James-Kerber--Representation Theory of Symmetric Groups

*

CHAPTER 2

Ordinary Irreducible Representations and Characters of Symmetric and Alternating Groups The present chapter contains a characterization of the ordinary irreducible representations of symmetric groups as common constituents of monomial representations induced from Young subgroups. Besides this, the ordinary irreducible characters are given as .l-linear combinations of permutation characters induced from Young subgroups, and an important recursion formula is derived. As an application, the ordinary irreducible characters and representations of An are given. It is shown furthermore, that Sn is characterized by its character table. An examination of the values of the irreducible characters leads to the notions of cores and quotients of partitions which later on are of great importance for modular purposes. Finally, the Littlewood-Richardson rule, which is one of the most useful results of the whole theory, is derived.

2.1

The Ordinary Irreducible Representations of Sn

By an ordinary representation of a group G we mean a finite-dimensional representation of G over the field C of complex numbers. It will turn out that each ordinary representation of Sn can be realized over the rational field, so that Q (and thus every field) is a splitting field for Sn' Hence it does not make much difference whether we assume that the groundfield F is of a characteristic which does not divide n! or we just consider the ordinary representations of Sn' ENCYCLOPEDIA OF MATHEMATICS and Its Applications, Gian-Carlo Rota (ed.). Vol. 16: G. D. James and A. Kerber. The Representation Theory of the Symmetric Group ISBN 0-201-13515-9 Copyright © 1981 by Addison-Wesley Publishing Company, Inc.. Advanced Book Program. All rights reserved. No part of this publication may be reproduced. stored in a retrieval system. or transmitted. in any form or by any means. electronic, mechanical. photocopying, recording. or otherwise. without the prior permission of the publisher. 34

2. \ The Ordinary Irreducible Representations of Sn

35

We know from general representation theory of finite groups that Sn has the same number of ordinary irreducible representations as conjugacy classes. 1.2.6 has shown that a conjugacy class of Sn is characterized by the cycle partition y~n of its elements. For each such partition y of n we constructed Young subgroups Sy and considered their trivial one-dimensional representations ISy ' the identity representation of Sy' and ASy ' the alternating representation of Sy' In Section 1.3 we expressed the intertwining number

a, f3~n, of the induced representations in terms of numbers of double cosets

and in terms of numbers of 0-1 matrices with prescribed row and column sums. The particular intertwining number

for example, is equal to the number of 0-1 matrices with row sums a, and column sums a~. But it is not difficult to see that there is exactly one 0-1 matrix which satisfies these conditions on its row and column sums, namely the matrix 2.1.1

(

~

1

: :.: : : : : : .1 . 10

~ : : : : : :~ ) 0

where the 1's are placed in each row as far to the left as possible. This yields 2.1.2

A careful interpretation of these equations yields several important results. First of all we remember that these equations hold whenever the groundfield F is of characteristic,,", 2 (cf. 1.3.7). Therefore let us interpret 2.1.2 first as a result on representations of Sn over the field a. Since the intertwining number i( " . ) is "bilinear" and has all its values in No, 2.1.2 implies that there is exactly one irreducible a-representation of Sn' say D, which is contained in both ISa t Sn and ASa. t Sn' Furthermore i(D, D)= 1, and D is contained in both ISa tSn and A Sa' tSn with multiplicity 1. Since i(D, D)= 1, D must be absolutely irreducible, so that it is also an irreducible C-representation of Sn' Altogether this yields the following basic result: 2.1.3 THEOREM. If a is a partition of n and if Sa and Sa' are Young subgroups of Sn' which correspond to a and ai, then the induced representations ISa tSn and A Sa' t Sn have exactly one ordinary irreducible constituent in common.

Ordinary Irreducible Representations and Characters 2.1

36

Furthermore this irreducible constituent can be realized over 0, and it is contained with multiplicity 1 in both IS" i Sn and AS". is,,.

As any two Young subgroups of SrI which correspond to the same partition a' ~ n are conjugate subgroups, the characters of IS" is,, and AS", i Sn depend only on the partition a of the integer n and not on the dissections of the set n which yield S" and S"" Hence the same must be true for the common irreducible constituent mentioned in Theorem 2.1.3; it depends only on a. We therefore denote this uniquely determined constituent, and also its equivalence class of representations, by [a], since there is no danger of confusing this with the corresponding Young diagram. Abusing the symbol n, we have 2.104

Two special cases are easy to identify, namely the identity representation ISn of Sn and the alternating representation ASn of Sn' We denoted by (I") the partition (I, ... , 1) of n. Since the only c.orresponding Young subgroup is S(I") = { 1}, both I S(I") i Sn and A SO") i SrI are equal to the regular representation RSn of Sn: 2.1.5

Furthermore from

S(,,)

=Sn it follows that

2.1.6

Thus 2.1.4 implies 2.1.7

It is important to perceive the close connection between [a] and [a ' ]. Since

and

(use the standard argument for characters J.L of G and v of H:;;;;; G that J.L0v i G=(J.L t H0v)i G), we have for each a~n: 2.1.8

This means that the ordinary irreducible representation [a' ] differs from [a] only on the odd permutations, and there just by the sign.

37

2. I The Ordinary Irreducible Representations of Sn

The set {[a]la~n}

2.1.9

is a subset of the set of equivalence classes of ordinary irreducible representations of Sn' We would like to show that 2.1.9 is already the complete set of these equivalence classes. A preliminary consideration shows 2.1.10 LEMMA. The multiplicity i(IS" iSn ,[,8]) of [,8] in IS" iSn is nonzero only if a~,8.

Proof 2.104 shows that i(ASw i Sn' [,8]) = 1. The assumption that i(IS" i SII' [,8])*0 yields therefore i(IS" i SII' ASf3 , 1SII )*0. Hence there exist 0-1 matrices with row sums a I and column sums p} f)I, which implies a ~,8. • Notice that in this proof we did only use the trivial part of the Gale-Ryser theorem, as we promised at the end of Section 104. We are now in a position to prove the main theorem of this section. 2.1.11 THEOREM. {[a]la~n} is the complete set of equivalence classes of ordinary irreducible representations of SII'

Proof We need only show that [a] = [,8] implies a =,8, for then the cardinality of {[a]la~n} is equal to the number of conjugacy classes of SII' so that this system must be complete. But if [a]=[,8], we can argue in the following way:

Thus by 2.1.10 we have both done.

a~,8

and

,8~a,

which imply a=,8, and we are •

Hence each ordinary irreducible representation of SII is realizable over C, so that C is a splitting field for Sn' This yields (apply (83.7) in Curtis and Reiner [1962]): 2.1.12

THEOREM.

Each field is a splitting field for Sn'

This is not true for alternating groups; A 3 provides a counterexample. Let us conclude this section with a numerical example. By 2.1.11, we know that [3], [2,1], and [13] are all the equivalence classes of ordinary irreducible representations of S3' 2.1.7 yields

Ordinary Irreducible Representations and Characters 2.2

38

Since the squares of the dimensions add up to 6, [2,1] has dimension 2. Since 0:: = (2,1) is self-associated, 2.1.8 yields that the character of [2,1] vanishes on the class of transpositions. This gives the character table of S3 up to one remaining entry: (2,1 )

-1

(3 )

a

*

1)

[I' J [2, I]

1

1

[3 J

The remaining entry turns out to be - 1 by the orthogonality relations, say. Hence the complete character table of S3 is (I')

(2. I)

-1

(3)

1)

a

-1

1

1

[! 'I [2,11

[3]

But we are far from being able to evaluate the character table for every n, so that we need to examine more closely the permutation characters induced by Young subgroups. Before we start doing this we should not forget to mention a result which follows from 1.1.26, 2.1.7, and the well known form of the composition series of symmetric groups: 2.1.13 THEOREM. All the ordinary irreducible representations of Sn are faithful except [n] (for n;;;'2), W] (for n;;;.3), and [2 2 ] (for n=4).

2.2

The Permutation Characters Induced by Young Subgroups

In the preceding section we constructed with the aid of the representations ISa i Sn a complete system of ordinary irreducible representations of Sn' The former are transitive permutation representations, and we would like to examine them more thoroughly. In order to do this, we put the p( n): = 1P( n)1 proper partitions o:i of n in lexicographic order: 2.2.1 and define a matrix 2.2.2 with the foHowing multiplicities as entries: 2.2.3

39

:.2 The Permutation Characters Induced by Young Subgroups

°

2.1.3 together with 1.4.8 and 2.1.1 shows that M n is an upper triangular matrix with l's along the main diagonal:

2.2.4

*

M= n

o

This has several important consequences. We denote by 2.2.5

the value of the character cycle partition f3, and by

~a

of [a] on the conjugacy class of elements with

2.2.6 we denote the value of the character

~a

of ISa i Sn on this same class. Then

2.2.3 yields

2.2.7 If we put the values

~;

and

~p

into matrices, say l~i,k~p(n),

2.2.8

then Zn is the character table of Sn' and En satisfies (cf. 2.2.7)

so that we obtain from det Mn *0 (cf. 2.2.4): 2.2.9 M n is a matrix over No and has determinant l; hence M n-

1

is a matrix over Z, so that by 2.2.9 each ordinary irreducible character ~a of Sn is a Z-linear combination of the permutation characters As there are exactly p( n) such permutation characters, this Z-linear combination is uniquely determined. Thus the following holds:

e.

2.2.10 THEOREM. The ring char(Sn)= EBfin/z~a' of generalized ordinary characters of Sn possesses besides its Z-basis {~alaf-n} the Z-basis aa/af-n}, which consists of characters of transitive permutation representations of Sn'

40

Ordinary Irreducible Representations and Characters 2.2

If we want to express fa in terms of this basis of permutation characters, we need to know M n- 1 (or M n ). It should be mentioned that M n can be evaluated using enumerations of double cosets. For M n is triangular and has l's along its main diagonal, while the scalar product of the ith and the jth row of M n satisfies (apply 1.3.5)

2. m ik m jk =i(ISa

2.2.11

i

iSn , ISa) iSn )

k

=

2.

1.

This number of double cosets can be found as a certain coefficient in II(1-x i y)-1 (d. 1.3.14). A numerical example illustrates this: M 3 is of the

form (notice (13)«2, 1)«3»

* 1

o

* *

and we know

2.

(IS(3)iS 3,IS(2,I)iS3 )=

1=1,

53,,5(2.1)

2.

( I S(3) i S3' I S(l J) i S3 ) =

1 = 1,

53 "5(13)

so that

*

1

o

)

Furthermore, (IS(2, I) i S3 ,IS(I) i S3) =

2.

1= IS3: S(2, \)1 = 3,

512 ,11"5(1'1

so that finally

M3 =

(6

2 1

o o

)

41

2.2 The Permutation Characters Induced by Young Subgroups

We have obtained from 2.2.11 and 1.3.14 the following result: 2.2.12 THEOREM. The matrix M n can be evaluated from the coefficients of the generating function IIi. k( 1- Xi Yk ) - I.

Since 1S(I") i Sn = RSn (d. 2.1.5), M n contains mensions

In

its first row the di-

2.2.13 of the ordinary irreducible representations [a] of Sn' And as i(IS(n) i Sn, 1Sa iSn)=i(ISn' 1Sa iSn)= 1, it contains in its last column only l's, so that we have

2.2.14

M= n

*

o

Furthermore the scalar product of the first row of M n with its ith row is just the index ISn: Sa'l, l";;;;i";;;;p(n). But the knowledge of M n alone does not suffice to evaluate Zn' We also need :::n' the matrix of the permutation characters. Since 1Sa i Sn is the permutation representation of Sn on the left cosets of Sa it is in principle possible to evaluate ~a( 17) by checking which left cosets of Sa remain fixed under left multiplication by 17. If we want to do this, it makes life easier to visualize the left cosets by so-called tabloids, which may be introduced as equivalence classes of tableaux. We call two tableaux t and t' (with the same diagram) row equivalent if and only if t' arises from t by a horizontal permutation: 2.2.15

t~t'

:=

317EH(t) (t'=17t).

The equivalence class of t is denoted by 2.2.16

{t} .

In other words: {t} arises from t by neglecting the order of points in the rows. We shall indicate this by drawing lines between the rows of t. It is obvious that the tabloids of shape [a] are in one-to-one correspondence with the left cosets of Sa and admit an action of Sn which is equivalent to the left

42

Ordinary Irreducible Representations and Characters 2.2

multiplication of Sn on the cosets of Sa' For example the (2, I)-tableaux are 1 3

2

1 2

2 3

3

2

3

3 2

1

3

2

1

while the tabloids with diagram [2, 1] are 1 2

3

3

2

3

2

I s3 leaves each one of them fixed, (12) leaves just the first one fixed, and (123) has no fixed tabloid. This enables us to evaluate ~(z, I): ~(z, 1)(1) = 3, ~(Z,I)«(12»= 1, ~(Z,I)«(123»=0. Thus we obtain for :::3' since

IS(13)

IS3 =RS3

and IS(3) IS3 =IS3, '=' _

-3-

(63 o1 1

1

We now obtain 2 3 in the following way:

-2

-1

o

1

o

1

-: )

A further result on M n can be derived from 2.1.10. For this lemma says that m ik ~O only if exi~exk. This means for the characters

2.2.17

~a =

r

a

+

~ (~a,

[P ] W~

f3r>a

and yields a lot of zeros in the matrix M n , n;,.6. We would like to derive a result, which in a sense reverses 2.2.17, and which will turn out to allow a representation theoretical proof of the Ruch-Schonhofer theorem. We have already seen that i(ISa I Sn' ISp I Sn) is equal to the number of matrices over No with ex as vector of row sums and p as vector of column sums. We can apply this if we happen to know the number of such matrices. The following lemma gives the number of matrices in a particular case:

2.2.18 LEMMA. If r l , rz , c I , and C z are nonnegative integers with the property r l +rz =c I +c z ' then the number of2X2 matrices over No with row sums r l , rz and column sums c l ' Cz is equal to 1+min{r p rz , c l ' cz}.

43

2.2 The Permutation Characters Induced by Young Subgroups

Proof If for example r\ =min{r" r2 , c" c2 }, then we have the following I + r I choices for the entries of the first row of such a 2 X 2 matrix:

(:

r\ -r) *

'

Each of these choices yields exactly one 2 X 2 matrix with row sums r, and column sums c" since r] was assumed to be the minimum, so that a suitable second row can be found and is uniquely determined. In the case when r2 , c" or c2 is the minimum, an analogous argument yields the statement. • This helps in the proof of 2.2.19 LEMMA. If a=(a" a 2 )t-n, a2 >0, and a*: =(a]

+ 1, a 2 -1),

then

(i) [aj+IS,," iSn = IS" iSn, (ii) the dimension of[ a j satisfies

r

"2

(iii) IS" iSn = ~ [n-p, pj. • =0

~"

Proof (i): 2.2.18 yields for the inner product of the generalized character - ~,," with itself: i(IS" i Sn' IS" i SJ + i(IS,," i Sn, IS,," i Sn) - 2i(IS" i Sn' IS,," i Sn) = (a 2 + 1) + (a 2 -1 + 1) - 2( a 2 -1 + 1) = 1.

e-

Hence ~,," is ± 1 times an irreducible character of Sn' Since (IS" iSn,[a])= 1, but (IS,," iSn,[a])=O (by 2.1.10), we obtain ~,,-~,," =K", as stated. (ii): The statement follows from the dimensions

by an application of (i). (iii): This part follows immediately from part (i), by induction on a2'

•

44

Ordinary Irreducible Representations and Characters 2.2

2.2.20 THEOREM. If a, {3~n. then there is a representation Df3a of Sn with ISf3 iSIl + Df3a =ISa iSIl if and only if a
Proof (i) If there exists Df3a with ISf3 i SIl + Df3a = ISa i SII' then (ISf3 i SIl' [{3]) = I shows that (ISa i SIl' [{3]);;;. I; thus a Q{3 by 2.1.1 O. Also a =/= {3. (ii) If a
This shows that it suffices to prove the existence of a Df3a of the required form only for the case a
for suitable i and j (cf. 1.4.10). If we denote by # the outer tensor-product multiplication of representations, then ISa iSn=([a,]#" '#[all])iSn = ([ a i ] # [a)] # (

#

[a v ] )

)

t Sn

V*l.)

which by 2.2.19(i) equals ( ([ a, . a)] + ([ a i + I] # [ a) -

I] ) i Sa, +,,)

)#

(

= ([ a,. a)] # ( #

vt.) [a v]))i SII [a,,] ) ) i Sn + ISfJ i Sn

"*1.)

where

• Expressed in terms of the matrix Mil this reads as follows: 2.2.21 COROLLARY. For given i and j. each entry m ik of the ith row of M n is greater than or equal to the corresponding entry m)k of the jth row of M n if and only if a ' Qa).

Furthermore we obtain from 2.2.20 and 2.1.10: 2.2.22

COROLLARY.

(IS" is,,.[{3])>O=aQ{3.

45

:3 Irreducible Characters as Combinations of Permutation Characters

Finally we obtain the Ruch-Schonhofer theorem:

For 2.2.22 yields the = part, while the other direction of the statement follows from the trivial part of the Gale-Ryser theorem. This completes the story and gives at the same time a representation theoretical proof of both the Gale-Ryser theorem and the Ruch-SchOnhofer theorem.

2.3

The Ordinary Irreducible Characters as Z-Iinear Combinations of Permutation Characters

An important result of the last section was Theorem 2.2.10, which says that each ordinary irreducible character ~a of Sn is a uniquely determined I-linear combination of the permutation characters It is the aim of the present section to derive a direct method for determining these linear combinations. The expression for ~a in terms of the is of a determinantal form. so we first introduce a multiplication of representations [a] of symmetric groups. If m and n are natural numbers, then Sm X Sn can be embedded into Sm+n in a natural way:

e.

e

2.3.1 since Sn ~Sm+n"m (where m+n: = p, ... , m+n}). Hence if af-m and f3f-n, then the (ordinary irreducible) representation [a]#[f3] of Sm XSn defines a representation of a subgroup of Sm+n which is isomorphic to Sm X Sn' and we can induce into Sm+n' The resulting representation is denoted by [a] [131 and called the outer product of raj and [f3J:

[a] [13] : = [ a] # [ 13 h Sm + n .

2.3.2

It is easy to see that this multiplication is associative (apply (43.3) in Curtis and Reiner [1962]), and it is commutative. Hence for example

2.3.3 With the aid of this multiplication we now define a determinant which corresponds to a=(a l ,a2, ... )f-n as follows: 2.3.4

l[ai+j-i]l=

[all

[at+l] [a l +2] [al+h-I] [ a 2] [ a 2 + I] [ a 2 + h - 2] ............................................ [ a 2 - 1]

[ah-h+l]

[a h -h+2]

[a h -h-1:"3]

[a h ]

Ordinary Irreducible Representations and Characters 2.3

46

when

2.3.5

ah+ 1 =

0, subject to the definitions

[r]:={~

if r=O (which is consistent with 2.3.3), if r
(We put I'[r]: =[r] and O·[r]: =0.) Note that 2.3.5 ensures that the determinant's definition is independent of the choice of h (subject only to ah+ I = 0). For example if a: = (3,1 2 )f-5, we get

[3]

[4]

I

[I]

0

I

[5] [2] =[3][1][1]-[3][2]-[4][1]+[5] [I]

[3]

[4] [1]

[5]

I

[2]

[6] [3]

0

I

[ I]

[2]

0

0

0

I

Since minus signs appear, the determinant 2.3.4 has to be interpreted as a generalized "representation". The character of our example is

If we had M s at hand, then we would see that this character is equal to ~(3.12l, so that the above determinant which corresponds to a=(3, 12 ) yields

the desired expression for ~a as a I-linear combination of the permutation characters And it is in fact true that for any partition a, the determinant 2.3.4 gives this expression, for short:

e.

[a] = I[a i +j - i ]1· It is our aim to provide a proof of this. In order to do this, we first generalize our definition of permutation characters ~a slightly. We defined an improper partition A of n to be a sequence of nonnegative integers whose sum is n. Thus, A may be regarded as an element of IN, the set of mappings from N into I. Now assume only that AE I Nand '2% I Ai = n (so that A still has finite support, but we no longer assume that its parts are nonnegative). Then A will be called a composition of n. (The reader will be encouraged to hear that we do not intend to introduce any more terms to describe elements of IN.) He who likes his definitions to be as general as possible may want occasionally to drop the assumption that '2~= lA, =n (e.g. in 2.3.6).

=3 Irreducible Characters as Combinations of Permutation Characters

47

Sometimes, when using compositions of n, we shall impose the restriction that Ai =0 for i>n (this works automatically for proper partitions)-a restriction which could, in fact, be avoided in all places where we make it, but this would force the introduction of some really unwieldy notation. To continue with the story, let A be a composition of n, and put 2.3.6

~

>... _

.-

{the character of IS>.. i Sn the function mapping Sn to 0

if AF n, otherwise.

Although it is unusual to consider "the zero character", as in the case where A is not an improper partition of n, these zero characters behave in the obvious way. The properties we need are these: (i) (the zero character of Sm) # (a character of Sd=the zero character of

Sm+k'

(ii) Inducing or restricting a zero character gives a zero character. These results follow by inspecting the processes involved; alternatively, they may be taken as the definitions of the restriction of a zero character, etc. Now regard Sn as the subgroup of SN consisting of the permutations fixing each point i>n. Then, by definition, every 7T which belongs to S", belongs to some Sn' (Remember that we chose to define a permutation to be a bijection fixing all but finitely many points.) Hence we may extend the function sgn to S", in a well-defined manner. For every composition A of n, and every 7T E S"" A0 7T is a composition of n, and

We define addition on .l N pointwise. Then 2.3.7

A-id+7T= (AI -I +7T(I), A2 -2+7T(2), ... )

is a composition of n. Using this notation, and having 2.3.6 in mind, we put for compositions A of n 2.3.8

x>..:=

~

sgn7T·~>"-id+".

"ESN

Since ~~= I AI = n, there exists hE'" 0 such that Ai = 0 for all i > h; then for all7TES N \Sh' A-id+7T has a negative part. Therefore, the sum in 2.3.8 is finite and 'Vi~h,

X>" = ~ sgn7T·g>..-id+". "ES,

48

Ordinary Irreducible Representations and Characters 2.3

Thus X A is a generalized character of Sn' In particular, if A: = (X~n, then X a is clearly the character of the determinant I[ (Xi +j - ill, so that it is our aim to prove Xa = ~a. We divide the proof of this basic result into several steps. A preliminary lemma gives an important property of X A:

2.3.9

LEMMA.

If A is a composition of n, and

Proof We put 7": =(i, i+ I)ESN . Then we have

if j"", i, i + I, if j=i, if j=i+ 1. This implies

2.3.10

~1'-id+?TT =e-id+?T,

LEMMA.

so that

Suppose that A is a composition of n=m+k. Then

(i) ~AiSmXSk=~I'Fke-I'#~I', (ii) X A iSm X Sk = ~I'F kXA-I'#~I'.

Proof (i): Both sides of the equation equal zero unless A is an improper partition of m+k. Assume. therefore, that AFm+k. (The sum over Jl has only finitely many nonzero terms, since ~A-I'#~I' =0 unless A- JlF m.) Mackey's subgroup theorem then yields

~A iSm X Sk =

~

I( (Sm XS k ) n7TSA7T -j) iSm XS k .

Sm XSk?TSx

By 1.3.10 the double-coset Sm X Sk7TSA is characterized by the 2 X n matrix

It is therefore uniquely determined by

2.3 Irreducible Characters as Combinations of Permutation Characters

49

The character of this representation is

and so (i) is proved. (ii): The definition 2.3.8 of XA yields

and by (i), this equals ~ ~ sgn'IT·~(A-I')-idh#~1'= ~ XA-I'#~I'.

I'F k 7T

I'F k

•

If A is a composition of n, and p. is a composition of k, we define sgn'IT·e- id -(I'-id)07T.

~

XVI':=

2.3.11

7TESN We claim that the following is true:

2.3.12 LEMMA. If A is a composition of m + k, then X A t Sm X Sk = " /3' k XA//3#X/3 • .:. Proof The proof of 2.3.1O(ii) has already shown that

xAt Sm X Sk =

~ sgn 'IT ~ e-id+7T-I'#~I'.

7TESN Replacing p. by

P.O'IT

we get (as

xAlSm XS k =

J1-Fk

~I'07T =~I')

~

sgn'IT ~ e-id-(I'-id)07T#~I'.

7T

~k

Now, for each p., P.i is eventually zero, and thus there exist uniquely determined 13 F k and a E S N such that

(13- id) and

0

a= p.- id

50

Ordinary Irreducible Representations and Characters 2.3

This enables us to proceed as follows (recall that all the sums are finite, so there is no problem about rearranging): x>'J.SmXS k =

sgn'7T

~ "ES",

=~

~ oES",

~~>'-id-(P-id)ooo"#~(P-id)oo+id

P

sgn(a'7T)sgna~e-id-(.B--idJoo"#e-id+o-1

".0

=~ P

P ~

sgn pe-id-(p-id)op# ~ sgnae- id +o- I o,=S",

pES",

= ~ X>'/fJ#X fJ P

= ~ X>'/fJ#X fJ . P,k

The last equation holds because xP =0 if Pi -i=Pi + 1 -(i+ 1) (by 2.3.9). • The lemma shows the importance of the generalized characters X>'/fJ of Sn-k'

2.3.13 THEOREM. If af-n, k";;;;n, and pf-k, then (i) xo./ fJ "",0 only if a; ~Pi for all i,

(ii) (xa/P,~(n-k»= {l ifal~PI~a2~P2~a3~"', otherwlse.

°

Proof (i): We consider the determinant by which X a / P is defined (cf. 2.3.11). It is (cf. 2.3.4)

As the sequences a-id and P-id are strictly decreasing, an entry [ai-i(Pj-j)]=O, i.e. a;-i-(p;-j)a; - i - (P, - i) = a; - Pi' Hence xa / P "'" implies P; .,;;;;a;. (ii): For each composition A of n-k we have (cf. 2.2.14)

°

(~\~(n-k»)=

{I

if A'F~-k, ° otherwIse.

Thus, if ~ denotes the Kronecker symbol,

TIsa, -;-(fJ.",-"U)).'"'O

~ ( X alP ,'i>/::.(n-k»)- "" sgn1T . "

I

2.3 Irreducible Characters as Combinations of Permutation Characters

51

If al ~f3l ~a2 ~f32 ~a3 ~ "', then this determinant has ]'s along its main diagonal and O's below, and hence it is equal to ]. Otherwise it is not of this triangular form, so that by the monotonicity of the sequences a - id and 13 - id two columns are equal, and hence the determinant must vanish in this case. •

This result turns out to be crucial in the proof of 2.3.14 YOUNG'S RULE (FIRST VERSION). For each AF n such that Ai ==0 when i>n and every partition a of n, we have that the inner product (x a , is equal to the number of (n - ])-tuples (13(1), .. . , f3( n- I» such that

e)

(i) 'VI ,,;;;;i";;;;n-] (f3(i)~2:~= ,A), (ii) 'V] ,,;;;;j";;;;n (f3/') ,,;;;;f3j2),,;;;; ... ,,;;;;f3?-IJ ";;;;a), (iii) 'Vj>], i~ I (f3p> ";;;;f3}!..-i », if we set 13(0): =(0) and f3(n): =a. '

Proof Applying 2.3.12, we have

which is by 2.3.]3

if the sum is taken over all the f3(n-l) which satisfy n-I f3(n-I)~ ~

Aj

and

al;;;"f3ln-I)~a2~f3in-I)~ ....

i=l

Another application of 2.3.]2 and 2.3.13 yields

where the sums have to be taken over f3(n-l) and f3(n-2) subject to the

52

Ordinary Irreducible Representations and Characters 2.3

following conditions: (i) ,8(n-I)~~7-IAj' ,8(n-2)~~;1-2Aj'

(ii) CX I ;;:,,8\n-l) ;;:'cx 2 ;;:,,Bin-l);;:, "', (iii) ,8\n-I);;:, ,8\n-2);;:, ,81 n- I);;:, ,81 n- 2);;:, .... Further iterations yield

(x",e)=

2: 2: ... 2:

p,n-I) p,n-2)

=

(XI1''',~(AI))

p(l)

2: .. ·2:1 p'n-I)

p(l)

2:

1,

(p'l), .... p'n-l))

where the sum is taken over all (n - 1)-tuples (,8(1), ... , ,8( n conditions described in the statement.

