P. Brenner, V. Thomee, and L. B. Wahlbin, Besov Spaces and Applications to Difference Methods for Initial Value Problems. II, 154 pages. 1975.
Vol. 404: Theorie du Potentiel et Analyse Harmomque. Edite par J. Faraut. V, 245 pages. 1924. Vol. 405: K. J. Devlin and H Johnsbr&ten, The Souslin Problem. VIII, 132 pages. 1974. Vol. 406: Graphs and Combinatorics. Proceedings 1973. Edited by R. A. Bari and F. Harary. VIII, 355 pages. 1974. Vol. 407: P. 8erthelot, Cohomologie Cristalline des Schemas de Caracteristique p > o. II, 604 pages. 1914. Vol. 408: J. Wermer, Potential Theory. VIII, 146 pages. 1974 Vol. 409: Fonctions de Plusieurs Variables Complexes, Semmaire Frangois Norguet 1970-1923. XIII, 612 pages. 1974. Vol. 410: Seminaire Pierre Lelong (Analyse) Annee 1972-1973 Vl, 181 pages. 1974. VoJ. 411: Hypergraph Seminar. Ohio State University, 1972 Edited by C. Berge and D, Ray-ChaudhuP. IX, 28? pages. 1974 Vol. 412: Classihcation of Algebraic Varieties and Compact Complex Manifolds. Proceedings 1974. Edited by H. Popp. V, 333 pages. 191'4.
Vol. 435: C. F. Duekl and D. E. Ramirez, Representations of Commutative Semitopological Semigroups. Vl, 181 pages. 1975. Vol. 436 : L. Auslanderand R, Tolimieri, Abelian HarmonicAnalysis, Theta Functions and Function Algebras on a Nilmanifold. V, 99 pages. 19'75. VoL 437: O. W. Masser, Elliptic Functions and Transcendence. XlV, 143 pages. 1915. Vol. 438: Geometric Topology. Proceedings 1974. Edited by L. C. Glaser and T. B. Rushing. X, 459 pages. 1975. Vol. 439: K. Ueno, Classification Theory of Algebraic Varieties and Compact Complex Spaces. XIX, 278 pages. 1975 Vol. 440: R. K. Getoor, Markov Processes: Ray Processes and Right Processes. V, 118 pages. 1975.
Vol. 413: M. 8runeau, Variation Totale d'une Fonctlon. XIV, 332 pages. 1974.
Vol. 441: N Jacobson, PI-Algebras. An Introduction. V, 115 pages, 1915 VoL 442: C. H. Wilcox, Scattering Theory for the d'Alembert Equation in Exterior Domain& III, 184 pages.,,1975.
Vol. 414: T. Kambayashi, M. Miyanishi and M. Takeuchi, Uni potent Algebraic Groups. VI. 165 pages 1974
Vol. 443: M. Lazard, Commutative Formal Groups. II, 236 pages. 1975.
VoL 415: Ordinary and Partial Differential Equations. Proceedings 1974. XVlI, 447 pages. 1974. Vol. 416: M. E. Taylor, Pseudo Differential Operators. IV, 155 pages. 1974. Vol. 417: H. H. Keller, Differential Calculus in Locally Convex Spaces. XVl, 131 pages. 1914. Vol 418: Localization in Group Theory and Hemotopy Theory and Related Topics. Batte}le Seattle 1974 Seminar. Edited by P. J. Hilton, Vl, 172 pages 1974. Vol. 419: Topics in Analysis. Proceedings 1970. Edited by O. E. Lehto, I. S. Louhivaara, and R. H. Nevanlinna. XIII, 392 pages. 1974. Vol, 420: Category Seminar. Proceedings 1972/73. Edited by G. M Kelly. Vl, 375 pages. 1974.
Vol. 444: F. van Oystaeyen, Prime Spl~ctra in Non-Commutative Algebra. V, 128 pages. 1975. VoI. 445: Model Theory and Topoi. Edited by F. W. Lawvere, C. Maurer, and G. C Wraith. Ill, 354 pages. 1975. Vol. 448: Partial Differential Equations and Related Topics, Proceedings 1974, Edited by J. A. Goldstein. IV, 389 pages. 1915. Vol. 447: S. Toledo, Tableau Systems for First Order Number Theory and Certain Higher Order Theories. Ill, 339 pages. 1975. Vol. 448: Spectral Theory and Differential Equations. Proceedings 1974 Edited by W. N. Everitt. XlI, 321 pages. 1975. Vol. 449: Hyperfunctions and Theoretical Physics. Proceedings 1973, Edited by F. Pham. IV, 218 pages. 1915.
VoI 421: V. Peenaru, Groupes Discrets. VI, 216 pages. 1974
Vol. 450: Algebra and Logic. Proceedings 1974. Edited by J. N, Crossley. VIII, 307 pages. 1975.
Vol. 422: J.-M. Lemalre, Algebres Connexes et Homologie cles Espaces de I~acets. XlV, 133 pages. 1974.
Vol. 451: Probabilistic Methods in Differential Equations. Proceedings 1974. Edited by M. A. Pinsky. VII, 162 pages. 1915.
VoI. 423: S. S. Abhyankar and A. M. Sathaye, Geometric Theory of Algebraic Space Curves. XlV, 302 pages. 1974.
Vol. 452: Combinatorial Mathematics III. Proceedings 1974. Edited by Anne Penfold Street and W. D. Wallis. IX, 233 pages. 1975.
Vol. 424: L. Weiss and J. Wolfowdz, Maximum Probability Estimators and Related Topics. V, 106 pages. 1974. Vol. 425: P. R. Chernoli and J. E. Marsden, Properties of Inhn~te Dimensional Ham~ltoman Systems. IV, 160 pages. 1974. Vol. 426: M. L. Silverstein, Symmetric Markov Processes. X, 287 pages. 1974. Vol. 422: H. Omori, Infinite Dimensional Lie Transformation Groups. XlI, 149 pages, 1974. Vol. 428: Algebraic and Geometrical Methods in Topology, Proceedings 1973. Edited by L F. McAuley. XI, 280 pages. 1974
Vol. 453 : Logic Colloquium. Symposium on Logic Held at Boston, 1972-73. Edited by R. Parikh. IV, 251 pages. 1975, Vel. 454: J'. Hirschfeld and W. H. Wheeler, Forcing, Arithmetic, Division Rings. VII, 266 pages. 1975, Vol. 455: H. Kraft, Kommutative algebraische Gruppen und Ringe. III, 163 Seiten. 1975. Vol. 456: R. M. Fossum, P. A. Griffith, and I. Reiten, Trivial Extensions of Abelian Categories. Homological Algebra of Trivial Extensions of Abelian Categories with Applications to Ring Theory. Xi, 122 pages. 1975.