I)

subject to the •

We are now in a position to prove the main result of this section: 2.3.15

THEOREM.

For each

cx~n

we have that

the character of the ordinary irreducible representation [cx] of Sn' Thus, in particular, the set

is the complete set of ordinary irreducible characters of Sn' Each expressed in the determinantal form ~"=

2:

~"

can be

sgn7T·~"-id+"

"ESn

as a linear combination of permutation characters

e, AF n, with coefficients

0, ± 1. We can therefore express the representation [cx] itself as a generalized "representation" in the following deterrninantal form:

subject to the conventions [r]: = I if r=O, and [r]: =0 if r
2.3 Irreducible Characters as Combinations of Permutation Characters

53

Proof (i) We prove first that for any AF n (with A; =0 for i>n), (X", f')*0 implies a~A: This follows from Young's rule, for condition 2.3.14(iii) yields /3i l ) =/3j2) = ... =f3,~n~ll =0. Hence /3(I)f-A 1 together with a l ~/3\I) gives a l ~AI (by f3i l ) =0); f3( 2l f-A I +A 2 together with a l ~f3\2), a 2 ~f3f) yields a l + a 2 ~ 13\2) + /3i 2l ~A I + A2 (by f3j2l = 0), and so on. (ii) (X", ~")= I: By Young's rule (X", ~") is equal to the number of (n-l)-tuples (/3(1), ... ,f3(n~I) which satisfy (see (i»

Thus (( a l ),( ai' ( 2 ), ... ,( a l , ... , an~ 1» is the only possible (n -I)-tuple. (iii) (X", X") = 1: If (X", ~"-id+")*O, then a - id +1TF n anda~a - id+'17, by (i). But a-id +'17~a, as it is very easy to see, and so a=a-id +'17 and '17= 1. Together with part (ii), this shows that ~" is the unique summand f' of X" for which (X", f')*0. Therefore, (X", X")=(X", ~")= 1. (iv) In order finally to identify X" with ~", we consider the determinantal form I[a; -i+jJI which shows that [aJ is contained (use 2.2.22) in the main diagonal term only. Thus (X", ~")= 1. But (iii) shows that X" is ± an irreducible character, and so X" 0::: ~". •

A numerical example is

But

has the character

for '17 E S3 induces on the left cosets of S2 in S3 the same permutation as on the tabloids I 3 Thus

2

13 2

T3 I

Ordinary Irreducible Representations and Characters 2.3

54

This yields again the character table of S3:

More generally we have by 2.3.15 for each

n~2:

(i) [n-l,l)::::[n-l][l)-[n],

2.3.16

(ii)

~(n - I, I) ( 7T ) :::: a I ( 7T ) -

1.

(This of course also follows from 2.2.l9(iii).) The representation [n-l][l], which has the character

is sometimes called the natural representation of Sn' Another useful corollary of the deterrninantal formula is the following result on the values of the ordinary irreducible characters of Sn on the class of elements which consist of a single n-cycle:

2.3.17 f"((l ... n»::::{~-lr

if a::::(n-r,lr),

O:;;;;r:;;;;n-l,

otherwise.

Proof The only Young subgroup which contains n-cycles is the Young subgroup Sp: ::::Sn' Thus 'n))"",O only if f3::::(n). The deterrninantal form therefore shows ~a«(l ·n))""'O only if a summand ±~(n) occurs in

e«(l .. "

But this is the case only if I[a;+j-ill is of the form

I[a, +j-i)l::::

+ h - 1:::: n (d. 2.3.4), i.e. h::::n-a 1 + 1, or equivalently a is of the form a::::(n-r, 1') for a suitable r:;;;;n-l.

(h minimal such that ah+ I :::: 0), so that we must have a l

2.3 Irreducible Characters as Combinations of Permutation Characters

55

If on the other hand a=(n-r, 1'), then ~a((l ... n))=(_l)h-l~(n)((l. .. n))=(_l)h-l

=(-1)',

•

as stated. This shows that

~a((l

.. ·n»*O if and only if [a] is a f-shaped diagram:

x

X

X

X

X

[n-r,l']= : X X

Such diagrams are called hooks and play an important part in the representation theory of Sn. We would like to conclude this section with a proof of a formula which expresses the dimension of [a] in terms of hooks. It was T. Nakayama who first introduced the concept of hooks, by which we mean specific f -shaped subsets of diagrams [a]. In order to make this precise, we call the node of [a] which lies in the i th row and j th column the (i, j)-node of [a]. Then we denote by

r

2.3.18 the (i, j )-hook of [a], which consists of the (i, j )-node, called the corner of the hook, and all the nodes to the right of it in the same row together with all the nodes lower down and in the same column as the comer: j

The (i, k)-nodes, k>j, form the arm of the hook, while the (I, j)-nodes, l>i, form the leg of the hook. The (i, ai)-node is called the hand of the hook, while the (a), j)-node is called the foot: comer

hand

I!' I!' X X···········X a Hi} -

foot?

X] . :

X

arm leg

56

Ordinary Irreducible Representations and Characters 2.3

The number

h~j

of nodes of Hi), i.e.

2.3.19 is called the length of Hi). The number I':) of nodes in the leg of Hi), i.e.

2.3 .20 is called the leg length of Hij. To Hi~ there corresponds a part of the rim of [a] which is of the same length. It consists of the nodes on the rim between the hand and the foot of 2 Hi), including the hand and the foot node. To H,Oj2.1 j for example there correspond the encircled nodes of [3,2,1 2 ] as follows:

x

0 0

o o o

0

This associated part of the rim will be denoted by

It is important to notice that the result

of removing

R~j

from [a] is again a Young diagram: e.g.

x [ 3 2 12 ]\R(3.2.1 "

1.1

2

)

=

X

X X

=X=[l].

X X

We intend to show that the dimension product of all the hook lengths: 2.3.21

THEOREM.

r

r

of [a] is just n! divided by the

=n!/IIih~j.' .j

Proof Notice that for each composition A= (A I' A2'" .) of n we have A

_

n!

g (l) - AI''A 2', ... ' if we put (cf. 2.3.5) 1/r!=0 when r
2.3 Irreducible Characters as Combinations of Permutation Characters

57

h nonzero parts) the determinantal formula gives

an h X h determinant. Now

Elementary transformations of the determinant on the right hand side of this equation yield the Vandermonde determinant

Therefore 2.3.22

r

This equation shows that can be expressed in terms of the hooks which have their corner in the first column. To complete the proof we show that for 1 ,,;;;,i,,;;;,h we have h (h"il -h")1 )II'""=1 h"i" =h"iI", II )=i+l

Hi~

(1)

On each side of (1) there are hfl factors. The factors hfl -hjl are strictly increasing as j increases, while the hf" are strictly decreasing as P increases. Furthermore the factors on the left-hand side are all ,,;;;,hfl' It therefore suffices to prove that the factors on the left-hand side are pairwise different. This can be done by a verification of (2)

for a suitable j, which depends on Thus

a)+ 1
P.

We put j: =

a~,

so that aJ ~ P, while

and

This proves (2), which implies (1), which then implies the statement.

•

58

Ordinary Irreducible Representations and Characters 2.4

r,

This formula relates the prime divisors of n!, and the hook lengths so it is helpful when we come to consider modular representations.

2.4

h~J'

A Recursion Formula for the Irreducible Characters

In the preceding section we saw how ~o: can be expressed as a I-linear combination of permutation characters so that we are now able to of the characters calculate the character table Z; of S, from the table :::n by suitable applications of the determinantal form of [a]. But this is still unsatisfactory; we would prefer to evaluate Z; either directly or recursively. To obtain a recursion formula for ~o: we need to know what happens when [a], a a partition of n, is restricted to subgroups Sm, m..is a composition of n, we put

e,

e

e

2.4.1

+" ...),

X- :=(A"

, Ai-I' Ai -I,

N+: = (A"

, A;_I' A; + I, AI+"''')'

Ai

e,'-

Then 2.3.10 yields (since =0 if Ai - is not an improper partition of n - I; in particular, =0 for all sufficiently large i):

e'-

2,4.2 LEMMA. 00

e~Sn-I=2:e-. i=1

Applying this to the determinantal form of

~o:,

we obtain

~O:~Sn_'= ~ sgn7T(gO:-idh~Sn_l) ?TES

n

= 2:sgn 7T2:g
/

?T

By 2.3.9 we have that X a'-7"=O only if

(\:j-I;;;'a

j

+,. Therefore only sum-

59

2.4 A Recursion Formula for the Irreducible Characters

mands for which cx;>cx;+I' and hence cx'-~n-I contribute to KatSn~I' This proves the first half of the following theorem, the second half of which follows by Frobenius's reciprocity law:

2.4.3 THE BRANCHING THEOREM. If cx=(cxl'cx 2 , ... ) is a partition of n , then we have for the restriction of [cx1to the stabilizer Sn-l of the point n

; ll:'j>a'+l

On the other hand, if S, denotes the stabilizer of the point n + I in Sn+ I' then we have for the induced representation

i lX;
(put cxo:

i-

1

= 00).

For example [3,2, 12 ] t S6= [2 2 ,

e ]+ [3, 1

3

]

+ [3,2,

I],

while

This branching theorem shows how to evaluate Ka(7T) if 7T contains l-cycles and if the character table of Sn-I is known. In the case where 7T does not contain l-cycles, we know Ka(7T) only if 7T is an n-cycle (cf. 2.3.17). Hence if 7T contains a k-cycle, say, then a formula would be welcome which expresses Ka(7T) in terms of certain KJ3( p), f3~n - k, p arising from 7Tby cancelling this k-cycle. We shall derive such a formula, and it should be clear from the foregoing that we need first to restrict from S, down to Sn-k X Sk' the subgroup of elements which leave the subsets n - k and nxn- k invariant. An immediate corollary of 2.3.1O(ii) is

2.4.4 LEMMA. Suppose that A is a composition of n = m + k. If 7TE S, contains a k-cycle, while p E Smhas the cycle type

then

X"(7T)= ~ X"-I"(p)~rk)' I"Fk

60

Ordinary Irreducible Representations and Characters 2.4

In order to apply this we remember that ~rk) *0 if and only if J.Lhas the form J.L=(O,oo.,O,k,O,oo.), "

~n

which case

~rk)

= 1. Therefore, 00

X A(

2.4.5

7T)=

~ X(A\, .. ,A,_\,A,-k.A,+\

....

)(p).

i=\

Now we consider the summands in 2.4.5 for the case when ;\.=a partition of n, An m-fold application of 2.3.9 yields

IS

a

2.4.6

°

It is not difficult to see that the existence of an m> such that the composition on the right-hand side of 2.4.6 is a proper partition of n - k is equivalent to the fact that the corresponding diagram is obtained from [a] by cancelling the rim part R~j of a (uniquely determined) hook Hi) of length k of [a]. In this case m=I'0, and we obtain

If no such m exists, it follows from the considerations above and 2.3.8 that the character in 2.4.6 vanishes. This yields the desired recursion formula for

t'(

7T):

2.4.7 THE MURNAGHAN-NAKAYAMA FORMULA. If a is a partition of n, if kEn, and 7TESn contains a k-cycle, while pES n - k is of cycle type (a l ( 7T),... , a k - I( 7T),a k( 7T)-I, ak+ I( 7T),... , a n - k( 7T)), then we have the following recursion formula for t'( 7T): t'(7T)=

~

(_I)/~J~[a]\RfJ(p)

i,j hfJ =k

(recall that ~ [0 ] = 1).

For example

,

~(3J)( (1234)( 56)(789)) = ~(23)( (1234 )(56)) - ~(3,2, 1)((1234)( 56))

+ ~(32)( (1234 )(56)) = _~(2,12)((1234))+~(22)((1234)) - ~(22)( (1234)) + ~(3. 1)((1234)) =-1-1=-2.

61

2.4 A Recursion Formula for the Irreducible Characters

It is clear that a diagram [a] has exactly one hook of length h II' Thus

and a repeated application of this argument yields

It is also clear that [a] does not possess any hook of a length >h l l, and that this implies 2.4.9 COROLLARY. ~;=FO=f3~(hll'"'' hick)' This shows once again the importance of the diagram lattice. Later on we shall need a few results on lowest dimensions of ordinary irreducible representations of symmetric groups. There are many results known, we would now like to derive a few of them by various applications of the branching theorem. 2.4.10 THEOREM.

(i) For each n, [n] and [I"] are the only one-dimensional ordinary representations of S". (ii) For 2,;;;;n 4, the lowest dimension 1 of an ordinary irreducible representation of S" is n - 1 (while for n: = 4, [2 2] has dimension 2 = n - 2). For 2';;;;n=F6,[n-l,l] and [2,1"-2] are the only ordinary irreducible representations of S" which are of dimension n- 1 (while for n: = 6, [32 ] and [23 ] are other irreducible representations of dimension 5 = n - 1). (iii) For 2';;;;n=F5there is no ordinary irreducible representation [a] of S" which is of a dimension j'" such that

'*'

'*'

n';;;;j'"';;;;n+2 (while for n: = 5 we have [3,2] and [22,1]. which are of dimension 5 = n, and [3,12 ] of dimension 6=n+ 1). Proof (i): This part is an easy consequence of 1.1.25 and 1.1.26. If we prefer to avoid this argument, we may proceed by induction on n as follows. If [a], o:f-n,is one-dimensional, the same holds for [a]l S"~ I' Hence, by the induction hypothesis, [o:]l S"_ 1 coincides with [n - 1] or (I" ~ I], and the branching theorem yields

[a] E {[ n l-[n- I, 1], [1" ], [2, 1"~ 2l}. But 2.3.21 has shown that both [n-l, which finishes the proof of part (i).

I] and [2,1"-2] are of degree n-l,

62

Ordinary Irreducible Representations and Characters 2.4

(ii): An inspection of the character table shows that the statement holds for n.::;;8. We can therefore assume a~n~9

and

1
(a) If [aU Sn-I contains one-dimensional constituents, we obtain from (i) and the branching theorem that

1), (2, In-2)},

aE {(n-l,

and 2.3.21 shows that both tn-I, I] and [2, I n - 2 ] are of dimension n-l. (b) If [a 1~ S, _ I does not contain anyone-dimensional constituent, then by induction it must be irreducible, for Sn-I has no irreducible representation [,8] such that I 2. But then

where none of the constituents can be one-dimensional. since n-2>6. Hence the induction hypothesis yields the contradiction

I"

~2(n- 3)~n+ 3.

•

(iii) follows immediately from the proof of (ii).

The proof of this theorem shows clearly that analogously one can derive further results concerning ordinary irreducible representations of low dimension. We next undertake to extend the Murnaghan-Nakayama formula so that it can be applied to the generalized characters XVI' introduced in 2.3.11. To do this we need the following result:, 2.4.11 LEMMA. If A is a composition of n + r, m+k=n, then

P

a composition of r, and

(i) XA/V ~ Sm X Sk = ~ /LFK X(A-I')/V#CI'r; , (ii) and if 7TE S, contains a k-cycle, while p E Sm is of cycle type (a 1( 7T), · .. , a k _ I( 7T), a k( 7T) - -l , ak+l(7T), ... , a m ( 7T»), then X A/V( 7T) =

L X(A-I')/V(p)~rk)' I'Fk

63

2.4 A Recursion Formula for the Irreducible Characters

Proof (i): We obtain from 2.3.11 that

tr

so that an application of 2.3.10 yields =

~

sgn 7T ~

=

~ (~Sgn7T·e-l'-id-(V-id)O'" )#~I' JLFk

=

e-l'-id-("-id)o",#~1'

JLFk

tt

tr

~ X(A-I')/V#~I',

i'Fk

the last equation coming from 2.3.11. (ii): As

we obtain from (i)

=~

•

x(A-I')/V(p)~rk)'

JLFk

Again we remember that ~tk) *0 if and only if }J. is of the form }J.= (0, ... ,0, k, 0, ... ), in which case ~rk) = 1. Therefore, the following must be true for A, v, 7T, and p as in 2.4.1 I: 00

X A/ V( 7T)=

2.4.12

~ X(A\ ... A,_\.A,-k,A,+\

... )/v(p).

i=l

In order to derive from this the desired generalization of the MurnaghanNakayama formula 2.4.7, we associate with X a / P, a~m+k, f3~k, f3i";;;ai, 1,,;;;j,,;;;k. a diagram which will be denoted by 2.4.13

[a/f3]

and called a skew diagram. It arises from [a] by deleting the first f3i nodes of its ith row, 1
[8,7

2,5/4.3

2

]

= [9,8

2,6/5,4

2

,

1] = X

X

X

X X X

X X X

X X X

X X X

X

64

Ordinary Irreducible Representations and Characters 2.4

As we can delete from a skew diagram [a / f31in an obvious manner all the parts R~!f3 of its rim which correspond to hooks Hi) such that R~J does not contain any node of the subdiagram [f31of [a], we can generalize 2.4.7 as follows (a, f3.7T,P as above, and d. the derivation of 2.4.7 from 2.4.5): 2.4.15

THE MURNAGHAN-NAKAYAMA

FORMULA FOR SKEW DIAGRAMS.

X"/f3(7T)= ~ (_l)liJX["/f3J\R~f\p). i,j hf j =k

The generalized character X"/f3 IS 10 fact a character. Therefore the diagram [a / f31corresponds to a representation, which will be called a skew representation. It will also be denoted by [a/f3]. More precisely, we shall prove next the following result (a~m+k, f3~k, as above): 2.4.16

[a/f3]=

~ ([a]'[f3][y])[y]. yem

\ Proof The statement is equivalent to

i.e., we have to verify that

2.4.17

~, ~ . "ES

X"/f3(7T)KY(7T)= m~kr m

••

(p.(/)ES

~ m

XS

~"(P(J)~Y(p)~f3((J) . k

But the right-hand side of this equation is by 2.3.12

which by the orthogonality relations is equal to the left-hand side of 2.4.17, as required. • Later on the Littlewood-Richardson rule will show how the multiplicities ([a],[f3][y]) in 2.4.16 can be evaluated.

2.5 Ordinary Irreducible Representations and Characters of An

2.5

65

Ordinary Irreducible Representations and Characters of An

As soon as a result on representation of S, is derived, it is usual to ask what follows for the alternating group An' The standard method to derive the corresponding result for An from that of S, is to apply Clifford's theory of representations of groups with normal subgroups. This theory is easy to use, since the factor group S, / A n is cyclic, so that we need not deal with projective representations. Moreover the factor group is of prime order (if n> I), so that the inertia group of any ordinary irreducible representation of An is either An itself or Sn' Let us first deduce from the system {[a]la~n}

of ordinary irreducible representations of S, a complete system of equivalence classes of ordinary irreducible representations of An' A preliminary remark follows immediately from 2.1.8. For each a~n we have:

2.5.1

[a] ~ An = [ a'] ~ An'

Thus [a] and [a'] are associated representations with respect to A n in the sense of Clifford's theory: their restrictions to An have irreducible constituents in common. (This is the reason for calling a' the partition "associated" with a.) Clifford's theory tells us (d. Curtis and Reiner [1962, § 49]) that the irreducible constituents of their restrictions are conjugates of each other. Moreover the restriction is just equal to a certain multiple of a complete class of conjugate irreducible representations of An" Let us consider an irreducible constituent D of [aHA n:

2.5.2

D is either self-conjugate with respect to S, - i.e. (assuming n> I) 2:5.3 where for each 'TTEAn

2.5.4

D II2)((12)'TT(12)):

=D( 'TT)

- or D is not self-conjugate, which means that D and DII2) are inequivalent irreducible representations of An' D is self-conjugate if and only if its inertia group is Sn' Then D can be extended to a representation 15of S, and gives

66

Ordinary Irreducible Representations and Characters 2.5

rise to the following two irreducible representations of Sn: 2.5.5 D is not self-conjugate if and only if its inertia group is An' in which case it yields by induction an irreducible representation of Sn:

2.5.6 Furthermore each irreducible representation of S, is of one of these forms D" i = 1,2,3, and both [a) and [a') arise from D in the way described. Thus [a) is of the form D I or D 2 if and only if [a) [a')' i.e. if and only if a*a'. (It is this fact that irreducible representations of S, which are associated with respect to S, form easily recognizable pairs, which does not hold if the characteristic of the field is 2, and in this case life is not as easy.) We therefore have

*

2.5.7 THEOREM. Suppose that a is a partition of n> I. (i) If a*a'. then [aHA n =[a'HA n is irreducible, while (ii) if a=a', then [aHA n =[a'HA n splits into two irreducible and conjugate representations [a)~ of An' i.e., [a) +(12), defined by

is equivalent to [a)- . A complete system of equivalence classes of ordinary irreducible representations of An is therefore

If for example n: = 3, then we obtain the system

This shows that the self-associated partitions of n form an important subset SA(n) of P(n): 2.5.8

SA( n): = {ala'=a~n}.

The next question which arises concerns the character table. We ask how the table of All can be obtained from that of Sn' We remark first that 2.5.7 implies that the only problem is the evaluation of the characters of An which are of the form ~a~.

2.5 Ordinary Irreducible Representations and Characters of A"

67

If a is a self-associated partition of n, then the diagram [a] is symmetric with respect to its main diagonal. This implies that the hooks Hi~' I ~i~ k: = (length of the main diagonal), of [a] have their arms and legs of the same length. The parti tion 2.5.9 formed by the lengths of these main hooks H,~ of [a], characterizes a conjugacy class of Sn' which splits over An (c.f. 1.2.10); we therefore call it a split partition and denote the set of all the split partitions by SP(n): 2.5.10

} and the nonzero parts of a SP n : = { a af-n ... . () I are pairwise different and odd

It is easy to see that ht- ) provides a bijection from SA( II) onto SP( II): 2.5.11 Hence h(a) falls under suspicion; it turns out that in fact the values of ~a~ on the classes ch(a)~ are the only values which require a special description, while on any other class ~a= has half the value of fa. The value of ii: is given by 2.4.8: 2.5.12 LEMMA. If a is a self-associated partition of n, and k denotes the length of the main diagonal of [a], then /,a ~h(a)

=(_I)(n-k)/2

.

2.5.13 THEOREM. If a = a' is a self-associated partition of n> I, then the values of ~ac+ are /,a=

-l(/'a 2

~h(a)+ -

+-V/'aII h". )

~h(a) -

~h(a)

iii'

for a suitable numbering of the constituents of [aHAn. while on all the other classes with cycle partitions y h( a) we have

*"

and

f; is an even integer.

Proof We consider the following generalized character of An:
(fa + - fa-) (fa+ _

~a- ),

68

Ordinary Irreducible Representations and Characters 2.5

where ~a'" ( 'TT):

= ~a'" ( 'TT- \ ) = ~a'" ( 'TT),

i.e., ~a± denotes the character contragredient to t:". We would like to show that cpais the restriction to An of a class function epaof Sn' which is in fact a scalar multiple of the indicator function of the class ch(a) of Sn' (i) The function epa. The definitions of [a] + and [a]- yield that )'a+=)'aand ~y)'a+=)'a~y~y+

~y+

for

yESP(n)

and for

yEP(n )\SP(n).

Therefore

and cpa is the restriction to A n of a class function epa of Sn, where epa:= !cpatS n • epais given by if yESP(n), if yEP(n)\SP(n). Since for each f3E SA( n)

we have foreach

f3ESA(n).

(ii) epaas indicator function. We know that epais nonzero only on splitting classes. Let E denote an element of SP( n) which is minimal with respect to the dominance order satisfying

69

2.5 Ordinary Irreducible Representations and Characters of An

By 2.4.9 and the minimality of e we obtain

1.3 (

yh

-a

cp ,~

_1(,)) = __ IC'I-ayh -1(,) = __ IC'I-a( -1 n! cp,~, n! cp,

)(n-}j/2

'

j being the length of the main diagonal in [h- 1( e)] (cf. 2.5.12). This proves

ICI v: -a -, E1.\{O}. n. On the other hand, since

I

-n!

qsavanishes _

~ fJ~n

I

IcfJlcpa = -. fJ 2 =

(I)

outside An' -

2

n!

~ fJeven

IcfJlcpa f3

±(r a+- r: ,r a+-r a-) I

='2. 2=1. Now, the definition of

(2)

qsagives ({3rn).

(3)

We obtain from (1), (2), and (3)

ICfJl-a _ 7 CPfJ-

{I if 0

{3=e, otherwise.

(4)

This shows that qsais a multiple of the indicator function of C. (iii) ra:oon classes other than CEo If y =1= e, then for classes of A n with cycle partition y, (4) gives o=ya+

~y

_ya-

~Y'

Since

as has been stated. (We show below that e=h(a).) (iv) ra:oon C. Since -a_

cp,-

n! -Iya+ ya-12 IC'I~,:o -~,:O ,

70

Ordinary Irreducible Representations and Characters 2.5

we have

(for the last equation d. 1.2.15; we use eESP(n), and note 1.2.10). Applying this to

we find ;-0."

~E-+-

=

= 12 (;-0.. 2::v/nee'X)

;-o.C;C

~E~

~f

/

for a suitable x E [0, 2?r). (v) e=h(a). Supposing e=l=h(a), we obtain from (iii) 1'0.+ - [;-0. ~h(o.)+ -Dh(n\

-

-

I (

'2 -

1)(n-kl/2

lf1I

E'IoI!\

7L ,

which cannot happen, since a character value which lies in Q is an integer. (vi) e 'X =Sh(a)' If ch(n)" is ambivalent, then Sh(~l" EIR, so that we have x=O, i.e.

for if o:=h(a),

then by 2.5.12,

=1,

where the last equation follows from 1.2.13. If C h ( 0)" then ;-0.+

is not ambivalent,

_;-0.+

~h(n)+-~h(n)-'

so that for o:=h(a)

and by 1.2.13 we get

This completes the proof.

•

71

2.5 Ordinary Irreducible Representations and Characters of An

The character table of A 3 therefore is (3)

I

1

[3] 2.5.14

(2,1) -

(2, It

[2,1]+

1(-I+/3i)

1(-I-/3i)

[2,1]-

-H-I-/3i)

1(-I+/3i)

Also for the alternating group we need a few results concerning the ordinary irreducible representations of low dimension. 2.5.15

THEOREM.

(i) For n=f'3,4 the representation [nUA n =[lnUA n is the only onedimensional ordinary representation of An (while [2, I] "', [22 ] Oc are other one-dimensional representations of A 3 , A 4 ) . (ii) For n =f'5, the lowest dimension =f'l of an ordinary irreducible representation of A n is n - I (while for n: = 5 we have the representations [3,1 2 ] = of dimension 3). For 2";;;n=f'3,6, [n-I, IUAn =[2, I n- 2UA n is the only ordinary irreducible representation of A n which is of dimension n - I (while for n: = 3 we have no such representation, and for n: =6 we have both [5, lUA 6 = [2, 14UA 6 and [32UA 6 =[2 3UA 6 of dimension 5). Proof (i): The statement is seen to be true for n,,;;;4 by inspecting the character tables of S,, n,,;;;4,and recalling 2.5.7. From 2.4.10 we get that for n~5 there is no a~n such that =2; in particular no such a exists with a=a'. The statement therefore follows from 2.5.7. (ii): Since 2.4.10 is true, we need only show that for 2,,;;;n =f'3,6 there are no partitions a~n with the properties

r

a=a'

and fa=2n-2.

That this holds for n";;; 8 we can see by looking at the character tables. Let us therefore assume a~n~9,a=a', andfa=2n-2. (a) If [aUSn-l is reducible, then it is of the form [a ] t s, - I = [ /3] + [ y ],

where ff3 = f 'Y = n - I. This follows from 2.4.10, since a = a' implies that no one-dimensional constituent can occur. But 2.4.10 also gives

/3,yE

{(n-2,

1),(2, In-3)},

72

Irreducible Representations and Characters of Symmetric and Alternating Groups 2.6

so that by the branching theorem

which contradicts a=a', for we assumed n;;;'6. (b) If [aHSn-l is irreducible, then a is rectangular, say a=(r a=a', we have r=s;;;'3, and therefore

Both these constituents are of dimension ;;;'(n-2)+3 contradicting j " =2n-2.

S ) .

Since

(apply 2.4.1O(iii)), •

2.6 S, is Characterized by its Character Table We denoted the character table of S, by Zn:

where i is the row index, k the column index, and i and cp, with II>between the sets of irreducible representations of S, and Gover C, and cp between the sets of conjugacy classes of S, and G, which have the following property: Putting

then for each a, f3~n 2.6.2

ti

denoting the value of the character of [aj' on ~i. Our method of proof will be to show that the representation [n - I, I] maps G isomorphically

73

2,6 S, is Characterized by its Character Table

onto a group of reflections which by its order must be isomorphic to Sn' (The theorem is trivial if n = or I, so that we may assume (n - L I )~n.)

°

(i) (n-I,

I] is a real representation.

2.2.19(iii) implies

[n - 1, I] + [n] = IS
= a I ( "IT)

-

1;:;;.- 1.

Thus by 2.6.2, for each g E G r
(g);:;;'-I,

so that 1

2.6.3

~ r
_

IGJgEe

A

(s")

-1,

for we can write > - 1 instead of ;:;;.- 1, since )'(n-)

~
, I)

=)'(n-l,l)=n-l>-I ~
.

It is well known that for any ordinary irreducible character expression

f

of G the

2.6.4 can take the values -1,0, and 1 only, depending on whether a representation with character f is not equivalent to a real representation, but has real character, or has a nonreal character, or is equivalent to a real representation. Since r4). Let us denote by a, a the class sums of C", c-, respectively. It is well known that the character table of G yields the coefficients aapy in

e e

eaep= ~ aapye y. yen

74

Irreducible Representations and Characters of Symmetric and Alternating Groups 2.6

This shows that the following equation (which holds for Sn, n;;.4):

gtves

The character table determines the orders of the conjugacy classes (apply the orthogonality relations), so that for n;;.4: (i)

IC(I")'I=IC(ln)I=I, 2

2.6.6

n

(iii) IC(3.1 (iv)

v

4

(ii) IC(2 y

)

,

3 )

1=IC(2'y

4JI=

n(n-I)(n-2)(n-3) 8 n(n-I)(n-2) 3

1=IC(3.1n-J)I=

,

'

'

n(n-l)

IC(2.1n-2) 1=IC(2.1n-2)I= 2

.

Let C denote the conjugacy class of G, which consists of the squares of the elements in the class C of G:

Since :ve have for their orders and the orders of the centralizers of g E

g2EC

C and

ICI=ICIICc(g2)1 ICc(g)1' ICI divides ICI·

n

,

2.6.5 shows that the class of squares of elements g E C(2.I - ' ) must be n 4 n 3 either C(I")',C(2',l - ) ' , or C(3.1 - ) • But 2.6.6 shows that for n>4, C(I")'is the only one whose order divides

IC'(2

In

- 2)'

1=

n(n-l) 2

.

Together with the corresponding [a], some of the [a] we have

are faithful, so that

75

2.7 Cores and Quotients of Partitions

and hence we can conclude

{g2/gEC(2Y')V}

= {IG}'

so that C(2. J"-')v in fact consists of involutions. 2 (iii) [n -1. 1( g), g E C(2y- )v , is a reflection (if n >4). In (i) we saw that [n - I, I( is a real representation of G, so we may assume that for each gEG the mapping [n-I, 11' (g) is a real orthogonal one. The eigenvalues of such matrices are of absolute value 1. Since by (ii) C(2.1"-')v n 2 consists of involutions, [n - I, I(g), g E C(2.l - )v , has eigenvalues ± I (since [n-I, Ij' (g)2 =id) ,only. But the sum of the eigenvalues of [n -1. II' (g) is its trace ~("-I.I) (g), for which we have by 2.6.2

Hence [n-I, II'(g), gEc(2.1" 'lV, has exactly n-2 eigenvalues + I and one eigenvalue - I. Thus each such mapping is a reflection. (iv) G~S"ifn>4. The character tableZ, of G yields the coefficients a;:,i:;,from which we can s~e that C I l . 1" ' ) generates G. Hence by (iii) for n>4 the image of [n-I, I] is a faithful representation of G as an irreducible group of reflections. These groups are known (see e.g. Benson and Grove, [1971, Theorem 5.3.1]), so that we obtain from IGI=IS"I=n! that

(v) G ~ S" if n ~ 4. This follows by inspecting the groups of orders I, 2, 6, 24 and their character tables. •

2.7

Cores and Quotients of Partitions

Let us return to an examination of the values ~"( 7T) of an ordinary irreducible character of SIloThe Murnaghan-Nakayama formula 2.4.7 allows us to evaluate ~"( 7T) recursively by removing parts R~/ from the rim of [a] and then from the rim of [a]\R~j' and so on. The lengths of these parts which have to be removed correspond to lengths k, I, and so on of cyclic factors of 7T,so that the evaluation of ~()( 7T)comes down to the evaluation of character values of S,,-k,Sn-k-l' and so on. We ask what can be said if we restrict attention to a particular cycle length, say q. While a hook H/} of length q will be called a q-hook, the corresponding part R~j of the rim of [a] will be called a rim q-hook. It is very natural to ask whether the process of removing rim q-hooks from [a] always ends up with the same diagram which does not contain any further q-hook.

76

Irreducible Representations and Characters of Symmetric and Alternating Groups 2.7

Diagrams [a] which do not contain any q-hook are called q-cores, a name which we shall also use for the partition a. For example the zero diagram [0] is the only l-core, while the 2-cores are just the triangular diagrams: [0] and the diagrams

x,

x

x,

X X

X

2.7.1

X X

X,

X

X X X

X X X

X X

X,

X

For q> 3, the complete set of q-cores is rather complicated to describe. We ask whether we obtain a uniquely determined q-core

[a]

2.7.2

from [a] by successive removals of rim q-hooks. We shall shortly see that this is indeed the case. For example the 3-core of [7,5,4,3,2] is [4,2], although the pictures below indicate at least two different ways of reaching it:

X

X

X

X

X

X

X

X-X

X

X-X

I

X-X-X

X

X

X

X

X-X-X

and X-X

X

X-X

I

I X

X-X-X

X-X

I

X

X

I X

I X

X

I X

X

I X-X

In order to prove that we always finish up with the same q-core, we shall give an algorithm which yields this core, and first of all we need an algorithmic description of the removal of a rim q-hook. It is clear that a diagram [a] = [a" ... , (lh] «(lh 7'=0) or its partition a is uniquely determined by its first-column hook lengths 2.7.3 For example, if we are told that h 4 : =2, h 3 : =4, h 2 : =5, and hI: =7 are all the first-column hook lengths, then we deduce that a = (4,3 2,2). The method of reconstructing a is simply to use the equations 2.7.4

ah-

1 =hh-l-

l , ... ,

Notice that in this way we get a partition (l = (aI' .... aT) from each

77

2.7 Cores and Quotients of Partitions

sequence of strictly decreasing nonnegative integers

2.7.5

/3, >/3 2 > ... >/3r

by putting

2.7.6

1.;;;i';;;r.

If /3r=0, we get a sequence having some zero parts a i at the end, which does not matter. Thus from 9,7,6,4,1,0 we obtain a=(4,Y,2), a partition which we also obtain both from 8,6,5,3,0 and from 7,5,4,2. We now formally define this generalization of first-column hook lengths. Let ~ denote a composition of some integer. In accordance with 2.3.8 we may define a generalized representation of a symmetric group by putting

2.7.7 subject to the conventions 2.3.5. This equals the determinant of a finite matrix, since the subdeterminant 2.7.8 does not depend on r provided that r~max{il~i*O}. It is clear that for each such r, 2.7.8 is uniquely determined by its last column, i.e. by the r-tuple

(x,-1

2.7.9

+r, ~2 -2+r,

... , ~r -r+r).

Each such sequence is called a sequence of /3-numbersfor ~. 2.7.3 shows that this concept generalizes the concept of first-column hook lengths of partitions. Now, if (/31'/32 , ... , /3r)is a sequence of /3-numbers for ~ and

then [fL]=

-[~].

Also,

is a sequence of /3-numbers for fL. (All we have done is exchange the ith and U+ l)th rows of the determinant; d. 2.3.9.) This means that [~] vanishes if two /3-numbers for ~ are equal. The same happens if some /3-number is negative. In all other cases, there is a unique permutation '1T such that ;;"0. If (/31T(I)' ... , /31T(r)is a sequence of /3-numbers /31T(1) > /31T(2) > ... > /31T(r)

78

Irreducible Representations and Characters of Symmetric and Alternating Groups 2.7

for a, then 2.7.10

af-~;\j

and

[;\]=sgn7T·[a].

The sequences of /3-numbers of partitions have the advantage that they are strictly decreasing: 2.7.11

a, -1 +r>a

2

-2+r>

... '>«, -r+r.

They can therefore be replaced by the set of these numbers, i.e. by 2.7.12

{a j -1

+ r, ... , aT-

r+ r}.

Each such set will be called a set of /3-numbersfor a. Thus each finite set of elements of ~ 0' being a set of /3-numbers, yields a sequence of /3-numbers and in this way a partition. Now the proof of 2.4.7 by an application of 2.4.5 shows clearly that the following is true: 2.7.13 LEMMA. Removing a rim q-hook Rfj from [a] means for each of its sequences /3" ... , /3rof /3-numbers that a suitable /3k is changed into /3k - q, and the resulting set {/3" ... ,/3k-,,/3k-q,/3k+I, ... ,/3r} is then a set of /3-numbersfor [a]\Rf}. And concersely:if /31' ••• , /3ris a sequence of /3-numbers for [a] such that for a suitable k we have O,,;;;;/3k -q=l=/3j,for all i=l=k,then {/31'... , /3k- l' /3k - q, /3k+I' ... , /3r} is a set of /3-numbersfor a diagram [y] which arisesfrom [a] by removing a rim q-hook.

/3-numbers can be conveniently recorded on an abacus. Imagine that we have an abacus lying on a table with the runners going north-south and the abacus is viewed from the south:

(

> ) )

( (

We assume that there are q runners, called the Oth runner, lst runner, ... ,(q-l)th runner, from left to right and that the length of the runners and the supply of beads are both sufficiently large for the abacus frame not to interfere with our calculations. The possible bead positions are determined by assuming that all the beads are initially at the top and that

79

2.7 Cores and Quotients of Partitions

we move beads only through one bead width at a time. Label the bead positions as below:

o q

q-2 2q-2

I

q+l

q-l 2q-l

For example, if q: = 3, we have 012 345 678

A bead configuration is associated with a set of f3-numbers (and hence a diagram or a partition) by letting the actual bead positions determine the f3-numbers. If again q: = 3, then the bead configuration

These are corresponds to the set of f3-numbers {0,1,2,5.6,9,11,13,17}. f3-numbers for the diagram whose first-column hook lengths are 2,3.6,8,10,14, that is, for the diagram [9,6.5,4,2 2 ] . We notice the following: 2.7.15. Given a bead configuration, the quickest way to find the first-column hook lengths of the corresponding diagram is to count the first gap as 0, and count on from there. We are now in a position to prove the desired theorem, for any q and every af-n there exists a uniquely determined is obtained from [a] by successive removals of rim q-hooks do this. no matter which of the removable rim q-hooks resulting diagrams we remove first: 2.7.16

THEOREM.

which says that q-core [ti] which as far as we can of [a] and the

Each diagram has a uniquely determined q-core.

80

Irreducible Representations and Characters of Symmetric and Alternating Groups 2.7

Proof By 2.7.13, removing a rim q-hook is equivalent to sliding a bead one space up. The bead configuration corresponding to a q-core of the diagram occurs when all the beads are as high as they will go. This configuration is clearly independent of the order in which we slide the beads. •

The 3-core of [9,6,5,4,2 3-core is (cf. 2.7.14)

2

]

is [3, 1], and the bead configuration of the

000 000

o

0

o

The partition a of the q-core [a] of [a] will be called the q-core of a. The number of rim q-hooks which must be removed from [a] in order to obtain [a] is called the q-weight both of [a] and a. An immediate corollary of the proof of 2.7.16 is 2.7.17 THEOREM. The number of partitions of n with q-core a depends only on the q-weight w of a and is equal to b(w):=

~

p(w o ) " 'p(Wq~I)'

(wO.···.Wq-I)

where the sum is taken over all the q-tuples (w o, ' " Wq~ I) of nonnegative integers such that ~i~d Wi= w, and where again p( w;) denotes the number of proper partitions of Wj'

Let us return to formula 2.4.5 and see how we can use this concept of a q-core in the evaluation of character values. For this purpose we introduce numbered bead configurations by giving each of the beads one of the

numbers 1, ... , r, if the bead configuration in question consists of r beads. When the beads are numbered in one way or the other, the bead configuration corresponds to an injective r-tuple (lJ' ... , Ir ) of nonnegative integers, Ii denoting the place number of the ith bead. For example, if we number 2.7.14 as follows:

then the corresponding r-tuple is (2,1,0,6,5,9,13,11,17).

81

2.7 Cores and Quotients of Partitions

Given such an r-tuple (II' ... , Ir), we define the r-tuple A by Ai:= Ir- i + 1r+i (l ~i~r). Then [A]*O. For example, 2.7.18 gives A=(9,4, 7,4,1,3, -2,0,2). Forgetting the numbering of the beads, we obtain the set {l!' ... , I.} and hence the corresponding partition a which satisfies [a] = ±[A]. The sign stems from the necessary exchanges, and it is equal to the sign of the permutation which maps the numbering of the beads which led to A onto the natural numbering of the beads in the configuration of [a]; e.g., the natural numbering of 2.7.14 is

Bearing this in mind, we consider an element p7TESn,p:=(i"oo.,i q) a q-cycle, 7TES n , leaving iI' i q fixed. If A is a composition of n, then by 2.4.5 00"

2.7.20

X h( p7T) = ~X/L( 7T), /L

where the sum is taken over all the compositions fL of n - q arising from numbered bead configurations obtained by moving exactly one bead one step upwards in the numbered bead configuration of A. If we start this procedure with the naturally numbered bead configuration for a partition a of q-weight w, and apply it w times, then we always end up with the same numbered bead configuration, no matter in which order we move the beads. This final bead configuration is still a numbered configuration if we keep the numbers when moving the beads, and it arises from a permutation a of the numbers in a naturally numbered bead configuration of Ii which contains the same number of beads. Let us illustrate this by an example before we go on.

82

Irreducible Representations and Characters of Symmetric and Alternating Groups 2.7

The configuration 2.7.19 yields

0 2.7.21

1

O2

Os

08

03 04 0' 7 09

06

while the naturally numbered bead configuration of the 3-core 2

[9,6,5,4,2

]

= [3,1]

IS

OJ 04 07

2.7.22

O2 Os

0 0 0

3 6 8

09 This yields, by comparing 2.7.21, 22, 2.7.23

a=

(~

;

3 3

4 5

= (45876).

5 8

6 4

7 6

8 7

~)

Let us denote the number of ways of moving the beads upwards step by step as far as they will go by 2.7.24 Then the considerations above show that if p consists of w q-cycles while acts on the remaining n - qw symbols,

1T

2.7.25 Now repeated applications of 2.4.15, the Murnaghan-Nakayama for skew diagrams, together with the foregoing arguments, yield

formula

2.7.26 so that we have proved the following:

2.7.27 THEOREM. If a~n is of q-weight w, and p E S, a product of w q-cycles, while 1TE S, acts on the remaining n - qw points, then

83

2.7 Cores and Quotients of Partitions

and

We shall show next how fqCY.can be evaluated. In order to do this we have to be a bit circumstantial, but we shall be repaid by meeting with another very interesting structure. We remember that fqCY.is the number of different ways in which a can be obtained from a by successive removals of rim q-hooks, i.e. by successive moves of beads in the corresponding bead configuration. But the beads are distributed into q runners. Examine each runner in turn. Since all diagrams on w nodes have s-weight w, it is clear how a bead configuration on one runner corresponds to a diagram. For example the three bead configurations

2.7.28

correspond to the diagrams [12 ], [3], [2, I]. Therefore we may naturally associate a bead configuration on an abacus having q runners with a sequence of q diagrams, by examining each runner in turn. Suppose, for the moment, that we impose the condition that the total number of beads on our abacus is a multiple of q. This ensures that each q-core is recorded on an abacus which is unique to within adding a constant number of beads to each runner. Suppose that to obtain the bead configuration corresponding to [a] from the bead configuration corresponding to its q-cote [a), we make a series of moves on the ith runner corresponding to the diagram [a(i)],O";;;i";;;q-1. Then the q-tuple of these diagrams, 2.7.29 is called the q-quotient of [a]. For example 2.7.14 and 2.7.28 show that [9,6,5,4,2 W]o[3],[2, lh. The proof of 2.7.16 gives 2.7.30

2

]

has 3-quotient

THEOREM. A diagram [a] is (for each qEf\!) uniquely determined by the pair ([ a]; (a]q) consisting of its q-core (a] and its q-quotient (a ]q.

84

Irreducible Representations and Characters of Symmetric and Alternating Groups 2.7

The desired number /qaof ways of getting [ti] from [a] is closely related to the q-quotient [a]q' For we recall from the proof of 2.7.16 that (if Wi denotes the number of nodes of [aU»)), this number is equal to W

(

wo , ... , wq _

w!

) I

-

W , ..•

o·

W

,

q-I'

times the product over i of the number of ways of moving the beads on the ith runner up as far as they will go. But by definition of the q-quotient [a]q of [a], this number of ways of moving the beads on the i th runner up as far as they go is equal to the number of ways [aU)] can be brought to [0] by removing a rim l-hook, On the other hand, the branching theorem yields that this number of ways of bringing [aU)] to [0] by removing l-hooks is equal to the dimension, 2.7.31 of [aU)], so that we finally obtain 2.7.32

2.7.33 COROLLARY. Let a~n be of q-weight w, and [a]q denote the q-quotient 1. If p consists of b q-cycles, of[ a], where [a(I)]consists of Wi nodes, O";;;;;i";;;;;qbssw, and 1Tacts on the points fixed by p, then

r (p1T) = { 0sgn a

(7 (

W

wo,,,,,w

q_

1

)f(O) .. 'f(q-

I))'';

~

(1T) if b=w, if b>w.

We wish next to give a pictorial description of the q-quotient. In order to do this we introduce the q-residue of the (i, j )-node of [a], which we define to be the least nonnegative integer r congruent toj-i modulo q:

r =j - i (mod q ),

2.7.34

For example, the 3-residues of [9,6,5,4,2

o 2

1 0 2 1 0

1

2

2 1

o

2

0

1

2

o

1 0

2

2

o

1

O";;;;;r
]

2 1

are shown below:

o

2

85

2.7 Cores and Quotients of Partitions

By construction, as we work round the rim of [a] from top to bottom, the q-residue decreases by 1 as we go from one node to the next. Thus: 2.7.35. The length of a hook is divisible by q iff the difference of the q-residues of the hand node and the foot node ==-1 modulo q.

Given a diagram [a], draw lines through the rows whose last node has q-residue i and through the columns whose last node has q-residue congruent to i+ 1, O~i~q-l. Then by 2.7.35 the intersections of these lines mark just those nodes of [a] which are comers of hooks of length divisible by q and which have hand nodes of residue class i. For example, if q: = 3, we get for i=O, 1,2 and a: =(9,6,5,4,2 2 ) 2.7.36

°

120 1 12 0 2 0 12

T° l

2

2

0

0

1 0 2 1

1 0

2 OIl 2 0 1

o1

012 2 0

+-r

It is clear that the intersections of these lines give nodes forming a diagram shape. Call the q-tuple of these diagrams the star q-diagram o/[a]. As 2.7.36 shows, the star 3-diagram of [9,6,5,4,2 2 ] is equal to the 3-quotient of [9,6,5,4,2 2 ] , and in fact the following is true:

2.7.37

THEOREM.

The star q-diagram is the same as the q-quotient.

Before proving this we need a couple of lemmas. 2.7.38 LEMMA. Provided that our abacus has a multiple of q beads (i.e our set of fJ-numbers has cardinality divisible by q), a bead on the i th runner corresponds to a row whose last node has q-residue i. Proof

(i) Suppose that a has precisely h nonzero parts, and the first gap in its abacus configuration occurs on the kth runner. Since the total number of beads is divisible by q, we have k+h==O (mod q).

86

Irreducible Representations and Characters of Symmetric and Alternating Groups 2.7

(ii) Suppose that hi corresponds to a bead on the ith runner. Then hi =i-k (mod q) (see 2.7.15). Therefore i=hi
i=hi -h=a

j

-j

(mod q),

•

as required.

2.7.39 LEMMA. The effect of removing the first column from [a] is recorded on the abacus by inserting a new bead in the first gap. This is clear from 2.7.15.

Proof of Theorem 2.7.37. Notice that if we count down a runner, the numbers we get being determined by the position of the beads on that runner, we obtain a set of f3-numbers for the diagram corresponding to the configuration on that runner. (Counting down the first runner in the example 2.7.14 gives 0,2,3 which is a set of f3-numbers for [12].) Choose an abacus for [a] which contains a multiple of q beads, and suppose that the first gap in the abacus configuration occurs on the k th runner. Adjoining a new bead in the first gap removes the first column from [a] (by 2.7.39) and from the kth part of the q-quotient (by the first paragraph of the proof). But removing the first column of [a] removes the first column of the kth part of the star q-diagram of [a] (by 2.7.38), and the result follows by induction. • Incidentally, Theorem 2.7.37 shows that

2.7.40. The q-weight of a diagram equals the number of hook lengths divisible by q. In particular, [a] is a q-core iff no hook length is divisible by q. We conclude this section with a combinatorial result which will be useful when constructing decomposition matrices of symmetric groups. Two diagrams, [a] and [f3], are said to have the same q-content if for each r, the number of nodes of [a] having q-residue r (cf. 2.7.34) equals the number of nodes of [f3] having q-residue r. Thus, for example, if

o12

0 1 2 0 2 0 1 2 0 120 1 [a]:O 1 2

2 0

o 1 201 20120 120 [f3]:0 1 2

2 0 1 2

o

o 1 201 2 0 1 2 0 120 1 [y]:O 1 2

2 0 1

o

then [a] and [f3] have the same 3-content (each contains eight O's, six I's,

2.X Young's Rule and the Littlewood-Richardson Rule

87

and seven 2's), but [y] has a different 3-content ([y] contains eight O's. seven ls, and six 2's). The discrepancy occurs because [a] and L8] have 3-core :4.2], while the 3-core of [y] is [22,1 2 ]. 2.7.41 THEOREM. Suppose that a and {3 are partitions of the same integer. Then [a] and [{3]have the same q-core if and only if[a] and [{3]have the same q-content.

Proof A rim q-hook contains precisely one node of each q-residue. But the q-core of a diagram is obtained by removing rim q-hooks as often as possible. If [a] and [{3] have the same q-core, they therefore have the same q-content. The other implication is not as easy. Suppose we have an abacus containing the same number of beads on each runner; in particular, our abacus has a multiple of q beads. The configuration where all the beads are at the top corresponds to the empty diagram. Now build the diagram [a] by adding one node at a time. Adding a node whose q-residue is i is recorded on the abacus by removing some bead from the (i - 1)th runner and putting it the same distance south on the ith runner (with the obvious modification to this rule if i=O), by 2.7.38. If [a] contains x nodes whose q-residue is i, and y nodes whose q-residue is i + 1, the number of beads on the ith runner has therefore increased by x - y, by the time we reach [a]. On the assumption that [a] and [{3]have the same q-content, we deduce that the number of beads on each runner is the same in the configurations for [a] and [In But, as we have already remarked, the q-core of a diagram is determined by the number of beads on each runner. Therefore, [a] and [{3] have the same q-core.

•

2.8 Young's Rule and the Littlewood-Richardson Rule The determinantal form 2.3.15 has shown us the way to write the irreducible character ~a in terms of permutation characters We now investigate how to reverse this process, and write as a sum of irreducible characters. Having done this, we shall seek the irreducible constituents of (see 2.3.2)

e

2.8.1

e.

[ a] [ {3] =[a] # [ {3] t s, + n '

com.Bvn, [a]#[{3] the corresponding irreducible representation of SmXSn •

An algorithm, the so-called Littlewood-Richardson rule, will finally show us how the diagrams of the irreducible constituents [y] of 2.8.1 can be constructed from the diagrams [a] and [{3]. First we consider particular cases.

88

Irreducible Representations and Characters of Symmetric and Alternating Groups 2.8

2.8.2 YOUNG'S RULE (SECOND VERSION). [a][n]=~[y], where the sum has to be taken over all partitions y~m + n such that for each i we have a i ~Yi ~ai-l (If we put aD: = 00).

Proof. By 2.4.16 ([a][n]'[ y ])=(x y / a, ~(n») =

(xy/a, ~(n»),

•

so that an application of 2.3.13 yields the statement.

2.8.3 COROLLARY. The diagrams [y] of the irreducible constituents of[a][n] may be calculated by adding n 2' s to the diagram [a] in all possible ways such that no two 2' s appear in the same column.

For example if a:=(3,2 2 ) and n:=2 we obtain: X X X

X X X

X 2 2

X X X

X X X

X 2

2

X X X

X X X

X 2

X X X

2

X X X

X

2

X X X

X X X

X

2 2

2

Thus

We can apply this process repeatedly in order to get the constituents of 2.8.4 For example, if we want to calculate [3][2][1], we first evaluate [3][2]: 122

1

1

1 1 1

2

2 2

2 This yields [3][2] = [5] + [4,1] + [3,2],

so that the constituents of [3][2][ 1] are obtained as follows: 111223

1 1 1 2 2 3

1 1 1 2 3 2

1 I I 2 2 3

1 I I 3 2 2

I I I 223

I 1 2 2 3

I 1 I 2 2 3

89

2.8 Young's Rule and the Littlewood-Richardson Rule

Therefore [3][2][ 1] == [6] + 2[ 5,1] + 2[4,2] + [4,1 2 ] + [3 2 ] + [3,2,1]. This algorithm justifies the introduction of tableaux having repeated entries. A generalized Young tableau of shape [a], a~n, and content '\,'\F n, arises

from [a] by replacing the nodes of the diagram by positive integers in such a way that the integer i occurs exactly '\; times; e.g. I 3

2 2

3

2

is a generalized Young tableau of shape (6,2) and of content (3 2 ,2). A Young tableau is therefore a generalized Young tableau of content (In), for a certain n. A generalized Young tableau is said to be semistandard if the numbers are nondecreasing along each row and strictly increasing down each column. For example

I 3

1 3

2

2

2

1 I 2 3

2

2

3

1

I

233

2 2

are the only semistandard tableaux of shape (6,2) and content (3 2 ,2). In terms of this concept we can formulate YOUNG'S RULE (THIRD VERSION). The multiplicity ([ 13 d ... [13 k ],[ a]) of [a] in ISfJ1Sn equals the number of semistandard tableaux of shape a and of content 13.

2.8.5

More generally the constituents [a],

a~m

+ n, of the representation l3~m,

8Fn,

are obtained by first adding to the diagram [13] 8] nodes in all the admissible ways, then to each one of the resulting diagrams 82 further nodes, and so on. This motivates the introduction of skew tableaux and of generalized skew tableaux. They arise from skew diagrams [a/ 13] if we replace the nodes by natural numbers; in the case of generalized skew tableaux repetitions may occur. When we are dealing with generalized semistandard skew tableaux we simply speak of semistandard skew tableaux. Let

2.8.7 be the number of generalized semistandard skew tableaux of shape a / 13 and

90

Irreducible Representations and Characters of Symmetric and Alternating Groups 2.8

of content A. For example, by Young's rule 2.8.8

If we are given a generalized skew tableau, say

2.8.9

I 2

2 3 we can read its entries from right to left in each row and one row after the other, downwards, obtaining a sequence of positive integers. For example 2.8.9 yields 2.8.10

2

3 2.

Such a sequence is called a lattice permutation if and only if the following holds: For each place} of the sequence the number of i's which occur among the first} elements is greater than or equal to the number of (i+ 1)'s, for each i. Hence 2.8.10 is a lattice permutation. For the sake of simplicity we now allow the nodes of a diagram or a skew diagram to be replaced by elements of an arbitrary totally ordered set. We prove a combinatorial lemma which will tum out to be crucial: 2.8.11 LEMMA. Assume a~nl +n z +b, n i , bEI\J, /3~nz, y~nl' Then the number of semistandard tableaux of shape alY which contain /3 i numbers i, I ";;;i,,;;;/3;, together with b symbols x such that the points i form a lattice permutation (if we read their entries along their rows from right to left and the rows downwards), is the same if the order is defined by

x
1<2< ...
Proof For O,,;;;k,,;;;/3; and the corresponding order 1< ...
the set of semistandard tableaux of shape alY which contain /3i numbers i, I ";;;i,,;;;/3;, together with b symbols x. The subset of M k which consists of such tableaux the points i of which form a lattice permutation will be

91

: ' Young's Rule and the Littlewood-Richardson Rule

ndicated by

Our method of proof is to construct bijections

\\hich satisfy

that the statement immediately follows by definition of Alli and A o from the fact that the I" are bijections. '0

(i) The bijections. We define a mapping I" on M" as follows: If TEM". then I,,(T) arises from T in the following way: (a) In each column of T which contains both x and k, we interchange x and k (remember that Tis semistandard, so that each column of T contains x at most once and k at most once, and k occurs just above x if the column contains both x and k), and afterwards (b) in each row containing both x's and k's left fixed under (a), we shift the x's to the left of the k's (remember that Tis semistandard, so that the x's occur in T just to the right of such k's). For example in the following situation: ~k

T:

~

-lor empty k ... k

x ... x

k x

k

I

k

k

x

I or empty xl' - - - - -;;;,k+ --------

x k

x k

x

we get

x ...

X

k ... k

X

x ... x k ... k

Obviously Ik(T) EM"_I' so that k M k -> M k- 1• I" is a bijection, since it is inverted by gk: M k- 1 -> M k , which is defined by its operation on an element of M"-l as follows. (a') Interchange x and k when they occur in the same column, and then (b') in each row which contains both x's and k's fixed under (a'), shift the x's to the right of the k's.

92

Irreducible Representations and Characters of Symmetric and Alternating Groups 2.8

(ii) 1,0 ... 0 Ipi [Api] CAo' Take a Tpi EA pi ' Suppose, inductively, that the i's in each one of the following tableaux form a lattice permutation:

We have to show that this is also true for

(Note that we do not prove/dAk]CAk_l; this is false.) The definition of Ik shows that the lattice property holds for all the numbers in Ik(Tk ), except, perhaps, between the numbers k and k+ 1. In particular,

/PI (Tp; ) EA pi - , . We may therefore assume that k<{3;. Suppose that there is a k+ 1 in the ith row of Tk • Then the required lattice property in Ik( Tk ) is true for this k + 1 unless there is an x to the left of it in the same row of Tk and this x lies below a k. We must examine the latter case more closely: jth column

t Tk:

~ k - 1 Ik ... x

x

k x

."-y-'

>0

k.. .

k

k+l. .. k+l

x. . .

X

X ...

k+l...k+ll ;;;.k+2

'---v------' '---v------'

a

~ithrow

b

Every k+ 1 in the ith row of Tk has a k or x immediately above it, since k
Irei', j')lk is in the (i', j') place of Tk and i'j} I

-I {(i', j')lk+ 1 is in the (i', j') place of Tk and i'j}l;;;.b. Therefore, (the number of k's in the first i-I rows of Ik(Td) -(the number of (k+ l)'s in the first i-I rows of Ik(Tk));;;.b+a. This shows that the

: R Young's Rule and the Littlewood-Richardson Rule

93

lattice property holds for all the (k+ l)'s in the ith row of fk(Tk ), and completes the proof of (ii). • (iii) g~; 0 • • • 0 g][Aol CA,B; follows quite analogously. 2.8.12 COROLLARY. The multiplicity ([.8][8d·· ·[8,1,[yJ), .8~m, 8~n, y~m+n, equal to the number of semistandard tableaux of shape y which contain .8i srmbols i, 8j symbols j~ 1,,;;;, i ~.8;, 1~j";;;' I, subject to the ordering 1< ... < 1< \< ...
15

Proof The considerations above have shown that

which is the number of sernistandard tableaux of shape y / /3, of content 8, i.e. containing 8j symbols}, 1~}~/. Now the generalized Young tableau

1·· 2

\ 2

/3; ... /3; is the only sernistandard one of shape and content .8 where the i's yield a lattice permutation. Thus k y /,B,13 is equal to the l!umber of sernistandard tableaux of sh_ape y w~ch contain .8i i's and 8j} 's, subject to the order 1< ... <.8; < 1< ...
Proof Let ga,By denote this number of semistandard skew tableaux. 2.8.12 yields, if 8~m, .8~n, y~m+n,

a~m

But afm

94

Irreducible Representations and Characters of Symmetric and Alternating Groups 2.8

so that for each c5~m:

~ kallga/ly=([c5l] .. '[c5,][fj],[y)) = ~ k all ( [ a ][ fj ], (y ]).

The statement now follows from the regularity of the matrix (k all ).

•

2.8.14 COROLLARY. The diagrams of the irreducible constituents [y] of[ a ][/1] (counted with their multiplicities ([a][fj],[y])) are obtained as follows. In all possible ways, add to a fixed tableau of shape [a] semistandard tableaux of content fj which yield lattice permutations; the resulting tableaux will have the shapes [y] of the constituents of [a][ fj]. An alternative procedure is to take a symbol aij for each i and) for which the (i, i)-node belongs to [fj], then (i) Add to the diagram [a] all the symbols alj" After the additions no row of the compound tableau may contain more nodes than a preceding row. (ii) Next add all the symbols a 2 ), according to the same rule, and so on, until all the symbols have been added. (iii) These additions must be such that for all i, if y<}, a iy goes in a later column than ail' and for all}, if x
[2,1] :

a l2

,

a 21

to [3,2]. From the first row, keeping in mind 2.8.14(i) and (iii), we get X X X al2all XX

X X X all XXa l2

X X X XX

all

X X X XXa l1

a l2

a 1Z

X X X XX

a lZ a l1

To these compound tableaux we add a 21 in all possible ways subject to 2.8.14 (ii) and (iii): XXXa l2 a 11 X X a Z1 X X X all X X a Z1

a l2

X X X al2all XX

a Z1

X X X all X X a lZ a Z1

X X X all X X alZ

a Z1

X X X all XX

X X X all XX

XXX X X all

alZ a Z1

a l2

alZ aZl

a Z1 XXX X X all

alZ a21

XXX XX

a l2 a l1 , a21

:

~

95

Inner Tensor Products

", that the required reduction is given by [3,2 ][2, I] = [5,3] + [5,2,1] + [4 2 ] + 2[ 4,3, I] + [4,2 2 ] + [4,2, 12 ] +[3 2 ,2]+[3 2 ,1 2 ]+[3,2 2 ,1].

2.9

Inner Tensor Products

The inner tensor product :.9.1 "f two ordinary irreducible representations [a] and [13] of Sn is an ordinary ~t'presentation of Sn which is in general reducible, so the question arises how

evaluate the decomposition of 2.9.1 into its irreducible constituents. No ',itisfactory answer to this question is known, but we can at least show how w tackle it by applications of the determinantal form and the LittlewoodRichardson rule. The determinantal form yields for the character ~a of [a]

:l)

:.9.2 1T

'0

that the character ~a0 ~fJ of [a][ I3l satisfies

2.9.3 1T

"ow as ~a-id+1T is induced from Sa-idh (recall 2.3.3), a well-known iormula on tensor products of induced characters (see e.g. Curtis and Reiner [1962, (38.5)]) yields 2.9.4 ~o

that 2.9.3 implies

2.9.5 1T

But the decomposition of ~fJ t Sa-idh can be evaluated in principle by applications of the Littlewood-Richardson rule, and induction from this decomposition into Sn can be performed using the LittlewoodRichardson rule again. Hence the whole procedure can be carried out mechanically; we do not need the character table. An example illustrates this. We would like to evaluate the decomposition of ~uitable

2.9.6

[3,2,1]0[3,2, I].

96

Irreducible Representations and Characters of Symmetric and Alternating Groups 2.9

First of all we remember from Section 2.3 that

2.9.7

[3]

[4]

[5]

[3,2,1)= [1)

[2] 1

[3] [1)

o

= [3][2][1] - [3][3]- [4][1][1] + [5][1), so that we have to decompose the restrictions 2.9.8

t(3,2,1) ~S3

2.9.9

t(3,2,

XS2 XS 1,

1) ~ 54 X SI X 51'

t(3,2,1) ~S3 t(3,2,

XS3,

1) ~ 55 X SI'

Boring but easy applications of the Littlewood-Richardson rule together with Frobenius's reciprocity show the following: 2.9.10 (i) [3,2, 1] ~ 53 X 52 X 51 i 56 = [3] [2] [ 1] + [3] [ 12 ][ 1] + 3[2, 1][2] [ 1] + 3[2, 1][ 12 ][ 1] + [ 13 ][ 2][ 1] + [1 3 ][ 12 ][ 1), (ii) [3,2, 1] ~ 53 X 53 i 56 = 2[3] [2, 1] + 2[2, 1][2, 1] + 2[2, 1] [ 13], (iii) [3,2, 1] ~ 54 X 5 I X 51 i 56 = 2[2, 12 ][1][ 1] + 2[2 2 ][ 1][ 1] + 2[3, 1][1] [1] , (iv) [3,2, 1] ~ 55 X 51 i 56 = [2 2, 1][ 1] + [3, 12 ][ 1] + [3,2][ 1]. Further applications of the Littlewood-Richardson rule therefore yield the desired result: 2.9.11 [3,2,1]0[3,2,1]=[6]+2[5,1)+3[4,2]+4[4, 12 ]+2[3 2 ]+5[3,2, 1] + 4[3,1 3 ] + 2[2 3 ] + 3[2 2 ,1 2 ] + 2[2,1 4 ] + [1 6 ]. Although we can evaluate the decomposition of t"0 t f3 in this way mechanically, the situation is still unsatisfactory, since we sometimes need more explicit information about the irreducible constituents of [a]0[j3]. In order to get it, we go back to the permutation characters ~" and consider their inner tensor products ~"0e. Now ~" is the character of the action of 5n on the left cosets of S", or, equivalently, on the set of tabloids of shape [a], i.e. on the set 2.9.12

ncr : = { ncr In" a dissection of n into subsets of orders aJ

(d. 1.3.2). Therefore ~"0e is the character of the natural action of

5n on

97

2.9 Inner Tensor Products

the cartesian product n",Xn ll

2.9.13 by 2.9.14

7T (n'" , nil

) : == ( { 7T [ nj] , 7T [ n;] , .. , }, { 7T [ nf] , 7T [ n1] , ... } ).

The orbit of (n"',nll ) is characterized by the orders 2.9.15 so that the stabilizer of this orbit is a Young subgroup S. corresponding to the dissection of n into the subsets n~ n nf This yields 2.9.16 LEMMA. ~"'®e ==~.~', if the sum is taken over all the kXk matrices v =(Vi) such that Vi} E 1'.1 0 and

~ Vi) =f3) and

~ VI} =ai , )

Hence at least the inner tensor product ~"'®e is explicitly given as a sum of permutation characters. For example

where the sum is taken over all the matrices

such that

Vi)

El'.l o and VII +V 12 ==V I1 +V 21

v 12 +v22

=V 21

==n-1,

+v22 == 1.

The only matrices of this form are

so that we obtain

=2t
3t
98

Irreducible Representations and Characters of Symmetric and Alternating Groups 2.9

It is sometimes useful to employ the following remark in place of Curtis and Reiner [1962, (38.5)]. Let H~G be groups, and suppose that every element of H is conjugate (in H) to its inverse, i.e., H is ambivalent. If ep and if; are characters of G, E the character of the identity representation of H, then 1

(EiG,ep0if;)=(E,(ep0if;)~H)= IHI

L

ep(h)if;(h)

hr=H

= (ep ~ H, if; ~ H) = L (ep ~ H, X)( if; ~ H, X), x

where the last sum is over the irreducible characters of H. This gives an alternative method for evaluating
Ua0 ~/3, e) = L Ua ~ Sy' X) (~/3 ~ Sy, X).

2.9.17

x The evaluation of ~a ~ Sy involves the Littlewood-Richardson rule again, and the calculation of ~a0 ~/3 is no quicker by this method than by the one already given. But viewing the problem this way is sometimes of assistance, for example in: 2.9.18

THEOREM.

Suppose that a, /3, y~n. Then (~a0~/3, gy»O only if n

la-/3I:=

L

la;-/3J';;;;;2(n-y,).

i=1

Proof (~a0~/3,e»O implies, by 2.9.17, that ~alS and ~/3lSy have a common constituent. Therefore, there are partitions 8U) of y; such that

and

By the Littlewood-Richardson rule, the diagram [8(1)] is a subdiagram of both the diagrams [a] and [/3]. For this to be possible, we clearly require la-/31~2(n-Y1)' •

Proof (~a0~/3, ~Y»O implies (~a0~/3, P»O, so that the statement follows from 2.9.18. •

:

~

99

Inner Tensor Products

2.9.20

COROLLARY.

([a]0[.8],[y]»0 implies

Proof 2.9.19 shows that the assumption implies

"ow by the triangle inequality

la-.8I=la\-.8\I+la 2-.821+ ... +la -.8 ;;;.Ia t -.8t I + In-a) - (n-.8 I )1 =2I a\-.8 I I, ll

,0

11

1

that we get

:\s ([ a]0[.8], [y]) is symmetric in a, that the following is also true:

.8, and y, we get from

this last inequality

and hence also

which completes the proof.

•

This result shows how we can estimate the irreducible constituents [y] of [a]0[.8] in terms of the number n-y, of nodes of [y] which do not belong to the first row. We call this number, 2.9.21 the depth of [y] or y. Using this notation, we can rephrase 2.9.20 as follows: 2.9.22 THEOREM. The depth d y of each irreducible constituent [y] of[ a] 0 satisfies the inequalities

[.8]

100

Irreducible Representations and Characters of Symmetric and Alternating Groups 2.9

Exercises 2.1 If P";;;;'Sn' then P induces permutation groups PI and P2 on the subsets of order k and on the subsets of order k - 1 of n. Using 2.2.19(iii), show that, provided 2,,;;;;,k";;;;'n/2, PI has at least as many orbits as has P2 • 2.2 Prove that the characters of the representations [a 1± of A n are real or non-real according as the number of nodes above or below the diagonal of the Young diagram [a 1is even or odd.

CHAPTER 3

Ordinary Irreducible Matrix Representations of Symmetric Groups

When investigating the representation theory of a given group, It IS normal to follow the calculation of its ordinary irreducible characters by a consideration of its irreducible modules, and since we now know the characters of Sn we should concern ourselves next with representations. We do this in the present chapter, by dealing with left ideals of the group algebra OSn' At the same time, we investigate some of the matrix representations which arise. In fact, the irreducible modules for Sn are more easily described as Specht modules, and a shorter construction of the matrix representations is given in Chapter 7, but we present here Alfred Young's classical approach using the group algebra, since it provides an excellent example of the ring-theoretical tools from general theory being applied in a practical case.

3.1

A Decomposition of the Group Algebra QSn into Minimal uft Ideals

We remember from general representation theory that we can obtain the ordinary irreducible matrix representations of a given finite group G at least theoretically as follows. The group algebra CG has a unique decomposition into simple two-sided ideals 1", say: 3.1.1

CG= $ 1".

ENCYCLOPEDIA OF MATHEMATICS and lts Applications, Gian-Carlo Rota (ed.). Vol. 16: G. D. James and A. Kerber, The Representation Theory of the Symmetric Group ISBN 0-201-13515-9 Copyright © 1981 by Addison-Wesley Publishing Company, Inc., Advanced Book Program. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system. or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the publisher. 101

102

Ordinary Irreducible Matrix Representations of Symmetric Groups 3.1

Each r is, by Wedderburn's theorem, isomorphic to a full matrix ring over C. Let R V denote one of these isomorphisms. Up to equivalence r has exactly one irreducible matrix representation, and hence this matrix representation must be equivalent to R V • Each such system of irreducible matrix representations R yields a complete system of pairwise inequivalent and irreducible ordinary matrix representations D of G as follows. If gEG, then 1c 'gECG, so that we obtain a decomposition of this element with respect to the decomposition 3.1.1 of CG. say V

V

3.1.2 Then we put 3.1.3 This shows that the crucial step is an explicit description of the isomorphisms R V of r onto a full matrix ring over C, the existence of which is guaranteed by the theorem of Wedderburn. The representation theory of S" is in fact one of the very rare cases in which such a description is available for a class of by no means trivial finite groups. Later on it will be shown that from the algorithm for symmetric groups we can derive algorithms for various other classes of finite groups, too. In order to describe this algorithm we first decompose OS" into its simple two-sided ideals. Even more, we give a decomposition of OS" into minimal left ideals (by 2.1.12 it suffices to consider OS" instead of CSn ). We shall do this by deriving a complete system of primitive idempotents, which generate minimal left ideals. For this purpose we recall the method which gave us the complete system of ordinary irreducible representations of S", in order to show how it fits into the general representation theory of finite groups and that it yields primitive idempotents. We started off with a pair Sa' Sa' of subgroups of S", say

We took their one-dimensional representations IHr

and AVf.

and considered the induced representations

A Decomposition of the Group Algebra QSn into Minimal Left Ideals

103

Ihese representations have the property that there exists exactly one double ,\)set V{'1THf such that the restrictions of IHf and A V{' coincide on the ntersection Hf n 1T V{'1T -I; this particular double coset is VjlXHf. Furthermore, for each aESIl\V1lXHf there exist by 1.5.8 transpositions - -= V{' and p E Hf such that 1Ta = ap. Since these elements 1T and pare :ranspositions, we have

This shows that G:=SIl and its subgroups H:=Hf and V:=V{', together .\ i th their representations IHf and A V{' over F: = 0, satisfy the conditions ,'I' the following theorem: THEOREM. Let G denote a finite group and F a field with char F+IGI. Furthermore let H and V be subgroups of G with one-dimensional characters 8 'JIld e such that

.1.1.4

(i) 81 Hn V=e 1 Hn V, and (ii) for each gEG\VH there exist hEH, vE V such that vg=gh, while 8(h)¥oe(v).

If we now put e:= ~ vE

~ e(v)8(h)vh, V hEH

then this element of the group algebra FG of Gover F generates a minimal left Ideal FGe, which affords an absolutely irreducible representation of G. Proof (I) We first prove that the following is true:

3.1.5

{aEFGI\ih,v(vah=e(v- I )8(h- l )a)}=Fe.

(a) Every Ke, KEF, satisfies for each vE V. hEH

vKeh=K

L

e(v')8(h')vv'h'h

h'. v'

=e(v- 1 )8(h- I )K

L e(vv')8(h'h)vv'h'h h' .c'

104

Ordinary Irreducible Matrix Representations of Symmetric Groups 3.1

(b) Conversely let a: = ~K gg E FG satisfy for each v E V, hE H the equation 3.1.6

~

Kgvgh=vah=e( V-I )S( h -} )a=e( V-I )S(h - I ) ~ Kgg.

g

g

Then we obtain a=Ke by comparing coefficients as follows: (0:) In the case when g is of the form g

=vh, 3.1.6 gives

(13) In the case when g E G\ VH, we take v E V, hE H such that vg = gh -- } and S( h - I) oF e( v) (such v and h exist by assumption). We obtain from 3.1.6 that

Since e( v )S( h) oF 1, this can be true only if Kg =O. (y) As any gE VH can be written in IHn VI ways as a product g=vh, we obtain from (0:) and (13) that

as stated. This completes the proof of 3.1.5. (2) A particular a E FG which satisfies for each v E V, h E H

is the element a: = e 2 , so that we obtain from 3.1.5

for a suitable KEF. Hence a proof of KoFO will show that e is essentially idempotent. In order to verify this, we consider the linear mapping
i.e. right multiplication bye. We shall evaluate

K

by calculating the trace of

which is adapted to the subspace FGe, i.e. the first dim F( FGe) elements of

105

A Decomposition of the Group Algebra OSn into Minimal Left Ideals

:he basis span FGe. Then cp is represented by a matrix

«1»

of the form

K

* K

°

o

°

This gives our first equation for the trace: trace cp= K' dim F ( FGe ). Secondly we take as basis of FG the elements 1F' g. Then cp is represented by a matrix «1>2 of the form

*

(Kl .

«I> _ 2-

* 11:) being the coefficient of IG in e= LKgg. Since K] second equation for the trace:

=IHn V I, we obtain as a

trace cp =IG 1'\ H n V I. Comparing the two equations for the trace, we get 3.1.7

IGI·IHnv\ K= dim F ( FGe) .

This is nonzero since IGI·I Hn (3) In (2) we proved that

VI "",0 by char F+IGI. ,

1

e:=-e K

is an idempotent of FG. In order to show that FGe=FGe is minimal, we need only prove that eFGe is a division ring. But again by 3.1.5 we obtain immediately that eFGe= eFGe = Fe:o: F. (4) Since by (3) eFGe is not only a division ring but also isomorphic to F, FGe is even absolutely irreducible. •

106

Ordinary Irreducible Matrix Representations of Symmetric Groups 3.1

This theorem elucidates the representation-theoretical background of our method of characterizing the ordinary irreducible representations of Sn as "in tersections" (d. 2.1.4) of monomial representations induced from Young subgroups. It shows furthermore that the elements ea:=,\:-acJc a :=

3.1.8

~

~

sgn7T'7Tp EQSn'

'irE V" pEH"

which we have defined already in 1.5.10, are in fact essentially idempotent and generate minimal left ideals in QSn' Putting (cf. 3.1.7) 3.1.9

K

a._ .-

n! fa.

we obtain 3.1.10 THEOREM. For each tableau t a the element

is a primitive idempotent in the group algebra QSn of Sn over Q. The minimal left ideal La: =QSn ea affords the absolutely irreducible representation [a] of Sn' If we let t a run just through the set of all the tableaux rf. a ~ n, (see 1.4.19) we obtain a set of irreducible left Sn-modules L'I. which by the results of Chapter 2 form a complete set. i.e. 3.1.11 COROLLARY. The minimal left ideals L'I: =QSnef. a~ n. comtirute a complete set of irreducible and painvise inequivalent left QSn-modules.

For a fixed a~ n there exist n! tableaux t a and therefore n! primitive idempotents e a, so we have obtained IP(n)ln! minimal left ideals La in QSn' while the desired direct decomposition of QSn into minimal left ideals consists of only La' minimal left ideals. We therefore have to pick a suitable subset out of the set of all the tableaux. For a given a ~ n we have to take exactly tableaux with diagram [a]. The question is which subset (if any) will do what we want. We get an idea which subset might be a suitable one by considering a combinatorial interpretation of For the branching theorem 2.4.3 tells us

nr

r

tt

r.

107

3.1 A Decomposition of the Group Algebra OS" into Minimal Left Ideals

that

fa =~a(1 ) . 5/1 is equal to the number of ways we can reach (0) from the diagram [a) by successively cancelling just one node and subject to the condition that after each cancellation a diagram (or finally [0]) remains. Or vice versa: is equal to the number of ways in which the nodes of [a] can be replaced by the points i En in such a way that the points of the resulting tableau in each row and in each column occur in their natural order (reading the rows from left to right and the columns downwards). These tableaux are called standard Young-tableaux. The standard (3,2)-tableaux for example are

r

3.1.12

I 2 5

1 3 5 2 4

3

4

134 2 5

124 3 5

123 4 5

This implies the following

3.1.13 COROLLARY.

r

is equal to the number of standard a-tableaux.

Let us denote the set of all the n ~ a-tableaux by t[ aJ: 3.1.14

t[ aJ: = {talta is an a-tableau},

and put

3.l.l5 We would like now to number the elements of st[ aJ with respect to the location of the point n, the point n - I. and so on. This ordering is a suitable one for later applications. We first notice that if t a Est[a]. then the point n occurs in la both at the end of a row and at the end of a column. Hence the tableau la*. which arises from t a by removing the point n is a standard tableau again. This allows us to define the last letter sequence tf.... , 1/0 of the elements of st[ a] recursively as follows:

3.1.16 DEFINITION. The numbers i and k of tr. tt: Est[al, satisfy i
tf
: ""'"

i
108

Ordinary Irreducible Matrix Representations of Symmetric Groups 3.1

For example 3.1.18 It is clear from 3.1.16 that the following is true:

3.1.19

LEMMA. For tt, tl: Est[a] we denote by [aU,})] and [ark,})] the diagrams of tt* .. *, tl:*' * (j asterisks), O,,;;;;j";;;;n. Then tt
O";;;;j";;;;h-l, while

This characterization of the last letter sequence of the elements of st[ a] will be used in the proof of 3.1.20

LEMMA.

If tt < tl:, then

eke;" =o.

Proof Let PI'"'' P" n be the points in the row of n in tl: and ql'"'' qs' n be the points in the column of n in tt, so that

3.1.21 where

:JC~*

has the obvious meaning, and analogously

'\I;"='V;a*(I-(q1n)- "·-(qsn».

3.1.22

(i) We shall first prove by induction on n that :JCZ'\I;" =0. (a) If [a(i,I)]=[a(k,I)], then tt
(b) If

*

then by 3.1.19, aU./)
[a(k,I)],

and hence

e"e" = _l-e"e" =0. k (K,,)2 k I

I

•

109

; 2 The Seminormal Basis of OS.

This shows that the

e;a, 1~i~r, are pairwise orthogonal. Thus

3.1.23 and hence the following holds: 3.1.24

THEOREM.

The minimal left ideals

rield the following decomposition of OSn into simple two-sided ideals

r:

where

fa

r =E9

1=1

L~.

The generating elements ef of the minimal left ideals nal primitive idempotents, so that in particular

3.2 The Seminormal Basis of

L~

are pairwise orthogo-

a Sn

It is our aim now to construct a O-basis 3.2.1 of OSn' which is adapted to the decomposition of OSn into simple two-sided ideals given in the preceding section, i.e. 3.2.1 should satisfy

r

3.2.2

fa

=EB OSnef = «e~k 11 ~i, k~r »0' 1=1

The elements of this basis will also satisfy the following orthogonality relations: 3.2.3 In order to find such a basis, we start as in the usual proof of Wedderburn's theorem by considering (r)2 isomorphisms between the direct summands

110

Ordinary Irreducible Matrix Representations of Symmetric Groups 3.2

of 3.2.2. We therefore define permutations 3.2.4

7Tl~ E

I ~i,

S" by

k~r.

It is clear that

3.2.5 With the aid of these permutations we put 3.2.6

in particular

e 1~ = e i' .

Since obviously 3.2.7 we have

3.2.8 This yields 3.2.9 Hence the elements e;'", I ~i, k~r, provide (fIY.)2 isomorphisms between the minimal left ideals QSAIY., I ~i~r. But unfortunately they do not satisfy the orthogonality relations 3.2.3. In order to force 3.2.3 we carry out a recursive orthogonalization procedure. For this we again denote by t~* the standard tableau which arises from t~ by deleting the point n, by t~** the one which arises from t~* by deleting n - I, and so on, until after n steps we arrive at the empty tableau 3.2.10 Let e~*,e~**, .. ",e~*,e~**, ... be the corresponding elements of the subalgebras QS,,_j,QS"_2'." of QSll" We are now in a position to define the seminormal units E~ recursively as follows (we put Kf* : = (n - I)! //;IY.*, K~** : = (n - 2)! //;IY.**, ... ):

3.2.11

III

;2 The Seminormal Basis of OSn

3.2.12 LEMMA.

(i) For each aEQSn we have

e0ae~k

=>\,K
,\: = [coeff. of Is" in e~Ja]. (ii) etef*et = K
Proof (i): From 3.1.5, 3.2.5, and 3.2.6 we get

This implies that for a suitable p, E Q

This yields by comparing the coefficients of Is" on both sides that

But for all b, cEQSn the following is true: 3.2.13

[coefi. of Is" in b· c] = [coefi. of Is" in c· b] .

Hence p,= [coefi. of Is" in e~je;'ja] = [coefi. of Is" in K
where A= [coefi. of Is" in etef*] . It remains to show that A= 1. We prove this by recursion. Another applica-

tion of (i) gives

where A*= [coefi. of Is" in et*ef*]. Since ef*EQSn-I' the coefficient of Is" is the same in both ej
Ordinary Irreducible Matrix Representations of Symmetric Groups 3.2

112

I (n:=2). ef*=et*=l s2 , so that I=A*=A. II (n>2). The induction hypothesis is

(I)

'f/f3~n-l,

This implies (use 3.2.Il):

This proves I = A* = A, and therefore (ii) is true.

•

We are now in a position to prove the main theorem of this section:

3.2.14

THEOREM.

The elements

satisfy the orthogonality relations

They form a Q-basis of QSn' called the seminormal basis of QSn' Proof (i) Since e:lc =et7Tik and e;" Er, we have for each a, i, and k e;'k Er = EB QSnet, i

so by the definition of efk

Elements of QSn which are contained in different simple two-sided ideals of QSn annihilate each other, so that we obtain

1\3

3.2 The Seminormal Basis of OSn

(ii) We shall now use induction on n to prove

I (n:=I): There is the standard tableau I only, so that i=j=h=k=1 and then

II (n> I): If 07"'e~je%k' then by definition of e~J we have

(I) so that necessarily ej*e%*7"'O.

Hence by the induction hypothesis j=h. And in this case, from (I) (use (ej*)2 = e;*),

and by 3.2.12

= -K"I e"*7rIJ e"7r"Jk e"* k U

I

J

This proves the orthogonality relations. (iii) The orthogonality relations imply that the e~k are linearly independent. Since there are n! = '1"U")2 such elements e~k> they form a O-basis of OSn' • Another formula, which will be of use later on, is

3.2.15 Proof From 3.2.11, e; = e't, and (ej*)2 = ej*, we get

114

Ordinary Irreducible Matrix Representations of Symmetric Groups 3.2

This yields E~e~kE'k

= E~E~*e~k E'k*E'k =K"E~E~kE'k

= K"E~k'

•

Let us now recall from general representation theory how we can obtain from such a basis {E~k} the corresponding matrices. The elements E~k' I ";;;i, k,,;;;j", form a Q-basis of the simple two-sided ideal 3.2.16 For each a EQSn we have uniquely determined a" EI", a r n, such that

a= ~ a". "rn

Hence by 3.2.16 there are uniquely determined (J/k( a) such that 1"

3.2.17

'"

a ,,- "" (Jik"()" a Eik , i.k=\

while 1"

3.2.18

a= ~

~ (J/k(a)e~k'

"rn i.k=\

The mapping 3.2.19 is a representation of I"; this follows easily from 3.2.14. It is irreducible and faithful, since by Wedderburn's theorem (J"[I"] is a full matrix ring. This irreducible representation is uniquely determined up to equivalence. Restricting it to the group elements, we get the desired matrix representation corresponding to [a]. We call it the seminormal form of the representation [a], since it is obtained from the seminormal basis. It should be mentioned that the basis consisting of the e~k also yields a decomposition of QSn into minimal left ideals: 3.2.20 so that in particular the following holds: 1"

3.2.21

(i) I" = k= EB \ QSn E"lk' (ii) QSnefk =«efk'·'" eh»Q'

liS

'J The Representing Matrices

3.3

The Representing Matrices

We are faced with the problem of evaluating the coefficients

The equation 3.3.1

a=

2 2 ai~( a hrk af- 11

;.k

yields (use 3.2.14) 3.3.2

3.3.3

LEMMA. a/iJ

7T) = K
of 7T - J in

etJ

Proof From 3.3.2 we obtain

(1) But [coeff. of 7T -I in etJ=[coeff. of I in 7Tet,J

which is by 3.2.13 = [coeff. of I in

er7Tef,J.

= [coeff. of 1 in

ai~(

which is by (1)

=ai~(7T)[coeff.

er

7T )erJ

of 1 in

en

But the coefficient of Is in is (K
er

erk

It is trivial that is represented under a
116

Ordinary Irreducible Matrix Representations of Symmetric Groups 3.3

elsewhere:

o o

3.3.4

o

o 1

o

o

o

i,

o k

while for /3'i'a, (Ja(efk) is the zero matrix. Putting ea. =

3.3.5

"'.

r

~

~

ea.ll'

i=l

we obtain an element of QSn which is represented by the r-rowed identity matrix If":

3.3.6

(Ja(ea)=If"'

This shows for example that ea is the identity element of furthermore

r.

Hence

lQSn=~ea.

3.3.7

acn

3.3.8

LEMMA.

For each af- nand 1~i~r we have ea * = I

~ efJ ""

(P,))

J

if the sum is taken over all pairs (/3, j), /3f- n, 1~j~ffJ, such that t1* = t:*, i.e. over all the standard tableaux which arise from t:* by adding an n-node. Proof 3.2.11 yields

If this is nonzero, then t1*=t:*=tf* by 3.2.14, so thatj=k and t1 arises

from t:* by adding an n-node. In this case we can proceed as follows:

'.3 The Representing Matrices

117

so that by 3.2.12

Hence we obtain from 3.3.4 and 3.3.2 that ef* is represented under (Jf3 by the same matrix as eJ. Since the representation (Jf3 of If3 is faithful, we are done. • It should be remarked that 3.3.8 is another version of the branching theorem 2.4.3. In order to make this clear, we keep the diagram [a] fixed for the rest of this section. For simplicity we skip the index a and denote by (Ji the representation corresponding to the diagram [a i -] (cf. 2.4.1) if a i - ~ n-1.

3.3.9 LEMMA. If Sn-I 0;;;; Sn denotes the stabilizer of the point n, then we have for each 17 E Sn-I the following decomposition of the matrix (J( 17): = (Ja( 17) into a direct sum of matrices:

where the summand (Ji( 17) is to be neglected when a'- is not a proper partition of n-1. Proof For 17ESn _ 1 we have both 17=

L L (Ji~( 17 )efk

and

17=

L L (Jt( 17 )eft·

Ihn-Ij,f

a~ni.k

By 3.3.2, 3.2.11, and the second equation,

This together with 3.2.14 shows that exist Ih n - I, j, and I which satisfy

(Ji~( 17)

is 0 except (possibly) if there

(I) Hence (Ji~( 17 )*0 only if both tf* and t:* have the same diagram [,8]. If on the other hand tf* and t:* have the same diagram [,8], i.e. if tf and tf have the point n in the same position, then we can use 3.3.3:

(Jik( 17) =

K

a[coeff. of 17 - I in eki] ,

which is by 3.2.14 and 3.2.6

= [coeff. of 17 -I in ek*e'k17kief*].

(2)

118

Ordinary Irreducible Matrix Representations of Symmetric Groups 3.3

From (1) we get that 1Tki =1T~ ESn _ l , so that in (2) only such summands of n fixed contribute to the coefficient of 1T - I. Thus

e~ which leave

which is by 3.2.15 = [coeff. of 1T -I in Kf3E~]

=aj!(1T),

•

by 3.3.3. Done.

This shows that our ordering of the seminormal basis of 1" with respect to the last letter sequence of the standard tableaux tt is adapted to the decomposition of the restriction of the seminormal form af< of [al to the subgroup Sn _ I: THEOREM. The irreducible matrix representation when it is restricted to Sn-I as follows:

3.3.10

af< IS ¥

n-I

=af<'-

+... +af
af<

of Sn decomposes

•

where the summand af<'- is to be neglected when a'- is not a proper partition of n-l. This shows clearly why we used the last letter sequence of the standard tableaux. Later on other orderings will be used, too. 3.3.11

LEMMA.

The matrix of e: is of the form h I 1

*

I

*

1 1

*

I

Kf<

1

*

o Proof (i) From 3.2.15 we get that

this yields the entry ahh(e:>=Kf<, as stated.

* h

_

119

3.3 The Representing Matrices

(ii) If k
7T:K i"7T -

I C\~"

= 0,

and therefore V'aE4JS n _

1

(e~ae't=O).

This implies that in particular e"e"e" = i h k

(~ )2 e"*e"e"*e"e"*e"e"* =0 ' K" , " h k k k

both if h
o * h

o "

0

h

The box with the (h, h) entry corresponds to the standard tableaux with diagram [a] which have n in the same position as has tho We next refine this argument by considering only those tableaux which have both nand n - I in the same position as has tho To do this we notice the following fact. If a standard tableau t has besides n the points PI'·'·' Pr (respectively q l' ... , qs) in one of its rows (columns), i.e. if

t=

120

Ordinary Irreducible Matrix Representations of Symmetric Groups 3.3

then

and 'Y='Y*(1-(q[,n)- "'-(qs,n)),

so that e='YX='Y*bcX*

for suitable b, cEQSn' Hence for a suitable aEQSn we have

J... )2 eC<*eC
eC
I

I

I

= (J... KC< )2 eC<*eC
I

I

I

I

=0 when t; and th have n in the same position but n -1 occurs in t; in a later row than in lh or in lh later than in If. An iterative application of this argument proves the statement. • We are now in a position to evaluate a C« 'IT). The transposItIons of successive points generate Sn (d. 1.1.17), so we need only evaluate the matrices which represent transpositions of successive points. We start with ac<((n - 1, n », for which we use the fact that T: =(n - 1, n) lies in the centralizer of Sn-2' the subgroup of Sn which stabilizes the points nand n - 1. The method of proof is of course induction on n, so that we can apply the induction hypothesis to a C« 'IT ), 'IT E Sn _ 2' using the branching as described in 3.3.10. The last letter sequence of the standard tableaux yields a subdivision of aC« 'IT) which corresponds to tableaux having both nand n - 1 in the same position, say in the ith and kth row, respectively, so that this box belongs to the diagram [O'(i-lk-j: to [O'i,kj for short. Hence

(ai, k( 'IT) has to be neglected when O'(i-lk- is not a proper partition of n - 2.) We notice that in this decomposition of aC« 'IT), ak,i( 'IT) occurs together with ai,k('IT) except for the case when either i=k or k=i-l and O' j - 1=O' j >O' j +["

All the other boxes occurring in 3.3.12 belong to pairwise inequivalent irreducible representations of Sn-2' since they correspond to pairwise different diagrams.

3.3 The Representing Matrices

121

We now subdivide the matrix a T), T: = (n - 1, n), correspondingly into boxes a i ,); k,/( T): U

(

(i,) .

3.3.13 (k,/)

Since 3.3.14 we obtain for the boxes of 3.3.13 3.3.15 Hence by Schur's lemma and a i')(7T)=a),i(7T) (if they exist): 3.3.16

a'.·.J ; k 'I( T)= {yi').] k,l

O-matrix

if

{i, j} = {k, I},

otherwise.

This shows that we are left with the problem of evaluating the scalar factors ym

(i) ai,i;i,i( T )=Yi:/'], (ii) ai, i-I; i, H

( T )

=

Y/:/~ i.]

3.3.17

... (ai');i')(T) ai');)'i(T)) (1l1) aJ,I;I,J . . . .() ....() = T aJ,I;J,1 T

(Y/:f.']

Yif])

.. y J ,',]

..

yJ,',] J,I

I,)

The first result concerning the entries is obtained from yields

T

2

.

= 1, a fact which

(i) (yi,i)2=(yi'i-l)2=1 1,1-1 , I,l

3.3.18

+yi,/,y/,i = (y/'i)2 +yi,),yJ,i = 1 (ii) (yi,))2 I,) J,l I,) J,l J,I I,) ,

In order to evaluate y/:/' we consider a standard tableau t)u which contains both nand n - 1 in its i th row. (Recall that if no such tableau exists, a i • i( 7T) does not occur in 3.3.12.) Then TEH)u and hence 3.3.19

122

Ordinary Irreducible Matrix Representations of Symmetric Groups 3.3

We use this in 3.3.20 We shall obtain Y/:/ by equating suitable coefficients of the matrices which represent the elements on each side of Equation 3.3.20. The matrix aa( Kaej) which represents the left-hand side of 3.3.20 has K a in the (j, j) place and zeros elsewhere (cf. 3.3.4). Let us therefore evaluate the corresponding coefficient of the matrix representing the right-hand side of 3.3.20. aa( ej*) has a 1 at (j, j) and zeros elsewhere in thejth row and thejth column, by 3.3.8. aa(ej) has Ka at (j, j) by 3.3.11. Thus aa(ej*ej) has the samejth row as aa(ej), while aa(nj*) has at (j, j) the coefficient Y/:/' and zeros elsewhere in the jth column. Hence

This proves 3.3.21

y/:/ =

1.

y/:/~i is obtained quite analogously. For in this case rE

Tja, for a suitablej,

so that reJa = - er and hence 3.3.22

i.i-I=_l . Y;.i-I

Let us now evaluate y/o/, y;I y/)' and yj/. To do this we choose k and t such that tf contains n-l in its ith row and n in itsjth row, and where rtf =tf·

Without loss of generality we can assume i t. Since r = 'lTt" we get 3.3.23 We again compare suitable coefficients of the matrices which represent the elements on each side of this equation. aa( r) contains the submatrix

at the intersections of the tth and kth rows and columns, and zeros elsewhere in these rows and columns. Comparing the coefficients (k, t) of

123

3.3 The Representing Matrices

the matrices representing the elements on each side of 3.3.23, we obtain 3.3.24

when

i
This together with 3.3.18 yields for the remaining coefficients:

(i) yi,}J = I,

3.3.25

_y},i j, I '

(ii) y"} = 1- (y!'i)2 = 1- (yl') )2. },I

},I

I,)

This shows that we have reduced our problems to the evaluation of y/J To complete the story, we introduce the concept of the axial distance between two points in a tableau in terms of which y/:} will be expressed eventually, Let t denote a tableau, and let the points rand s occur in t in the positions (ir' ir) and (is' J~), respectively. Then the absolute values

measure the distances of rand s from the main diagonal: I,

},

t:

If we take i r- ir and is -is instead of their absolute values, then the sign of this difference indicates whether r (or s) lies above or below the main diagonal. If i r- ir < 0, then r lies above the main diagonal. Hence the

difference 3.3.26

d ( r , s ) : = ( is -is) - ( i r- ir ) = (ir -is) + ( is - i r)

counts the number of steps we need to go from r to s, if steps to the left and downwards contribute + 1, while steps to the right and upwards contribute -1. The resulting integer d(r, s) is independent of the path we choose; we call it the axial distance between rand s in t. The axial distance between r and s in t}a will be denoted by 3.3.27

124

Ordinary Irreducible Matrix Representations of Symmetric Groups 3.3

It satisfies 3.3.28 Using this notation, the final result will be: 3.3.29 THE SEMINORMAL FORM OF [a] (cf. 7.3.13). If ti, ... , tfa is the last letter sequence of the standard a-tableaux, then the matrix

a Q(( m - I , m )) = ( a/k(( m - I, m ))) which represents the transposition (m- I, m), I
( (1~((m-

I, m») a:((m- I, m»))=(-diQ(m- I, m)-l

aki((m-l, m»)akk((m-l, m»

1

l-~t(m-l, ml\-Zj, d i (m-l, m)

(iii) a/k«m-l, m»=O everywhere else.

Proof By induction on n. (I) For n =2 the statement holds, as is very easy to see. (2) Now let n be greater than 2, and assume the statement to be true for Sn-l' so that it holds for 7": =(m- I, m), m
I ' n ) -1 Yr.r,sS = -dQ(nI

when

r
where tt denotes a standard tableau with n in its rth row and n-l in its sth row. Let} be defined by tt=(n-l,n)tr Together with tt and tjQ, we consider

tZ: =(n-2, n- I)tf, (if they exist) and use

(n-2, n)=(n-l, n)(n-2, n- I)(n-l, n) =(n-2, n-l)(n-l, n)(n-2, n-l). We shall equate the (j, i) coefficient of the matrices representing these

125

3.3 The Representing Matrices

two expressions for (n - 2, n). The induction hypothesis tells us that a"((n-2, n-l)) has the following submatrix at the intersections of the ith and jth rows with the ith and jth columns:

l'k or It exist, these are not the only nonzero entries of these rows and columns, but the others do not matter in the ensuing argument. If If or It do not exist, then this is still the correct submatrix.) Hence a"((n-l,n))a"((n-2,n-I)) has at these places the submatrix (put y:== (If

y;'n

l-y2) (-d f (n-2,n-1)-1 ° -y

== ( -yd f(n-2, n-1)-1

-(I-y2)dj"(n~2,n-~)-1).

-df(n-2, n-1)-1

+ydj"(n-2, n-l)-

Thus aj~(n-l, n )(n-2, n-l)(n-l, n»)

== -yd j"(n-2, n-1)-1 +ydj"(n-2, n-l)-l,

while ~~(n-2,n-I)(n-l,n)(n-2,n-I»)

==df(n-2, n-l) -ldj"(n-2, n-l) -I. This implies

1.y == -d"(n-2 J

'

n-I)+d"(n-2 n-l) I

,

== -df(n-2, n)+df(n-2, n-l)

==-df(n-l,n), the last equation by 3.3.28. This proves 3.3.30, and we are done.

•

Ordinary Irreducible Matrix Representations of Symmetric Groups 3.4

126

For an example we take a: =(3,2). The last letter sequence of the standard tableaux is 3.3.31

1

3

2

4

5

1 3

2 4

5

1

3 5

2

4

2 5

1 3

4

1 4

2 5

3

so that we obtain

-1 -1

3.3.32

1

I

while 1

3

2

4

_1 2

3.3.33

I

3

2

4 I

2

-1 3.3.34

0(3.2)((34))= I

R

3

"9

_1 3

and I

3

2

4 1

3

2

3.3.35

4 _1 2

This shows how easy it is to write down oa«m-l,

m».

3.3.36 COROLLARY. For each 7TESn , Oa(7T) is a matrix over Q.

3.4

The Orthogonal and the Natural Form of [a]

For certain purposes (like applications in science) it is more suitable to have orthogonal matrices as representing matrices. A. Young gave such a form, too:

1.4 The Orthogonal and the Natural Form of [a]

127

3.4.1 THE ORTHOGONAL FORM OF [0:]. We obtain a representation wa which is equivalent to aa bur which has orthogonal representing matrices if we replace (he submatrix 3.3.29(ii) by

-2 I \ Vl-d;a(m-l.m)

Proof We construct a matrix H a which satisfies

In order to do this. for a given standard tableau t we denote the last point of its ith row by a l and assuming that O:=(O:t ..... O:h) with O:h >0, we put

3.4.2

if

n=aj

,

)
Furthermore we define

.3 .4.3 Jnd so on, so that we can put 3.4.4 If i
have

\1oreover

so that altogether (check how 3.4.5

~.

.---!...

~k

(j/,~( m)

and (j/,,( m - 1) differ)

= I - d~( m - L m )-2 .

128

Ordinary Irreducible Matrix Representations of Symmetric Groups 3.4

Putting

Fh

0

H(J.:=

3.4.6

o we find that

differs from a(J.((m-l, m))just in the submatrix 3.3.29(ii) and has the stated form. It is clear that this matrix is orthogonal. • For example

-1 -I

I

2"

w(3,2)((2,3»=

tf3

tf3

1

2"

tf3

1

2"

tf3 3.4.7

I

2"

-1 w(3,2)( (3,4»

=

I

"3

ifi tf3

I

2"

=

2"

tf3

I

"3

tf3

I

W(3,2)( (4,5»

il2

_1 2

tf3

I

2"

1 3.4.8 COROLLARY. w(J. provides an irreducible matrix representation of Sn onto real orthogonal matrices, It corresponds to the basis consisting of the orthogo-

3.4 The Orthogonal and the Natural Form of [a]

129

nal units

This basis

{ai~

Ia ~ n,

I,,;;;; i, k ";;;;/"'} will be called the orthogonal basis of IR SIl'

Later on in this book we shall consider modular representations associated with the ordinary irreducible representations [a] of SIl' Since we shall be interested mainly in their irreducible constituents, it does not matter which associated modular representation we take. Therefore it will be easiest to start with a matrix form of [a] such that its matrices are matrices over lL, so that we need only reduce the coefficients modulo the prime characteristic p in order to get an associated p-modular representation. In practice, we shall do this directly, but it is interesting to see how to perform the calculation inside OSIl' For this reason we shall now derive Young's natural form p" of [a], which will turn out to be a matrix representation over lL. To do this we recall from 3.2.12 and 3.1.24 that the elements 1 e"7T"e" e i"k.·-- e i"7Ti"-7T"e" k - ik k-- - a I ik k-- K "e-"7T"e-" i ik k K

satisfy 3.4.9

where

JL~J: = [ coeff. of 1 in e,U .

We put these coefficients into a matrix M":

3.4.10 and prove 3.4.11 LEMMA. M" is a lower triangular matrix. i.e., JL~J =0 (f r
Proof By induction on n. If we partition M" into boxes according to the last letter sequence, the boxes along the main diagonal are lower triangular and have l's along their main diagonal by the induction hypothesis. When the index pair (r, s), r
Ordinary Irreducible Matrix Representations of Symmetric Groups 3.4

130

This shows in particular that MO: is invertible, say L 0:: = ("Ajk):

3.4.12

= (MO:) -I.

We notice that L 0: is a matrix over 7L by 3.4.11. Using the entries of L 0: we define natural units by 3.4.13 and prove that they also satisfy the orthogonality relations:

Proof

__1_ ~ -

/3

0:

K K

'0:

'/3

0:

/3

.c.. f\rjf\sleire ks '

r.s

which is by 3.4.9 _

~ ( KO:I

-Uo:/3

)2 .c..~ f\rjf\sIK '0:

'0:

0:

0:

0:

}J-kreis

r. s

=80:/3 Klo:

~"A~/( ~}J-~r"A~j )e~ s

r

• 3.4.15

LEMMA. Vo:,i,k(y/ic~O).

Proof 3.4.9 shows that y/ice ki

=e~ ~O.

•

This together with 3.4.14 yields: 3.4.16 THEOREM. The natural units form a basis of QSn' which is called the natural basis.

The corresponding matrix representation 1'0: of Sn' 3.4.17

v,1( 'TT ) : = [ coeff. of 'TT in y,~] ,

~

The Orthogonal and the Natural Form of [a]

131

, called the natural form of [a]. It remains to show that it is integral. .l4.18

LEMMA.

Va,i,k,7T

(Pi~(7T)El.).

Proof From 3.4.14 we obtain (cf. the proof of 3.3.3) that

Pi~( 7T) = K"[ coeff. of 7T - I in Yki] , = [coeff. of 7T - I in

7eki\Ji] ' •

Exercises 3.1 Read the papers of Malzan [1969], Puttaswamaiah [1963], Puttaswamaiah and Robinson [1964], Specht [1938], and Thrall [1941] in \1rder to see how the irreducible matrix representations of the alternating group can be obtained and what their main properties are. 3.2 Derive an algorithm which directly yields a rational integral form of the representing matrices, or study the corresponding sections in the first edition of Boerner's book [1955].

CHAPTER 4

Representations of Wreath Products

Since the ordinary irreducible characters and representations of Sn are now at our disposal, we should try to apply them to the representation theory of a wider class of groups. Apart from the alternating groups, which we have already discussed, the most natural groups to consider are the complete monomial groups GwrSn , G being a given finite group. These groups are wreath products, so we introduce the concept of wreath products and derive their representations over algebraically closed fields.

4.1

Wreath Products

Let G be a group and H a subgroup of Sn' Denoting by GO the set of all mappings from n= {l, ... , n} into G: 4.1.1

GO:= Ulj:n->G}.

we put 4.1.2

Gwr H: =Go XH= {(f; 7T)1 f: n-> G and 7TEH}.

For JEGo and 7TEH we definej" EGo by 4.1.3 ENCYCLOPEDIA OF MATHEMATICS and Its Applications. Gian-Carlo Rota (ed.). Vol. 16: G. D. James and A. Kerber, The Representation Theory of the Symmetric Group ISBN 0-201-13515-9 Copyright l' 1981 by Addison- Weslev Publishing Company, Inc.. Advanced Book Program. All rights reserved. No part of this publication may be reproduced. stored in a retrieval system, or transmitted, in any form or by any means. electronic. mechanical. photocopying. recording, or otherwise, without the prior permission of the publisher. 132

\33

4.1 Wreath Products

It is easy to check that for 1f' EH, 4.1.4 Furthermore we introduce a multiplication on G" by 4.1.5

( ff') ( i ) : = f( i ) .f' ( i ), i En.

Using this, one easily verifies that Gwr H together with the composition defined by 4.1.6 forms a group, the wreath product of G by H. Its order (if G is finite) is

4.1.7 If we define eEG" by

4.1.8

iEn

and for fE G" the mapping f 4.1.9

f

-1

-1

E G" by

(i): =f( i) -

J,

iEn,

then we obtain for the identity element in Gwr H and for the inverse of (/; 1f)EGwr H:

4.1.10 where 4.1.11

f - J . - ( J7T-f 1

J7T- 1 • -

)-I-(f-l) -

11-\'

We shall sometimes describe (f; 1f) E G wr H more explicitly by displaying the values of f, i.e., we write

(/(l),·.·,f(n);1f) instead of ( f; 1f) if it seems advisable. The following normal subgroup of Gwr H is of particular importance: 4.1.12

G*:= {(/; I H )I fEG"}.

G* is called the base group of Gwr H. It is the direct product of n copies G,

134

Representations of Wreath Products 4.1

of G, where 4.1.13 The subgroup 4.1.14

H': = {(e; 7T )17TEH} c:=H

is a complement of G*, so we have

(i) GwrH=G*·H', 4.1.15

(ii) G* =

IT Gi~Gwr H, i

(iii) G*nH'= {lGwrn}= {(e; In)}· Another interesting subgroup is the diagonal subgroup of G*: 4.1.16

diag G* : ={(f; In )1 f constant}

= {(g, ... , g; In )lgEG} c:=G, so that 4.1.17

diag G*· H' ={(f; 7T )1 f constant}

= {(g, ... , g; 7T )lgEG, 7TEH} c:=GXH. This shows that we have nice embeddings of G, H, and GXH in Gwr H. If G is a permutation group of finite degree, say G,,;;;Sm, then we obtain a permutation representation 1J; of Gwr H as follows: 4.1.18

(j-I)m+i ) 1J;:GwrH ...... Smn :(f,7T)...... ( (7T( ·)-I)m+f(7T( ·»(i) }

}

.

I

.'

"""'m. I "'J"'n

This faithful representation of GwrH. where G~Sm and H~Sn' is in a sense a natural one, since 1/;[ Gd acts on {I, ... , m} C {I, ... , mn} in the same way as G acts on {l, ... ,m}, and the restriction of1/;[G 1J to {I, ... ,m} is just G. Furthermore 1/;[G2 J acts on {m+ 1, ... ,2m} and G acts on m in similar ways, etc. Also 1/;[H'J permutes the subsets {I, ... , m}, {m+ 1, ,2m}, ... , {(n-l)m+I, ... ,mn} in the same way as H acts on {I, , n}; e.g., the element

( f; 7T) : =((12), ( 123), I ; (23) ) E S3 wr S3

135

4.\ Wreath Products

is mapped under tf; onto (12)(456)

(47)(58)(69) = (12)(475869) E S9'

~

'-y-------'

tf;((j; IH))

tf;((e; '17))

As an application of this permutation representation we obtain a nice description of the centralizers of elements in Sn' First of all we notice that if Cm : = «(1, 00" ~ Sm' then tf;[ Cm wrSnl is just the centralizer of the permutation

m»

(1'00" m)(m+ 1,00.,2m)·· '((n-l)m+ 1'00" nm)ESnm

(this follows from 1.2.1 and 1.2.15). In order to derive therefrom an expression for the centralizer of a general element '17 E Sn we need some more notation. If p~ Sri, P' ~ Sri" where n n n' = 0, then we understand by the direct sum PtB p' of these two permutation groups the following group acting on nUn': The underlying set is the cartesian product P X p', and the action is defined by

'17(W") ('17, '17')( w"): = { '17'( w")

if w" En, if w" En'.

Hence for example Sa = EB iSa' where Sa acts on suitable subsets. Using this notation we obtain for the ce~tralizer of '17 E Sn

4.1.19

CsJ '17)~

EB

tf;[ CiwrSa,('IT)]'

Q;('IT»O

Another important application of the concept of wreath products is the construction of Sylow subgroups of Sn' In fact this construction was mentioned by A. Cauchy in Vol. III of his Traite d' Analyse which was published in 1844, although Sylow did not prove his theorem until 1872. If p denotes a prime number and pi a p-Sylow-subgroup of Sp;, then for Cp :=«(1, ... ,p» we have

and it is easy to see that this is the order of the p-Sylow-subgroups of SpH I. Hence tf;[piwrCpl is ap-Sylow-subgroup of SpHI. Thus by induction we get

i

factors Cp

136

Representations of Wreath Products 4.1

If now /

n= ~ bipi,

O~bi
i=O

is the p-adic decomposition of n, then the exponent I'p(n!) of the maximal power of p dividing n! satisfies 4.1.21

I'p(nl)=b,I'p(p!)+b 21'p(p2 !)+ '" +b,I'p(p'!) =b 1 +b2(l +p)+ .. , +b,(l +p+ .. , +p/-l).

This shows that the following is true: 4.1.22. If P" denotes a p-Sylow-subgroup of S" and n = 2.~=Obi pi, O~bi
then

where (J)b,p': =piffi . " ffiP' (b i direct summands). An easy check verifies that the wreath product is associative following sense:

In

the

4.1.23 G~S/' H~Sm' I~S"

=

1/J[1/J[GwrH]wrI]=1/J[Gwr1/J[HwrIJ].

Hence we can abbreviate 1/J(1/J(GwrH]wrI] by

1/J[GwrHwrI]. Applying this to 4.1.20, we notice that

4.1.24 This together with 4.1.22 shows that each p-Sylow-subgroup of S" is a direct sum of wreath products of the form 1/J( wr 'Cp ]. cf. A third kind of subgroup of symmetric groups which will occur later on is the normalizer of a subgroup of the form

" Sm : = Sm ffi

(J)

... ffi S", ~ S",,, .

'------r--'

" summands

137

4.1 Wreath Products

The number of subgroups of Smn conjugate to EBnSm (which are exactly the subgroups of Smn similar to EBnSm) is

-,n-,

1 (mn) ( mn - m ) ( m ) ( mn ) ! --. m m ... m = =ISmn: I/;[SmwrSn]l· n. m. n.

Hence INsjEBnSm)l=mlnnl=II/;[SmwrSnll. Since I/;[SmwrSnl is similar to a subgroup contained in this normalizer, we have

4.1.25 Thus the permutation representation I/; of GwrH, where G,;;;Sm and H,;;;Sn' allows a detailed description of centralizers of elements, p-Sylow-subgroups, and normalizers of subgroups of form EBnSm in symmetric groups. Another permutation representation () of Gwr H is important in the theory of enumeration under group action in combinatorics. () maps Gwr H into the symmetric group on

mn :=_

4.1.26

{cplcp:n~m}

as follows: 4.1.27 where cp' E mn is defined by 4.1.28

cp'(i): =f(i)( cp( 1T -l(i))),

iEn.

The image of () is denoted by 4.1.29 and called the exponentiation of G by H. The subgroup 4.1.30

G H : = () [diag G*· H']

is called the power of G by H. Another important subgroup which has no particular name is 4.1.31

138

Representations of Wreath Products 4.2

We notice 4.1.32

and remark that 0, as well as tfi, is a faithful representation. Finally it should be mentioned that the series of Weyl groups (but not the exceptional Weyl groups) are special cases of the groups defined above. The Weyl groups are defined to be certain permutation groups acting on certain subsets ("root systems") of euclidean spaces. It is not difficult to show that the following holds: 4.1.33. (i) The Weyl group of type An (n;;;'l) is similar to Sn+l' (ii) The Weyl group of type Bn (n;;;.2) is similar to tfi[S2wrSnl, as is the Weyl group of type Cn (n;;;' 3). (iii) The Weyl group of type Dn (n;;;'4) is similar to tfi[S2wrSnlnA2n'

Therefore the results which follow apply in particular to the series of Weyl groups.

4.2 The Conjugacy Classes of Gwr Sn The wreath products of the form GwrSn, nEN, are of particular importance, and they are called the complete monomial groups over G. This name points to the fact that GwrSn can be considered as a group of monomial matrices which arise from the n! permutation matrices of dimension n by replacing the l's of the matrices by the elements g of G in all possible ways. The multiplication of these matrices is defined with the aid of the multiplication of G. We shall describe the conjugacy classes of these groups, and to do this we consider an element (f; 7T) E GwrSn , assuming that G contains countably many conjugacy classes C 1,C 2, .... If c(,,)

7T =

II

(J.7T(j.) ... 7T" -l(j.) )

.=1

in cycle notation (so that 1.1.15 holds), then we can associate with its vth cyclic factor (j." '7T,,-I(j.)) the uniquely determined element 4.2.1

g.(j; 7T): = f(j. )f{ 7T -l(j.)) .. -J{ 7T -I, + l(j.)) = ff" ..

-J"/('H (j. )

139

4.2 The Conjugacy Classes of GwrSn

of G, which we call the p th cycle product of (f; 1T) or the cycle product associated with (jv" '1T 1,-I(jv» with respect to f. (Here and below, for typographical convenience we sometimes write I( p) instead of lv' etc.) We have thus c( 1T) cycle products which are associated with the cyclic factors of 1T with respect to f. a k(1T) of them arise from the ak( 1T) cyclic factors of 1T of length k. Now let aik(f; 1T) be the number of cycle products which are associated with a k-cycle of 1T and which belong to the conjugacy class C' of G. We put these nonnegative integers a ik(!; 1T) together into a matrix 4.2.2

a(f; 1T): = (aik(f; 1T )).

The matrix has n columns and as many rows as there are conjugacy classes in G. Its entries satisfy the following conditions:

(i) aik(f; 1T) EN o,

4.2.3

(ii) La ik (f;1T)=a k(1T), (iii) Lka ik (f;1T)=n. i,k

We call this matrix a(f; 1T) the type of (f; 1T), since it generalizes the cycle type of the elements of Sn which was introduced in Section 1.2. One notices that for each matrix (b ik ) of the stated size the coefficients of which satisfy

4.2.4

(i) bik EN o , (ii) Lk'bik=n, i,k

there are elements of GwrSn which are of type (b ik ). It is our aim now to show that two elements of GwrSn are conjugate if and only if they are of the same type. The first remark concerns the definition 4.2.1 of the cycle product. Since we are merely interested in the type of (!; 1T), we need only determine a cycle product up to conjugation within G. Therefore we show first that the convention that jv in 4.2.1 denotes the least symbol of the cycle in question is unnecessary here, i.e. we show

Proof We can assume O';;;s.;;;l v -1. Then

140

Representations of Wreath Products 4.2

But in a group G each product ab of elements a, bEG is conjugate to ba, so that we can proceed to deduce

Another useful remark is 4.2.6

a(f;'7T)=a((e;'7T')(f:'7T)(e;'7T,)-I)=a((f';l)(f;'7T)(I';l)-I).

Proof Since (e; '7T')(f; '7T)( e; '7T') -I = (/".,; '7T''7T'7T,-I), ( f'; 1) ( f;

'7T ) (

f'; 1) -

1= (

f' ff~ -

I ; '7T ),

it suffices to check that the elements on the right-hand sides of these equations are both of type a( f; '7T). (i) '7T''7T'7T,-1 contains the cyclic factor ('7T'(jp)'" '7T''7T 1,-I(jp». Its cycle product with respect to f"., is by 4.2.5 a conjugate of

(ii)

gv( I'ff".,-I; '7T) = (/'ff".,-I )(I'ff".,-I)".' .. (I'ff".'-') ".""-' (j.) =I'f'" f",/",'I'-l(jp) =I'(j.)gv(f; '7T)I'(jp)-I~gp(f;'7T).

•

A final lemma before we prove that the type characterizes a conjugacy class: 4.2.7 LEMMA. If a(j; '7T)=a(j'; '7T'), then there is a '7T"ESn satisfying '7T"'7T''7T,,-1 and with the property that for each P, gv(j; '7T)~gJf~,,; '7T).

Proof a(j; '7T)=a(j'; '7T') implies (d. 4.2.3 (ii» that '7T =

if'7T 'if -

'7T~'7T',

say

I.

The set of all these if forms a right coset of the centralizer of

(e; ir )(f'; '7T')(e: ir )-1=(/~; '7T),

'7T.

Since

'7T=

141

4.2 The Conjugacy Classes of GwrSn

4.2.6 yields

(I)

a( f; 'IT) =a( /'; 'IT') =a{ f~; 'IT).

But

while

Since (l) is valid, there is a 'IT* in the centralizer of 'IT, so that for each have gJ f; 'IT) ~ gJ f;.;; 'IT), i.e. (cf. 1.2.1)

Thus 'IT": ='IT*ir fulfills the statement.

I'

we

•

We are now in the position to characterize the conjugacy classes: 4.2.8 THEOREM. Two elements (f; 'IT) and (f'; 'IT') of GwrSn are conjugate if and only if a(f; 'IT)=a(f'; 'IT').

Proof (i) If (f; 'IT )~(f'; 'IT'), then there are /" and 'IT" such that

(f'; 'IT')=(f"; 'IT")(f; 'IT )(/"; 'IT,,)-I

=(/"; 1)(e;'IT")(f; 'IT )(e; 'IT,,)-I(f"; 1)-1, and hence by 4.2.6 we have a(f'; 'IT')=a(f; 'IT). (ii) If on the other hand a(f; 'IT)=a(f': 'IT'), then by 4.2.7 there is a 'IT"ESn satisfying 'IT='IT"'IT''IT,,~I and gp(f; 'IT)~gp(f;,,; 'IT). Now (f;,,; 'IT)= (e; 'IT")( f'; 'IT)( e; 'IT") -I. Hence it suffices to prove

(1)

Keeping such gj,' 10;;;; 1'0;;;; c( 'IT), fixed and starting with r: = 0, we obtain from the equations

(2 )

142

Representations of Wreath Products 4.2

elements g,,~'-J(j,) which are uniquely determined (1.,;;.r.,;;.ly -I). Having done this for every cyclic factor of 7T, we define the mapping f* : n ---> G by

f*(i}: =gi'

iEn.

It is easy to check that this mapping satisfies

•

(cf. (2», which proves (1).

This characterization of the conjugacy classes of GwrSn by types gives the number of conjugacy classes of GwrSn • There are as many conjugacy classes as there are types, i.e. matrices (b ik ) which consist of n columns and as many rows as there are conjugacy classes in G, and whose elements satisfy 4.2.4. We derive a formula for this number. 4.2.9 LEMMA. If s E ~ is equal to the number of conjugacy classes of G and if p( m) denotes the number of partitions of m, then the number of conjugacy classes of GwrSn is equal to

~ p(n 1 } ' " p(n.} (n)

if the sum is taken over all the s-tuples (n I •.• ,n $) over ~ 0 such that ~n i = n. Proof If (a ik ) is a type of GwrSn , then the integers

form an s-tuple (n I"'" n $) with the properties

And if (a ik ) runs through all the types, then each admissible s-tuple arises in this way. The n i defined above is the sum of the elements of the i th row of (a ik) weighted by their column number. Therefore if the other rows are kept fixed, there are exactly p( n i) possibilities for the i th row to be the row of a type of GwrSn • This proves the assertion. • If we wish to derive the order of the conjugacy class of GwrSn which is characterized by the type (a ik ), G being a finite group with s conjugacy

143

4.2 The Conjugacy Classes of GwrSn

classes, then we put lo;;;;kO;;;;n. There are n!jIIkka*ak! elements of type (all"" an) contained in Sn (cf. 1.2.15). We fix one of them; call it '1T. The a k cyclic factors of 7T which are of length k can be distributed into the s conjugacy classes of G in the following number of ways in accordance with the given type (a ik ):

N ow let f: n -. G be a mapping which yields such a distribution of the cycle products. It remains to show how much freedom of choice is left for the values of f. In order to ensure that

we may choose the values f(J.), f( 7T -1(J.)), ... ,f( 7T -I, +2(J.)) at will, since we can always find a suitable f( 7T -I, + 1(J.)) E G such that the complete product g.( f; 7T) is an element of C' C G. Hence there exist

mappings f: n-. G such that a(f; 7T )=(a ik ) for our fixed 7T E Sn' Multiplying this number of mappings by the number n!jIIkka*a k ! of admissible 7TESn , we obtain the desired result: 4.2.10 LEMMA. If G is a finite group consisting of s conjugacy classes, then the class of elements of type (a ik ) in GwrSn is of order

If we want to evaluate the order of a single element (f; 7T)EGwrSn , we consider the equation

4.2.11 It implies that the order of (f; 7T) is a multiple of the order of 7T. The order is the least common multiple of the lengths of the cyclic factors of 7T (cf.

of'1T

144

Representations of Wreath Products 4.2

1.2.14). Now let the exponent r in 4.2.11 be a multiple of the length of each cyclic factor of 'Tr. If i E nand J. is the least point in the cyclic factor of 'Tr which contains i, say 'Tr/l(J~)=i, then

if r=w·l•. This shows that the following holds (cf. 1.2.14): 4.2.12. 1«/; 'Tr)I=lcm{kw,laik(/; 'Tr»O}, elements in C c;:;; G. 4.2.13

LEMMA.

If

Wi denotes the order of the

If G is ambivalent, then GwrSn is ambivalent.

Proof If

in cycle notation, then

in cycle notation. Hence 1 -1)-rr- I ( . )-( g. ((f ','Tr )-1)-g. (r~l. J",-l, 'Tr - J " , - l . • 'J",-/('l J. - g. (f','Tr ))-1 .

If G is ambivalent, then (g.( f; 'Tr)) -I is a conjugate of g,,(/; 'Tr), so that in this case we have a( f; 'Tr) =a« f; 'Tr) -I). •

This yields the first result on the character table of a complete monomial group: 4.2.14 COROLLARY. If G has a real character table, then GwrSn also has its character table over ~.

A special case is 4.2.15 lent.

COROLLARY.

For each m and n the wreath product Sm wrS" is ambiva-

For general wreath products Gwr H we notice the following: 4.2.16 H.

LEMMA.

Ambivalency of Gwr H implies the ambivalency of both G and

145

4.2 The Conjugacy Classes of GwrSn

Proof If Gwr H is ambivalent, then for every (f; 7T) E Gwr H there exist (/'; 7T')EGwrH which satisfy

This implies that 7T is conjugate to 7T - I, i.e. H is ambivalent. Choosing a constantfEGo, say f(i):=g, l~i~n, then for 7T:=1 we obtain from the equation above that there exist (/'; 7T')EGwrH which satisfy f'ff'-I = f' f'11' f' - I = f -I. But this implies that g and g - I are conjugates. Hence G has to be ambivalent as well. • A numerical example is provided by S3 wr S2:

1/;[S3 wr S2] = ({ 1, (12), (13), (23), (123), (132)}

EB {I ,( 45),(46),(56), (456), (465)})' {I, (14)(25)(36)} ~ S6' Numbering the conjugacy classes of S3 in the following way: CI:=Ps,},

C 2: = {(12), ( 13), (23 )} ,

C 3 := {(123),(132)},

then the types of S3 wrS 2 are

H(~ H(~ H(i H(1 H (: H(~ n(~ n(~ H (~

Images 1/;«(/; 7T)) of representatives (/; 7T) of these classes are the elements I s6 '

(12)(45),

(123)(456),

(12),

(12)(456), (14)(25)(36),

(123),

(15)(2634),

(153426).

The orders of the classes are 1, 9, 4, 6, 4, 12, 6, 18, 12. For later purposes it is useful to notice the following fact concerning the cycle partition of the image 1/;«(/;(1" 'n))), (/;(1" ·n))ESmwrSn in Smn: 4.2.17. If a = (a l , ... , ah)~m denotes the cycle partition of gl(/; (1 ... m )), then

146

Representations of Wreath Products 4.3

This is already suggested by the preceding examples and is easy to check. 4.2.17 shows clearly that the pth cyclic factor U.·· ·7T'·-JU.» of p contributes to the cycle partition a( 1/;((/; 7T»), (/; 7T) E Sm wr S" , just the summand 4.2.18 so we obtain the cycle partition of 1/;(( j; 7T» as the following sum of cycle partitions of the cycle products: c( 7T)

a(1/;((f;7T)))= ~ 1.·a(g.(f;7T)).

4.2.19

4.3

.=

J

Representations of Wreath Products over Algebraically Closed Fields

Let G denote a group, F a field, and D a representation of Gover F with representation space V. We show first how D yields in a natural way a representation of Gwr H, H a given subgroup of S". In order to do this we form the n-fold tensor power 0"Vof the vector space V with itself: 4.3.1

o" V: = V®

F .•.

® F V,

n factors.

We recall from multilinear algebra that the additive group of this vector space is generated by the tensors v\® ... ®v", i.e.

o" V=(v ® ... ®vnlv E V).

4.3.2

i

J

Let {b\, ... , bm } be an F-basis of V: for short,

For each mapping

q:J:

n -> m, i.e.

n

q:J E m

,

we put

4.3.3 and obtain in this way a basis of 0" V: 4.3.4

o" V= «b

The representation D turns V into a left G-module: 4.3.5

'VgEG,vEV gv:=D{g)v,

4.3 Representations of Wreath Products over Algebraically Closed Fields

147

so that 0 n v turns into a left Gwr H-module if we define the action of (f; 'fT) E Gwr H on an element bcp of the basis by

This yields for the action of (j; 'fT) on a generating element:

This last equation shows clearly the irrelevance of the chosen basis of V. 4.3.6 defines a representation of Gwr H with representation space 0 n v. Its restriction to the base group G* is essentially the outer tensor power #nD of D with itself. This suggests denoting the representation defined in 4.3.6 by 4.3.8

A numerical example illustrates this. We would like to evaluate the matrix representing «12),1,(123);(132» under the representation (#3[2,1J)- of S3 wr S3' We notice first that

(~[2,1] r((12),I,(123);(132)) =(

~[2,I]r«(l2),I,(l23);lsJ(~[2,I]r(e;(132)).

The seminormal form of [2,1] yields (cf. 3.3.29)

[2,1]«12))=(

-6

~),

[2,1](23))=(

~

-i),

so that [2,1](123))= ( -

~

-~ _1

) .

2

Let {b l' b2 } be the corresponding ordered basis of the representation space Vof [2,1]. Then (e;(l32» acts on the 8 elements bcp of the basis of

148

Representations of Wreath Products 4.3

@3 Vby place permutation as follows:

b
......

bt@b,@b, =b
b
......

b t@b 2 @b, =b
b
......

b2 @b t@b, =b
b
......

b2 @b 2 @b, =b
b
......

b,@b J @b 2 =b'P2'

b'P6: =b2 @b t@b 2

......

b t@b 2 @b 2 =b'P4'

b'P7: =b2 @b 2 @b,

......

b2 @b,@b 2 =b'P6'

b'P8: =b2 @b 2 @b 2

......

b2 @b 2@b 2 =b'P8'

Hence with respect to this ordered basis {b'PI'"'' b'P8} of @3 Vwe obtain

(it [2,1] f(e;(132))=

1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 0 1 0

0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1

Furthermore we have

(it [2, 1] ) - ( (12), 1, (123); 1) = ( - b I

2

1

0

0

I

0

4

-1

2

0

0

, 2

0

0

-1

0

0

0

0

~) @(b

0

0

n@( -~

~)

-

_1

2

0

0

0

0

0

0

0

0

0

3

, 2

0

0

0

0

0

0

0

0

0

_1

3

0

0

-2

0

0

_1

4

0

2

4

1

1

0

0

0

0

0

0

0

0

0

0

0

0

2

3

-4

_1 2

149

4.3 Representations of Wreath Products over Algebraically Closed Fields

so that altogether

(k [2,1]) -((12).1,(123);(132)) I

0

"2

-I

0 I

0 0

"2

0 0 0

-1

0

0

_1

0 0

0

0

0

0

0

2

0 0 0 0 0

I

0

0

I

-2"

0

3

0

0

0

I

"2

0

0

0

0

3

0

0

I

"2

0

0

0

3

0

4

0 0 0 0 0

4

4 I

0 0 0

"2

0

0

_2

0

_1

4

2

We would like to know the character of the representation (#nD)-. If again gv( f; 1T), I,,;;; /I";;; c( 1T), are the cycle products of 1T with respect to f (see 4.2.1), we obtain the following result, which is very useful:

Proof By 4.3.4 the trace of ( f; 1T) is just x(#n D)- (f;

1T) =

L [coeff. of bcp in (f; 1T )bcp]. 'P

Let [) denote the matrix representation corresponding to D with respect to the ordered basis {b!,,,,, bm } of V, and put 'VgEG

((d'k(g)):=[)(g)).

Then by 4.3.6

(f; 1T )bcp = ( L d il . cp(7T- 1(I)/ f(l) )b'l ) 0 ... 0 ( L d'n. 'P(7T- (n»)(f( n) )bin ) 1

11

L I ~jl •...• in~n

In

d il .'P(7T- '(I»(f(l))·· ·d,,,.cp(7T- '(n)(f(n ))b iI0 ... 0b'n

Representations of Wreath Products 4.3

150

In particular the coefficient of

b
in this sum is equal to

the factors of which can be rearranged in order to obtain C("")

II

d
• =1

Taking the sum of these expressions over all desired character x(#nD)-(j; 1T)=

II

f( 1T -1( J~») )....

we obtain for the

trace lD(g.(j; 1T»

•

•

as was stated. Let us illustrate 4.3.9 by a few examples:

(i) f(#n D)- : =:x(#" D)- (e; 1) =m n, (ii) X(#nD)-(e; 1T )=m

C

(7T),

n

(iii) X(#"D)-(j; 1)= 4.3.10

~

r,

(iv) X(#"D)-(g,

, g;

(v) X(#"D)-(g,

,g;(I·· ·n»=:XD(gn),

OO.MCA"",--+ (vi) X(#n D)- (g,

I '" "th"

II XD(j(i », ;=1

1) =XD(g

, g; 1T)

= IT

XD(gk

r

k (7T).

k= I

The restriction of (#nD)- to the complement H' of the base group G yields a representation of H which depends only on the dimension m of V. We therefore denote it by Pnm : 4.3.11 This is in fact a representation of Sn' 4.3.1O(ii) shows that we have for its character: 4.3.12

4.3 Representations of Wreath Products over Algebraically Closed Fields

151

Besides this restriction of (#nD)- to H' or S~, we have the restriction of

( # n D) - to the diagonal subgroup diag G* of the base group (d. 4.1.16) which is, as a representation of G, the n-fold inner tensor power 0 n D of D with itself: 4.3.13 This shows that we have: 4.3.14 'VgEG,7TESn

((# D

r(

g, ... , g;

7T )

=

®D ( g ) . P

m n ( 7T )

= Pnm ( 7T ).

®D ( g) ).

Later on we shall return to this equation. In the earlier part of this section we saw how a representation D of G yields in a very natural way a representation (#nD)- of Gwr H. It is even easier to see that a representation D" of H gives a representation D' of GwrH, for we need only put for (f; 7T)EGwrH: 4.3.15

D'U; 7T): =D"( 7T).

Hence given representations D of G and D" of H yield in a very natural way the inner tensor product of certain representations of Gwr H: 4.3.16

(D;D"):=( #D f0D',

If again I denotes the identity representation of the group in question, then we have in particular

4.3.17

(# D

r

=(D; 1)

and D'={I; D").

It is our aim now to show that each irreducible representation of Gwr H over an algebraically closed field can be induced from a suitable product of such representations (D; D"). Let us therefore assume for the rest of this section that the ground field F is algebraically closed and that G is finite. Let D J, ... , Dr denote a complete set of pairwise inequivalent and irreducible representations of Gover F with corresponding representation modules V', ... ,vr and corresponding matrix representations O', ... ,or. Then each irreducible F-representation of G* is of the form 4.3.18

D*:=D J# "'#Dn =: # D" i

152

Representations of Wreath Products 4.3

where Di E {D 1, ••• , Dr}, with representation module

V*:=V1# '" #Vn =: #

4.3.19

v"

I

where

v,: = Vi, if D =DJ. The underlying vector space is

4.3.20

VI 181 F' . ·181 F

i

v" =:

Q9

v,.

I

If ni denotes the number of factors Di of D* which are equal to DJ, I ~j~r, then

4.3.21 is called the type of D*. We denote by Sn ) the subgroup of Sn C~ H) consisting of just those . permutations which move at most the indices i of factors Di equal to DJ, if n}>O, or Sn :={lsJ if nj=O. Now the inertia group of D* is defined as 4.3.22 where ~ denotes equivalence of representations and D*(/:"') is as usual the representation conjugate to D* as follows: 4.3.23 Since

G*~Gwr H D .,

the inertia group is obviously a product

4.3.24

GwrHD·=G*·H'o.

of G* with a subgroup H'o. of the complement H' of G*. H'o. will be called the inertia factor of D*: 4.3.25 We notice that

4.3.26 and would like to show how H'o. can be expressed in terms of the subgroups Sn introduced above. }

4.3.27

LEMMA.

If we put

S~

)

:={(e; '7T)I'7TESn

' . -S(n)'

II S' . }

}, I

s::S'n'

n· -.-:::: )

I~j~r,

and

4.3 Representations of Wreath Products over Algebraically Closed Fields

153

then we have for the inertia factor H'n. of D*

so that for the inertia group of D" the following holds:

Proof We have [}*(e;1T)((f; I))=[)*((e; '17)-'(f; I){e; '17)) =[)*((f1T-1; I))

=[)I(J('17{I)))X ···X[)n(f{'17{n))). The question is for which '17 E H this representation is equivalent to [j) *. Since both [j)* and [j)*(e; 1T) are irreducible representations, we can use a character-theoretical argument. (i) If '17EHnS(n)' then the trace of [j);(f('17(i))) is equal to the trace of so that in this case for each f:

[j) 1T( i)(f( '17(i))),

trace [j)*( (f; I)) = trace

[j)*(e; 1T)((f;

I))

and hence (Hn S(n))',,;;;;;H'n•. (ii) On the other hand if (e; '17) belongs to the inertia factor, we have for eachf trace [j) * ((f; l)) = trace [j) *(e; I'T) ((f; I)) = trace [j)* ((f1T -I; I)). If we choose (f; I)EGi , i.e. f(j)=l for each j=l=i, then we obtain the equation (put f;: = dim( D)):

This holds for each f(i)EG. Let us consider the associated Brauer characters (if char F= p) or the traces (if char F= 0), rpr Then we can simplify the equation to obtain f{i)EG.

Since characters and Brauer characters of irreducible modular representations are linearly independent, we obtain rpi =rp1T-1(i)' so that Di ~ D1T (i) and hence'17EHnS(n)' •

154

Representations of Wreath Products 4.3

According to Clifford's theory of representations of groups with nontrivial normal subgroups, we now have to extend D* to a representation of its inertia group. It is not always possible to extend a given linear representation of a normal subgroup to a linear representation of its inertia group. But in our case here, where G* is the normal subgroup, we are fortunately able to extend D* to a linear representation of Gwr H D ., as we already saw at the beginning of this section when we extended #nD of G* to (#nD)- of its inertia group G wr Sn in G wr Sn' The corresponding extension procedure yields from the left G*-module #iV;, which affords D* a left Gwr HD.-module (#;V;)-, where the action of (f; 7T)EGwr HD' is defined as in 4.3.7:

We denote this extension of D* by

It is important to notice what this means for the corresponding matrix representation (D*)-. If 4.3.28

D*((f; l»=D,(f(l»X

= (d}'h(f(l»

XDn(f(n»

di:Jn(f(n»)

then by 4.3.7 for each (f;7T)EGwrHD •

which means that 4.3.30. (D*)-«f; 7T)) arises from D*(f; 1) by a suitable column permutation. Now let D" be an irreducible F-representation of HnS(n), and define D' by 4.3.31

D'( (f; 7T»: =D"( 7T),

for each (f; 7T) E Gwr H D" Multiplying these two representations yields the inner tensor product

with the representing matrices 4.3.32

(D *) - @ D'( ( f; 7T ) ) = (D *) - ( ( f;· 7T ) ) X D' (( f; 7T ) ).

4.4 Special Cases and Properties of Representations of Wreath Products

155

Clifford's theory says that (D*)-(j9D' is an irreducible F-representation of Gwr H D *, and that the induced representation

is an irreducible F-representation of Gwr H, but even more (d. Clifford's original paper [1937]): 4.3.33. D: =(D*)-(j9 D' i Gwr H is irreducible, and every irreducible Frepresentation of Gwr H is of this form. It remains to investigate which representations D* and D' (or D") have to run through in order that D shall run through exactly a complete system of pairwise inequivalent and irreducible F-representations of Gwr H. Using Clifford's notation, we call two irreducible representations of Gwr H associated with respect to G* if their restrictions to G* have an irreducible constituent in common. From Clifford's theory we know that there is a I-I correspondence between the classes of associated representations of Gwr H and the classes of representations of G* which are conjugate with respect to Gwr H. (Two representations D* and D** are conjugates with respect to GwrH if there exist (f;7T)EGwrH such that D*U;'1T)_D**.) And this correspondence is as follows. The restriction to G* of every element out of a class of associated representations is (up to a multiplicity) just the corresponding class of conjugate representations. Hence it suffices that in D the representation D* runs through a complete system of representatives of the classes of conjugate representations of G*. Moreover Clifford's theory yields that associated representations differ only in the facto!" D'. This yields the following main theorem:

4.3.34 THEOREM. The irreducible F-representation D=(D*)-(j9D'iGwrH runs just through a complete system of pairwise inequivalent and irreducible F-representations of Gwr H if D* defined by 4.3.18 runs through a complete system of pairwise not conjugate but irreducible F-representations of G*, and, while D* remains fixed, D" runs through a complete system of pairwise inequivalent and irreducible F-representations of H n S(n)"

4.4

Special Cases and Properties of Representations of Wreath Products

In the case when H=Sn' two representations of G* are conjugates if and only if they are of the same type. Hence D runs through a complete system of irreducible F-representations of the complete monomial group if D* runs through a complete system of irreducible F-representations with pairwise

156

Representations of Wreath Products 4.4

different types and D", while D* is kept fixed, runs through a complete system of pairwise inequivalent and irreducible F-representations of S(n)' Furthermore each irreducible F-representation D" of S(n)' where (n)= (n I ' " ' ' ns)' is equivalent to a representation of the form D"=D;'# .. , #D;',

where D;' is an irreducible F-representation of Sn" 1,,;;;;;i,,;;;;;s. Since 4.4.1

GwrS(n) "'" X GwrSn " J

we have in this case (cf. 4.3.16): 4.4.2

... # (D' ( D*)~®D'= (D 1 ., D")# ] s' D")=' s'

# (D' D") ;

I'

I

'

so that we obtain

4.4.3 THEOREM. The representation

runs just through a complete system of pairwise inequivalent and irreducible F-representations of GwrSn if D* = #;D; runs through a complete system of representations of G* which are of different type (n) = (n I' ... , n s)' while if D* of type (n) is kept fixed, D" = #; D;' runs through a complete system of irreducible F-representations of S(n)'

In accordance with 4.2.9 we obtain the following corollary on the number of ordinary irreducible representations: 4.4.4 COROLLARY. The number of ordinary irreducible representations of GwrSn is equal to ~p(nl)" ·p(nJ (n)

if s denotes the number of conjugacy classes of G, p( m) the number of partitions of m, p(O): = 1, and the sum is taken over all the types (n) = (n l ,. .. , nJ. As an example we derive the ordinary irreducible representations of S3 wrS 2 ·

...4 Special Cases and Properties of Representations of Wreath Products

157

(i) The representations of the base group Sj, their types, inertia groups and inertia factors. Sj "" S3 X S3 has the following ordinary irreducible representations:

[3]#[3], [3]#[2,1], [3]#[1 3], [2,1]#[3], [2,1]#[2,1], [2,1]#[1 3], [1 3]#[3], [1 3 ]#[2,1], [13]#[1 3]. With respect to the ordering D 1 : =[3], D 2 : =[2,1], D 3 : =[1 3] of the ordinary irreducible representations of S3' the types of these representations are (2,0,0), (1,1,0), (1,0,1), (1,1,0), (0,2,0), (0,1,1), (1,0,1) (0,1,1), (0,0,2). Hence a complete system of irreducible ordinary representations of Sj with pairwise different types is

The corresponding inertia groups are S3 wr S2'

s;,

Sf, S3 wr S2'

s;,

S3 wr S2 .

The inertia factors are S2'

s;, s;,

S2'

s;,

S2'

(ii) The representations of S3 wrS2: The ordinary irreducible representations of S2 are [2] and [12], the only one of S, is [1]. Thus we get for the ordinary irreducible representations of S3 wrS2 ([3]; [2]) = ([3]#[3])-0 [2]' =([3]#[3])-, ([3]; [12]) = ([3] # [3]) - 0 [1 2]', ([3]; [1 ])([2,1]; [I]) = «[3] # [2, 1]) - 0 ([ 1] # [1 ])')j S3 wr S2 =[3]#[2,IJj S3wrS 2' 3 ([3]; [1])([ 1 ]; [1])=«[3]# [1 3])- 0 ([ 1]#[ II)') i S3 wrS 2 = [3]#[13] i S3 wrS 2, ([2,1];[2])=([2,1]#[2,1])-0[2]'=([2,1]#[2,1])- , ([2,1]; [1 2]) = ([2,1]# [2,1 D- 0 [1 2]', ([2, 1]; [1 D([ 13]; [1]) = «[2,1] # [13]) - 0 ([ 1]# [1 ])')j S3 wr S2 = [2,1]#[ 13] i S3 wrS 2, ([ 13]; [2]) = ([ 13]# [1 3]) - 0 [2]' = ([ 13]# [1 3]) - , ([13]; [12]) = ([ 13]#[ 13])- 0 [1 2]'.

158

Representations of Wreath Products 4.4

Their dimensions are 1,1,4,2,4,4,4, I, I in agreement with

(iii) Representing matrices. ~~~cl

As a numerical example we shall evaluate '

([ 2, 1] ; [I2]) (( e; (12)) ) . First of all we notice that

[2,1]#[2,1]((e;I))=(6

~)

I

_ -

(

0 0

o

n

(6 o

o o

I

o o

~)

I

o

so that (use 4.3.29)

0) 0

oI o o

0 I

(we have to transpose the second and the third columns). From [12]((12))= ( - I), we obtain now

([2,1)# [2, I) po [I'

]'« e; (12)))~

(~

0 0

0

I

0 0

0 -I

0 0 0

I

0 0 -I

0

'lX(-ll 0 -I

0 0

0 0 0 -I

Another case of particular interest is the wreath product Gwr H, G being abelian, e.g. the wreath products Cm wrSn , Cm a cyclic group of order m, the

4.4 Special Cases and Properties of Representations of Wreath Products

159

so-called generalized symmetric groups. In this case the procedure for getting D from D* becomes much simpler, since D* is one-dimensional, so that (D*)-(j;

1T

)=D*(j; 1)= IlDJf(i)). ;

If 1T\, ... , 1TIH: Hns(n,1 is a complete system of representatives of the left cosets of the inertia factor HnS(n) of D* in H, and D" is again an irreducible representation of HnS(n)' we use the notation (0 *) -

0 0 ' ( ( f",~ I; 1T; :=

11T1Tk ) )

{(O*) - X 0 ' ( (f"i~ I; 1T; -l1T1Tk ) ) O-matrix

if 1T;-l1T1Tk E S(n) n H, otherwise.

With this notation in mind we obtain for the representing matrices:

o ((j; 1T )) = 4.4.5 =

( (0 *) -

@0 ' ( (J"i~ I;

1T; - 11T1Tk )) )

((O*)-((J",-I; 1))· ~,,( 1T;-I1T1Tk )),

where ;,." ( lI.JI

1T;

- I ) .= 1T1Tk •

{ 0" ( 1T; -

11T1Tk )

O-matrix,

if 1T; -l1T1Tk EH n S(n)' otherwise.

Since on the other hand

we have 4.4.6 LEMMA. If D* is one-dimensional, then we have for the representing matrices of D=(D*)-f2)D ' iGwr H:

i.e., the IH: Hns(n)1 2 submatrices of this matrix O«(f; 1T)) are, up to their numerical factors O*«(f" ~J; 1)), just the submatrices of the induced representa' tion (0" i H)( 1T).

160

Representations of Wreath Products 4.4

Another conclusion which can be drawn from the fact that ([)*) -«(f; '7T» arises from [)*«f; 1» by a suitable column permutation concerns the question whether Gwr H is an M-group. A finite group is called an M-group, if every irreducible representation of G over an algebraically closed field F with char Flj G I is induced by a one-dimensional representation of a suitable subgroup. 4.4.7 THEOREM. If G and the subgroups H n S(n) are M-groups then Gwr His an M-group, too. Proof If F is algebraically closed and char FlIGI, then every irreducible representation [)' of G (being an M-group), and hence also of G*, is equivalent to a representation whose matrices contain in every row and column exactly one nonzero entry. This holds also for [)" of HnS(n), and hence both [)* and [)' and therefore ([)*)-@ [)' and even [) = ([)*)-@[)'i Gwr H can be assumed to be of this monomial form. Since [) is irreducible, we obtain from a well-known theorem (cf. Huppert I, [1967] V, 18.9) that this implies that [) is induced from a one-dimensional representation of a suitable subgroup. •

We notice from this proof that properties carry over from G and the groups Hn S(nl to Gwr H if they remain valid under forming outer tensor products and inner tensor products of representations as well as under column permutations and induction. For example: 4.4.8 THEOREM. If KeF is a splitting field for G and all the subgroups Hn S(n), then K is a splitting field for Gwr H, too, i.e. every irreducible F-representation of Gwr H is realizable in K. 4.4.9 COROLLARY. Each field is a splitting field for Sm wrSn·

Further properties show up as soon as we consider the characters. In the same way as we got 4.3.9, we obtain for the character X(D')-0D' of (D*)-0D' C(1T)

4.4.10

X(D')-0D'((j; '7T»=X D'('7T)

II

XD"')(gv(j; '7T».

v=1

Since by 4.3.34 each irreducible character of Gwr H arises from such a character by induction, we obtain various corollaries on the characters of GwrH, a typical one being (cL also 4.2.16 and 4.4.8): 4.4.11 COROLLARY. If the ordinary irreducible characters of G as well as of the subgroups HnS(n) of H are rational integral (real), then all the ordinary irreducible characters of Gwr H are rational integral (real) as well.

4.4 Special Cases and Properties of Representations of Wreath Products

161

Exercises 4.1 Describe the centralizer CGwrS((j; 7T)) of (j; 7T) in GwrSn in order to determine which conjugacy classes of GwrSn split over GwrA n and which split over

for a given subgroup N of index 2 in G. 4.2 Deduce from Exercise 4.1 a characterization of the conjugacy classes of the Weyl groups of type Dn (cf. 4.1.33). 4.3 Which of the ordinary irreducible representations of S2 wrSn split over (S2 wrSn)A" and which of them form pairs of associated representations? 4.4 Prove along the lines of Exercise 4.3 that the character tables of Weyl groups of type Dn are rational integral.

CHAPTER 5

Applications to Combinatorics and Representation Theory

In this chapter we consider some applications of the preceding results on the representation theory of symmetric groups and of wreath products to combinatorics and to the representation theory of general finite groups. The applications to combinatorics cover the so-called P6lya enumeration theory which is an important part of combinatorics and which has at present its main applications in the theory of enumeration of isomorphism types of discrete structures such as graphs and boolean functions. This theory is covered because a quite general problem in the theory is concerned with the action of the wreath product [G]H=O[GwrH] on the set mn of all the mappings from n into m. The applications to the representation theory of a general finite group G arise from the fact that the direct product G X Sn is nicely embedded in GwrSn. Hence for any ordinary representation D of G we have that &rD breaks up into representations D 0 [a], a ~ n, of G, the so-called symmetrized products of D with the ordinary irreducible representations [a] of Sn' Conversely each irreducible constituent D; of Q!) n D yields an ordinary representation D 8 n D; of Sn' the so-called nth permutrized product of D and D;. The values of the characters of these representatiotls D 8 n D; appear quite often in the representation theory of G, so that we can derive certain results on the representation theory of G by simply recognizing that these numbers are values of characters of symmetric groups. We shall return to the theory of symmetrized products in Chapter 8, and more concrete examples of the concept are given there. Another application is provided by the notion of the plethysm of representations of G,,;;;;;Sm and H,,;;;;;Sn which arises from the fact that Gwr H isa ENCYCLOPEDIA OF MATHEMATICS and Its Applications, Gian-Carlo Rota (ed.). Vol. 16: G. D. James and A. Kerber, The Representation Theory of the Symmetric Group ISBN 0-201-13515-9 Copyright © 1981 by Addison-Wesley Publishing Company, Inc., Advanced Book Program. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the publisher. 162

163

5. J The Polya Theory of Enumeration

subgroup of Smn in a natural way. Furthermore we can apply the representation theory of Sn to the representation theory of multiply transitive subgroups of Sn by showing that certain representations remain irreducible when we restrict. We illustrate this in the case of the Mathieu groups. Also, characterizations of multiple transitivity are given in terms of the cycle structure of the elements of the group in question.

5.1

The P61ya Theory of Enumeration

This theory is concerned with the set mD:={cpjcp:n->m} and certain equivalence relations defined on mD with the aid of two given permutation groups G~Sm and H~Sn' The problem is to get as much information as possible on the equivalence classes. Before we formulate a typical example which is taken from graph theory, where the main applications of enumeration theory lie, we introduce three equivalence relations - i, i = 1,2,3, on mD which cover most of the problems in this area: 5.1.1

(i) CP-Ilf; :=

3wEH

(CP=lf;oW- I ), (CP=golf;oW-'),

(ii) CP-2lf;

:=

3wEH,gEG

(iii)cp-31f

:=

3{f;w)EGwrH 'ViEn

(cp(i)=/U)(If(W-I(i)))).

The definitions 4.1.26 up to 4.1.32 of the groups E H, G H , and [G] H show immediately that the classes of -), -2, and -3 are the orbits of these permutation groups in their action on mD. First of all we want to know the total number of orbits; later on we shall ask for details like the number of orbits with specific properties or for a complete system of representatives and so on. Before we try to answer such questions we provide a typical example. We compute the number of nonisomorphic graphs with p points. We consider a graph with p points as a mapping

where p[2J:={I, ... ,(~)}

is taken to be the set of the (~) pairs of points of the graph, and cp( i) : = 1

164

Applications to Combinatorics and Representation Theory 5.1

if and only if the i th pair of points is disconnected, while q?( i) : = 2 if and only if the ith pair of points is connected. For H we take the group

which is induced by Sp (acting on the p points) on the set

p[2]

of the (~)

pairs of points. It is clear that an orbit of the group

consists of a complete class of isomorphic graphs q? Hence the number of nonisomorphic graphs with p points is equal to the number of orbits of this group, which can be evaluated by an application of the Cauchy-Frobenius lemma (which is usually called "Burnside's lemma"; we follow Neumann [1979]) as follows: If 17ESp induces 17[2] on p[2], then 17[21 leaves exactly

mappings q? fixed, for it is easy to see that q? is left fixed if and only if q? is constant on each cyclic factor of 17[2]. Hence the Cauchy-Frobenius lemma yields the following answer to our problem: 5,1.2. The number of nonisomorphic graphs with p points is equal to

If p : = 4, a complete system of representatives of the isomorphism classes IS

5.1.3

Ixi I_: [/: :xI j'< 1-.

[Xl

IxI

This is a typical problem where the acting permutation group is of the form E H. We call it a Polya-type problem. An obvious generalization of 5.1.2 is 5.1.4 POLYA'S mn is equal to

THEOREM, CONSTANT FORM.

_1_ '"

.:.. m IHI 1TEH

C(77)

The number of orbits of E H on

.

165

5.1 The P61ya Theory of Enumeration

Problems where the group is of the form G H are called de Bruijn-type problems. A typical problem of this kind arises from the foregoing one by asking for the number of classes of graphs with p points which are isomorphic up to complementation. (The graph cp is said to be the one complementary to 1/; if cp(i)= I if and only if 1/;(i)=2.) Representatives of these classes for p : = 4 are (d. 5.1.3)

•

•

•

•

e_e

e __ e

e

e

e_e

e_e

e_e

e

e

I

e_e

e __e

1

e

1/

In order to solve this problem we need only take H: = Sf] as before and put G: = S2' It is obvious that the orbits of

are just the desired equivalence classes of graphs. Since G H =8[diagG*·H'], the representation of G H on ron is just the restriction 5.1.5 if N denotes the natural representation of G which has the character 5.1.6 Thus our character formulae 4.3.9 and 4.3.1O(vi) together with the CauchyFrobenius lemma yield the following two theorems: 5.1.7 POWER-GROUP ENUMERATION THEOREM, CONSTANT ber of orbits of the power group G H on ron is equal to I GIIHI

I

~

.:..

FORM.

The num-

n

IT (k )a k(7T)

._ a l g

(g.7TjEGXH k-l

.

5.1.8 EXPONENTIATION-GROUP ENUMERATION THEOREM, CONSTANT FORM. The number of orbits of the exponentiation group [G]H on ron is equal to

An immediate corollary of 5.1.7 is the following solution of the graph

166

Applications to Cornbinatorics and Representation Theory 5.1

enumeration problem introduced above: 5.1.9. The number of classes of graphs on p points which are isomorphic up to complementation is equal to

For

II a l(gk k

rk(,,[21)

=

{2

C ("12

1)

if g= (12) and each a2i+l( 77[2 J )=0, or g= 1, otherwise.

0

A combination of results on de Bruijn-type problems with results on P61ya-type problems yields further theorems of interest, as the following example shows. It is easy to see that twice the number of classes mentioned in 5.1.9 minus the number of classes in 5.1.2 is equal to the number of isomorphism classes which remain fixed under complementation. Thus 5.1.10. The number of self-complementary isomorphism classes of graphs on p points is equal to I

~

p!

2c (,,[2)).

"ESp 21k

=

ak(,,[2I)=o

Enumeration problems where the exponentiation group [G]H is involved are very rare. Furthermore, each such problem can be reduced to a P61ya-type problem, since it is not G which really matters but the orbits of G (see the exercises). However, problems often occur which are P61ya-type but where we have an exponentiation group in the exponent: H=[P]Q. A typical example is the enumeration of classes of boolean functions. Up to now we have used only the Cauchy-Frobenius lemma together with our results on characters of wreath products. But there is of course much more theory involved in enumeration under group action. Having evaluated the total number of orbits, one asks next for the number of orbits of a given content, say for the number of isomorphism 1 classes of graphs with p points and e edges. The graph ({J E 2 P[2 has e edges if and only if ({J takes the value 2 e times, i.e.

Hence more generally we say the ({J E mn is of content "A: = ("A I' ... , "A m) if and

167

5.1 The Polya Theory of Enumeration

only if for each i E m

Asking for the number of classes of functions same as asking for the generating function

!p

of content A is therefore the

where the monomial

has as its coefficient the number of classes of functions of content A. For example, by 5.1.3 this generating function for the enumeration of graphs of 4 points by content is just the polynomial 5.1.11 We would like to describe a systematic approach to this enumeration of classes by content. One starts with a weight function 5.1.12 which yields the mapping 5.1.13 We call the functions w* which arise in this way multiplicative weights on mn • When w is constant on the orbits of G the resulting multiplicative weight w* is constant on each orbit of [GjH, i.e. w* is [GjH-compatible, and hence also G H- and E H-compatible. (In fact w* is E H-compatible for each weight function w; we need not assume that w is G-compatible in the case of a P61ya-type problem.) Using such a compatible weight w* on mn , we can define a weight Won the set ~ of orbits w by putting for any

!pEw.

It is our aim to choose w in such a way that the polynomial 5.1.14

~ W(w) wEQ

168

Applications to Combinatorics and Representation Theory 5.1

is the generating function for the enumeration problem in question. Putting w = 1 for example, 5.1.14 yields just the number of orbits, while the generating function 5.1.11 for our graph-theoretic problem introduced above is obtained by putting w(l) : = Z I' w(2): = Z 2' The orbits of functions are sometimes called patterns, while the generating function 5.1.14 is called the pattern inventory. We need a systematic approach to the evaluation of the pattern inventory. We remember that P6lya's theorem, the power-group enumeration theorem, and the exponentiation-group enumeration theorem were obtained by an application of our results on characters of wreath products and the Cauchy-Frobenius lemma to the exponentiation group [G1 H =O[Gwr Hl; we would like to show that the same character together with a variation of the Cauchy-Frobenius lemma yields the corresponding theorems in weighted form. The variation of the Cauchy-Frobenius lemma is: 5.1.15 CAUCHy-FROBENIUS LEMMA, WEIGHTED FORM. Let 8: A->SM denote a permutation representation of the finite group A on the finite set M. Let furthermore w*: M->(P[x l , ... , xrl denote a mapping which is constant on the orbits w of 8[Al, W( w) denoting this value. Then 1

~W(w)=18[A]1 w

_1

IAI

~

'7TE8[A]

~

w*(m)

mEM '7T(m)=m

~

~

aEA

mEM 8(a)(m)=m

w*(m).

The proof of this lemma is essentially the same as the usual proof of the Cauchy-Frobenius lemma in its constant form via a double count of the entries of a matrix (bma ), where b

.={w*(m)

maO

0

if 8 ( a )( m ) = m , otherwise.

We leave the proof to the reader. From the equations of 5.1.15 we see that in order to apply this weighted form we have to determine the fixed points of 8(a). aEA, and evaluate their weights. The first part of this has usually already been done if we have evaluated the permutation character of 8. We consider the example

0: Gwr H ->Smn.

169

5.1 The Polya Theory of Enumeration

cp E m" is left fixed by ()((f; is true (see 4.1.28):

71

», (f; 71) E Gwr H, if and only if the following

5.1.16. (i) For each 1', cpUv) is a fixed point of gv(f; 71), and (ii) the further values of cp on this cycle of 71 are obtained from cpUv) as

follows: cp ( iv ) =f( iv ) ( cp ( 71 -

1(

iv ) ) ) = f( iv )f( 71 -

1(

iv ) ) ( cp ( 71 -

2(

iv ) ) ) = ....

In order to evaluate the weight of such a cp we recall that w is assumed to be constant on the orbits of G, so that

w( cp ( i,. ) ) = w( cp ( 71 -

I(

iv ) ) ) = ... = w( cp ( 71 -I, + 1(iv )) ).

Hence such a cp, which is left fixed by 8((f; 71», has the weight

w*( cp) =

II w( cp(Jv ))/'. v

This gives (as we can freely choose the values cp(J,,); d. 5.1.16(i»

5.1.17

~ If!

B«j.1T»(
w*(f)=

cW

I

~

.W(J)/').

v-I)

\ g,(/: 1T)(j)=)

This together with the weighted form of the Cauchy-Frobenius lemma yields: 5.1.18 EXPONENTIATION-GROUP ENUMERATION THEOREM, WEIGHTED FORM.

If w is a G-compatible weight function on m, then the pattern inventory of the exponentiation group [G] H acting on m" is equal to

Restricting our attention to the subgroup diag G* . H' yields the corresponding theorem for the action of G H : 5.1.19 POWER-GROUP ENUMERATION THEOREM, WEIGHTED FORM. If w is a G-compatible weight function on m, then the pattern inventory of the power

170

Applications to Combinatorics and Representation Theory 5.1

group G H acting on m" is equal to

A further restriction down to E H yields: 5.1.20 POLYA'S THEOREM, WEIGHTED FORM. For any weight function w'on m, we obtain as the pattern inventory of E H acting on m":

The next definition enables us to systematize our approach. If H,,;;;; Sn' then the following element of the polynomial ring Q[x" ... , xnl is called the cycle index of H:

5.1.21 We shall sometimes display the indeterminates by writing Cyc(H;

Xl"'"

X,J

instead of Cyc(H). Using this last notation we define what we understand by Palya insertion of a polynomialp=p(z" ... , zm)EQ[zp ... , zml in Cyc(H): 5.1.22

Cyc(Hlp ):=Cyc(H; P(ZI"'" zm)"'" p(z~, ... , z:;,))

This yields a simple formulation for 5.1.20. 5.1.23. The pattern inventory of E H is Cyc(HI~j=lw(j)).

For our graph-theoretic enumeration problem we get: 5.1.24.

(i) The number of isomorphism classes of graphs with p points is equal to

Cyc( S~2] 12) =Cyc( S~2J; 2,2, ... ).

5.\ The P61ya Theory of Enumeration

171

(ii) The number of self-complementary isomorphism classes of graphs with p points is equal to Cyc(

S1 21 ; 0, 2, 0, 2, ... ).

We list the cycle indices of some particular groups: 5.1.25. (i) Cyc({ls })=xr, (ii) CYC(Cn :"=«(l,oo.,n»)=;2: i1nlI>(i)X;/i, where II> denotes the Euler function, i.e. lI>(i): = I{ k ENI k';;;i and (k, i) = I}!,

rrnk=l I ! ...) Cyc(Sn )-'" (1ll -";'a ak

(XTk )a ' k

(iv) CYC(An)=i:a(l+(-I)a2+a4+···)rr;;=la:!

(:k t,

lx x(n-I)/2 -) 2 1 2 (v) C yc(Dn)-}Cyc(Cn)+ { l( n/2 2 (n-2)/2) 4 X2 + X 1 X2

If n is odd, If

n is even.

An important case is the weight function

where we obtain the so-called group reduction function 5.1.26

Grf( H ) : =Cyc ( H

I~ Z i ) I

n

'" = I- ~

IHI

II (ZIk + ... +Zmk )a

k

(")

•

"EH k==l

Returning to representation theory, we define the generalized cycle index of H with respect to an ordinary character XD of H by 5.1.27 It is very easy to see that the following is true for H,;;;Sn: 5.1.28

Cyc( H)=Cyc(Sn' IH iSn ) = ~ (IHiSn,[a])Cyc(Sn,[a]). afn

so that in particular 5.1.29

172

Applications to Combinatorics and Representation Theory 5.1

The polynomials

are called Schur-functions or S-functions for short. They are very important. There is in fact a way of deriving most of the results of the ordinary representation theory of Sn by considering these symmetric polynomials, i.e. a polynomial approach to the representation theory of Sn' A few remarks concerning the evaluation of the cycle-index polynomial Cyc(H) are in order. It is known as soon as we know the cycle type for each conjugacy class of H, and the latter can be calculated using the permutation character a I( . ). Obviously for each 'TT E H we have 5.1.31

\fkE~

(al('TT k )=

~i.ai('TT))'

ilk

so that by an application of Moebius inversion we obtain 5.1.32 Thus a J 'TT) can be obtained from the permutation character of H as soon as we know to which conjugacy class 'TT k belongs, for each k E ~. Another useful remark is that we obtain the cycle index of a direct sum of permutation groups just by multiplication: 5.1.33

Cyc( HffiK) =Cyc( H)· Cyc( K).

Furthermore if HI ";;;;Sn" H 2 ";;;;Sn" then they define a group H/i9H2 acting on n I X n 2 as follows. The underlying set of H/i!J H 2 is HI X H 2 , and the action is defined by

Then we have, as it is not difficult to check, 5.1.34

The cycle index of the composition 5.1.35

G [ H ] : = tJ; [ H wr G ] ,,;;;; Sm n

173

5.1 The P
of G~Sm and H~Sn (cf. 4.1.18) follows directly from 4.2.19. It is the following "composition" of the cycle indices of G and H:

Cyc(G)[Cyc(H)]:=-I~1 ~ kI] 1 I~I ~ IIx~,:
5.1.36

"EG

(

Il

-

)0;<")

pEH ' -

Our description of the basic ideas of combinatorial enumeration theory would be incomplete without mentioning J. H. Redfield, who published in 1927 (ten years before Pblya's famous paper appeared) an article which was overlooked for many years, although it anticipated the most important results of the whole theory. The trouble with Redfield's results was that they were formulated in terms of his "cup" and "cap" product of polynomials, which at first glance seem to be quite obscure. (But with the aid of representation theory they can be elucidated without much effort.) This may also be the reason why Redfield's results play their own role and usually they either need special consideration or are left out of the discussion. But since they are very useful, we would like to include them by giving the outline of an approach which unifies both Redfield's results and the preceding way of describing Pblya's and de Bruijn's theorems as well as the exponentiation-group case. Besides this, the more general problem which will be formulated and solved may turn out to be helpful for further enumerative purposes. Redfield's paper deals with the so-called superposition of isomorphism classes of graphs. We don't need a precise definition of this kind of problem, since an example gives quite a good idea of what is meant. The different superpositions of the two isomorphism classes

5.1.37

are

0 /

5.1.38

o _-,

,

/'

'"

"-

"-

(

)

\

I

\

I \

I \

I

\ I ~------I

'

I

\

/'

/ /

I

I /

We would like to formulate an enumeration problem of which the question of the number of superpositions of isomorphism classes of graphs can be shown to be a special case. For given m, n EN and an s where 1~s~m, we consider the set 5.1.39

174

Applications to Combinalorics and Representation Theory 5.1

of all the n X s matrices

the rows of which are injective s-tuples over m, i.e. each row consists of s different elements of m. We notice that 5.1.40

In order to define a suitable group which will act on MIS)' we introduce the following generalization of the wreath product: 5.1.41 DEFINITION. If H~Sn has the orbits w1, ... ,WrCD, and if corre-

spondingly G I' ... , Gr are subgroups of Sm' then we call the following group the generalized wreath product (G I' ... , Gr ) wr H of G I ' " . , Gr with H. The underlying set (which of course heavily depends on the given numbering of the orbits Wi) is defined to be

{(f; 17)1 f: D->

U

Gk , 'ViED (iEw)

=>

f{i) EGJ, 17EH }.

k

The multiplication law remains as it was:

(f; 17 )(f'; 17') : = (ff~; 1717'). A permutation representation follows:

8(s)

of this group on MIS) is defined as

I t is clear that for s: = I and G I : = ... := Gr : = G we obtain the representation 8 of Gwr H which was mentioned above. Besides 8(s)' we now consider a permutation representation '1/ of Ss on M(s):

5.1.43 Two elements in the same orbit of '1)[ Ss l are called column-equivalent matrices, for obvious reasons. The crucial fact is that for each A E M(s) and every (/; 17) E (G" ... ,Gr)wrH, crESs' we have 5.1.44

175

5.1 The P61ya Theory of Enumeration

That this holds is obvious, and it will allow us later on to apply the following result: 5.1.45 DE BRUUN'S LEMMA. If p.: P---'SN and v: Q---'SN denote permutation representations of finite groups P and Q on a finite set N such that p.[ Pl is contained in the normalizer of v[ Ql, i.e. 'VpEP,qEQ,iEN

3q'EQ

p.(p)v(q)(i)=v(q')p.(p)(i),

then we can define a permutation representation

Ii of P on the set N of orbits {,

i EN, of v[ Ql on N as follows:

Ii:

P ---. SN: pt-+ ( P. ( ;-) ( i)

).

The character of this permutation representation satisfies Xii (p )=al(1i (p ))=

I~I ~

I{iliEN and p.( p )(i)=v(q )(i) }j.

qEQ

= I~I ~al(p.(p-l)v(q)). q

Proof The check that Ii defines a permutation representation is very easy. To derive the character we notice that

-,-' p.(p)(i)=i

And if v[QL denotes the stabilizer of i in v[Q), then we have

__ i

_

p.(p)(i)=i

I

IQI

_ _i_ _ q /l(p)(i)=i v(q)(i)=i

1

IQI =_1

IQI

--'- q /l( P )(i) = i v( q)(i) = p.( p)( i)

~

~

q

i p.(p)(i)=v(q)(i)

1.

•

Applications to Combinatorics and Representation Theory 5.1

176

5.1.44 shows that p.: = 8(5) and v: = 'I) satisfy the conditions of de Bruijn's lemma, so that we can use it in the proof of 5.1.46 THEOREM. 8(s) and 'I) yield a representation ~s) of (G j , ••• , Gr)wr H on the set M(s) of classes of column-equivalent matrices A EM(s)' Its character satisfies

-

.

_ J..-

c(,,)

s (a k (g(f'1T))) v ' I, , Gk(a"") ak(a l ,) ak(a ).k .

al(8(s)((f,1T)))-S!a~s,vDlkDl

It remains to prove the stated identity for the character. By 5.1.45 we have to evaluate the number of matrices A=(aik)EM(s)' the coefficients a,k of which satisfy

(1) In order to do this we discuss the degree of freedom we have in choosing such a,k' If (in cycle notation) c( ,,)

1T=

II

v=\

(J~1TUv)" '1T 1,-jUJ),

then by iteration of (1) we get

so that in particular the following must hold:

(3) Let us now denote by SVA' I ,,;;;'\,,;;;c(a- I ,), the least points in the cyclic factors of a -I,. Equations (2) and (3) show clearly that the coefficients

(4) determine the matrix A completely. Thus we can freely choose at most these entries (4). But still we have to take (3) into account, which says that aj,.s" must occur in gv( f; 1T) and in a -I, in cycles of the same length (recall that the rows of A are injective). Finally (2) shows that, for .\=Fp., aj,.s" and a},. s '. must occur in different cyclic factors of gv< f; 1T). Thus for a fixed v the entries aj,.s" in the row with number)v are obtained as images If;(svA) of the SVA under a mapping If; from s into m which must be injective and which has to map cyclic factors of a -I,. onto cyclic factors of gv(j; 1T). Conversely each such injection defines a matrix which satisfies (1), and different injections yield different matrices A.

177

5.1 The P61ya Theory of Enumeration

As the entries (4) determine A uniquely, the number of such A is equal to the product (over v) of the number of injections as they are described above, i.e. it is equal to

•

This yields the statement.

In order to derive from this the most important results of Redfield we need only to put s: =m. in which case we get

This can be simplified considerably, for a summand on the right-hand side of 5.1.47 is "'" 0 only if all the I( v) have the same cycle type as has a, in which case the summand in question is equal to

Applying this to 5.1.47, we obtain 5.1.48

a\(~m)((f; l)))={o(~ak!kak )"-1

if "tv (a(f(v))=(a, ..... a",)) otherwise.

This yields Redfield's method of calculating the number of orbits of O(m)[(G, •...• Gn)wr{ls)J (which is, by definition of superposition, the number of superpositions of n isomorphism classes of graphs on m points and with automorphism groups G] .... ,Gn ). Redfield introduced the cap operation n on the set Q[x\ •.... xmJ as the linear extension of the following operation on n~ 2 monomials:

5.1.49

.X c,\

e .. 'XCn> n x d\ , ••• .xdn> m' m n ., . nx I , •• ·xen> m

otherwise.

Applications to Combinatorics and Representation Theory 5.1

178

In terms of this notation an immediate consequence of 5.1.48 is 5.1.50 REDF1ELD'S ENUMERATION THEOREM. The number of orbits of ~m)[(Gl,... ,Gn)wr{lsJ] is equal to the cap product

n cyc(G I

i )·

For an example we take G j : =G z : =Ds, the dihedral group of order 10, which is the automorphism group of the following isomorphism class of graphs:

The cycle index is

so that by 5.1.50 the number of superpositions of two isomorphism classes of graphs of this kind is Cyc( DJ n Cyc( Ds )= 160 (x?

nx? + l6(x s nx s)+ 25(xlx~ nx,xO)

= 16o(5!+16X5+25X8)

=4,

in accordance with 5.1.38. Redfield's enumeration theorem provides the solution of various enumerative problems, e.g. 5.1.51. The number of orbits of the power group G H , G,,;;;Sn, H~Sn on the set of the bijective mappings f{! E rI' is equal to the cap product

Cyc( G) n Cyc(H). _Proof. The set of bijections f{! E nn is in a natural way bijective to the set of classes of column equivalent 2Xn matrices with injective n-tuples over n as rows. This natural bijection maps an equivalence class of mappings onto an orbit of O(n)[(G, H)wr{ls)]=G H (restricted to the set of bijections). Thus 5.1.50 yields the statement. • M(n)

179

5. I The Polya Theory of Enumeration

But bijections cp E on are just the permutations of n, i.e. the elements a E Sn' Thus 5.1.51 can be rephrased in the following way: 5.1.52. Cyc(G)nCyc(H), G'TTH in Sn'

G~Sn ~H,

is equal to the number of double cosets

This yields the desired representation-theoretical interpretation of the cap product of cycle indices: 5.1.53 COROLLARY. If G,

H~Sn'

then

Cyc( G) n Cyc( H) = (IG i Sn' IH i Sn)' In order to deduce from Redfield's theorem the number of isomorphism classes of graphs with p points and e edges, we take G=SeffiSn-e'

and

H=S[21 p

in 5.1.51. Given a bijection cpEon , we associate with it a graph onp points and e edges by saying that the ith pair of points is connected iff cp(i)~e. Then, under the action of G H , two bijections are in the same orbit of G iff they are associated with the same graph, and they are in the same orbit of G H iff the corresponding graphs are isomorphic. Thus, 5.1.51 and 5.1.53 give: 5.1.54. The number of isomorphism classes of graphs with p points and e edges is equal to

Cyc(spl)ncyc(Se EBS p )=(IS) 21 i S p),I(Se EBS p _JiS(p)). (2) (2 (2) 2 More generally we obtain in the same way 5.1.55 THEOREM. The number of classes (under -I) of mappings cpEmn which consist of elements of content A. = (A. I , ... , A. m) 1= n is equal to Cyc( H) n Cyc(S·,..)= (lH iSn , ISh iSn ).

This, together with our previous knowledge of the representations of Sn induced from Young subgroups, yields various results on the monotonicity of such numbers, e.g. 5.1.56 COROLLARY. If /-'.;;;) A., then the number of classes (under -I) of mappings of content A. is less than or equal to the number of classes of content /-'.

180

Applications to Combinatorics and Representation Theory 5.1

Thus for example the number of isomorphism classes of graphs on p points and with e edges is nondecreasing for

as e increases. Besides the cap operation n Redfield introduced a cup operation U as the linear extension of the following operation on ;;;. 2 monomials: 5.1.57

Obviously the following holds for G, H ~ Sn: 5.1.58. Cyc( G) n Cyc(H) is equal to the sums of the coeffiCients in Cyc( G) U Cyc(H).

This indicates that an examination of the cup product of cycle indices might turn out to be useful. In fact one can write each cup product of cycle indices as a sum of cycle indices of stabilizers. This will be proved with the aid of the following lemma. (There is also an easy representation-theoretical proof of it, and the cup product has a natural representation-theoretical interpretation as well; this is left to the exercises.) 5.1.59 REDFIELD'S LEMMA. Let <'J: G-.Su denote a permutation representation of a finite group G on a finite set M; x: G-. Q[xl''''' x n ]. n EN, a class function; wi' ...• w, the orbits of <'J[G] on M; m/ Ew/; and

Gi : = {glgEG and <'J(g )(m i )=m/}. Then we have

Proof We consider the IM IX IG I matrix (a mg). where a

.={x(g)

mg'

a

The sum of all its entries is equal to

(1)

<'J ( g )( m ) = m • otherwise. if

lSI

5.1 The P6lya Theory of Enumeration

expressed as the sum of column sums. On the other hand the sum of its row sums is (2) m

g

gEG,

Ii(g)(m)=m

(the last equation holds because elements in the same orbit have conjugate stabilizers and since X is a class function). Comparing (1) and (2), we obtain the statement. • DECOMPOSITION THEOREM. If G I , ... , Gn ~ Sm, and denote the orbits of 8(m)[( G" ... , Gn)wr {Is,,}], AI E Wi' while

5.1.60 REDFIELD'S WI"'"

Wr

then

Cyc(GI)U ... UCyc(Gn )= ~ Cyc(Pi )· i=1

Proof We apply 5.1.59 to 8(m) and the class function (GI, ... ,Gn)wr{Is,) .... G I X··· XGn ->4)m[x 1.... ,x m], defined by

x:

otherwise. Since, by 5.1.48,

otherwise, the statement follows from Redfield's lemma.

•

The representation-theoretical interpretation of this result is clear from the fact that a transitive permutation representation is equivalent to the representation induced by the identity representation of a stabilizer. This may suffice concerning enumeration and enumeration by content of classes of mappings q;Emn with respect to symmetry groups E R, G R, [G]R. But some remarks about the construction of systems of representatives of these classes are well in order, since this is the most difficult part of the theory and not very much is known about this kind of problem. It can be solved for P6lya-type problems, i.e. in the case when E R is the symmetry

182

Applications to Combinatorics and Representation Theory 5.1

group, and therefore also for the case [G] H, since such problems can be transformed into P6lya-type problems. In order to show how a complete system of representatives of the classes of mappings
ri;,

say

5.1.63 Then 5.1.64

n=

l.:J

nA

and

I

Then with 1T E Sn we associate a 5.1.65



=j

'iliEm

/if/=A l

,.

Em" by putting

if 1T ( i) En).

It is clear that


5.1.66

3uESrf.EB ... EBS""m (r=u1T). I

This yields 5.1.67. The inverse image of
An immediate consequence is

5.1.68

5.1 The P61ya Theory of Enumeration

183

We have therefore proved the following: 5.1.69 THEOREM. A complete system of representatives of the double cosets SA7THr;;;,Sn is mapped under 7Tf->CP" (see 5.1.65) bijective~v onto a complete system of representatives of the orbits of mappings cp EO mn of content 'A under E H . This describes in particular a redundancy-free algorithm for the construction of such representatives. But since the evaluation of representatives of double casets is very cumbersome, one does not get very far. The graphs can be evaluated for up to 8 points (there are 12346 classes of them) using a big computer. To conclude with an example, we consider 'A:=(3,2) and H:=Ds = «(12345), (13)(45». We have to evaluate a system of representatives of the double cosets

Each such double coset is a union of right cosets SAP' and we remember that these can be represented by tabloids, in our case by the tabloids 123 4 5

124 3 5

125

134

135

3

2

2

145 2 3

234 I 5

235 I 4

4

5

245 I 3

4

345 1 2

As the present 'A = (3, 2) is 2-rowed, we need only take notice of the last row, i.e. we need only consider 4

5,

2

3,

3

5,

3

5,

4,

2

4,

5,

2

3,

4, 2.

Applying D s = {((13)(45)t(12345)b I 0,,;;; a";;; I, 0,,;;; b,,;;; 4}, we see that from 4 5 we obtain 4 while from 3

5,

5,

2,

2

3,

3

4,

3,

2

4.

5 we obtain 3

5,

4,

2

5,

184

Applications to Combinatorics and Representation Theory 5.2

Hence there are exactly 2 double cosets; they are represented by 1 4

2 5

3

and

1 3

2 5

4

i.e. by Iss

and

(34).

Applying w...... ep", we get 1...... 1 2 ...... 1
and

1...... 1 2 ...... 1 ep2 : 3 ...... 2 4 ...... 1 5 ...... 2.

These mappings represent the two classes of mappings ep E 25 under E D of content (3,2) which can be interpreted as colorings of a regular pentagon or "necklaces" with 5 beads in 2 colors, and of content (3,2): j

3

•

40/ "".2

\0--./ 5

1

This shows how tabloids can be used for orbit calculations. (It is clear that we could have avoided the double cosets, as a tabloid already can be considered as a function of a certain content.)

5.2

Symmetrization of Representations

In order to discuss, in this and the following sections of the present chapter, a few applications to general representation theory of finite groups, we return to 4.3.14, where we noticed that the representations (g) nD of G and Pnm of Sn afforded by (g)nV centralize each other: /l

/l

The reason for this very useful fact is that the subgroup diag G* . Sn ' (cf. 4.1.17) provides an embedding of the direct product GXS/l into GwrS/l: 5.2.2 Since 5.2.1 holds, certain decompositions of 0/lD and p/lm are closely related as it will be shown next. The argument which we shall use holds under suitable assumptions on the groundfield whenever we have centraliz-

185

5.2 Symmetrization of Representations

ing representations of two groups acting on the same vector space. In order to show this, we assume that we are given two groups G, and G2 with matrix representations [}\ and [}2 on a vector space V such that

5.2.3

'VgiEG i [])1(g,)o[])2(g2)=[])2(g2)o[}l(g,).

Furthermore we assume that []) I is completely reducible into absolutely irreducible representations [])) of G 1• Without loss of generality we can assume that this reduction of [}, has already been carried out and that an adapted basis has been chosen, so that []) I looks like

o o The direct summand of V which affords n i times the irreducible constituent [])) is called the homogeneous component of tvpe [])) of V (as a left G I-module). I t is a uniquely determined subspace, and if G I is finite and the characteristic of the groundfield does not divide IG II, it is equal to

(

5.2.4

~

(trace of [])/ (g - 1 ))g ) V.

gEG ,

This subdivision of [])I(gl) into boxes along the main diagonal yields a subdivision of the whole matrix [])l(gl) into boxes. We subdivide [}2(g2) correspondingly into boxes [])1!t(g2) of the same sizes: 5.2.5 [])2(g2)=

[])\\(g2)

[])\~,(g2)

[])\t(g2)

[} ;;,,( g2)

[}~:,(g2)

[])ll (

)

[])~~, (g2 )

[])12

[])fl(g2 )

[}f,~,(g2)

[])ff(g2)

[}f~2( g2 )

[]),;;, (g2)

[}21 (

[]);;, (g:)

[])22 (

"1"1

g2

","I g2

)

"I", ( g2 )

"2"2

g2

)

186

Applications to Combinatorics and Representation Theory 5.2

Using this notation, we obtain from 5.2.3 that for a fixed for each gl EG 1, l~k~ni' 1~I~nJ

g2

E G2 we have

5.2.6 Since OJ is assumed to be absolutely irreducible, we obtain from Schur's lemma if i =/= j,

5.2.7

if i=j,

where I denotes the identity matrix whose number of rows equals the dimension of oj and where d k/(g2) is an element of the groundfield F. Using the constants dk,.(g2)' we define 5.2.8 From 5.2.5 and 5.2.7 we obtain that there exists a permutation matrix T such that for each g2 E G2 we have

o

o where f' denotes the dimension of oj. We notice in particular that the multiplicity n j of OJ in 0 I is equal to the dimension of 0], while the dimension f1 of oj is equal to the multiplicity of 0] in 0 2 . An application of this quite general argument to 5.2.1 instead of 5.2.3 yields 5.2.9 THEOREM. Let G denote a group and D an ordinary representation of G on the vector space V, while n EN. Then the homogeneous component of type [a], a~ n, of ~nv (as left Sn-module) affords, if it is nonzero, times a representation of G. We denote this representation by Dc::J[a] and call it symmetrization of D by [a]. Its dimension is equal to the multiplicity (Pnm , [a])

r

5.2 Symmetrization of Representations

187

of [a] in Pnm , and it satisfies, for each gEG, 7TESn , and with respect to a suitable basis of ® n v, n

®[}(g)OPnm (7T)=

+

([}~[a](g)X[a](7T)).

IHn

(We allow [} ~ [a] to denote the zero" representation" of G if [a] is not an irreducible constituent of Pnm , so that the corresponding homogeneous component of type [a] is the zero subspace. Thus [}~[a]is defined for each a ~ n, but the corresponding summand in the equation above may have to be neglected.)

An immediate corollary is that the representation Pnm of Sn on ®nV induces a direct decomposition of ®nD into representations D~[a] (which are in general reducible) as follows (put 7T: = lSn in the equation of 5.2.9): n

@D= ~ r·(D~[aD.

5.2.10

al-/1

It is clear that the above argument can be reversed in order to get from ® n D a decomposition of P,';'; this will be discussed in the following section.

5.2.10 implies for example that

(i) 5.2.11

D=D~[l],

(ii) D@D=D~[2]+D~[12], (iii) D@D@D=D~ [3] +2( D~ [2,1 D+ D~ [1 3 ].

and so on. If again ~a denotes the character of [a], then the equation in 5.2.9 together with 4.3.1O(vi) yields

5.2.12 An application of the orthogonality relations yields for the character of D~[a]:

5.2.13

We would like to discuss a few basic results on the dimension and irreducibility of D~[a].

Applications to Combinatorics and Representation Theory 5.2

188

5.2.14 THEOREM. The dimension fDO[a] of DO[a] equals the number of semistandard a-tableaux with entries ";;;f D. A more explicit formula is

II

I.j l~l:S;a[ I~j~a(

Proof (i) 5.2.9 shows that ~

fDO[aJ=(P,(,[a])=

(IS/liSn,[a]),

/3=({3,'····{3r D )FIl

so that Young's rule immediately yields the first part of the statement. (ii) By definition of symmetrization,f DO[a] is just the multiplicity of [a] in P,':' (recall that m=fD), which is (r)-I times the dimension of the homogeneous component of type [a] of 0"Vas a left S,,-module, V being the representation space of D. On the other hand. the dimension of this homogeneous component is just the trace of the corresponding projection operator

(1) which projects 0 n V onto this homogeneous component. Furthermore this centrally primitive idempotent E a is the unit of the simple component I a of the group algebra CS" and hence it is the sum of the minimal idempotents e~: f"

f"

a

E = ~ e a = " L,ya'}(a a ..£.J .~ n! (

(2)

I

1=1

I'

t=1

This yields

(3)

I

I

f"_

=- ~ trace'VaJ{a f DOla ]= -traceE a an! ~ 1 I' f

;=1

so that we shall be done once we have shown that the following holds:

(4)

\;j k

trace 'Yka ')( ~ =

II U

D -

i +j ) .

t,j

In order to prove this we first notice that by 4.3. IO(ii)

(5 )

trace'Yka '}(% =

~ sgn p' (jDt

7TEHk pE vt

P7T ).

189

5.2 Symmetrization of Representations

Now we proceed by induction on n. (a) If n : = 1, then a: = (l) is the only partition, hence f" = 1, and in fact, since 'If":JC" k k = 1S,' we have trace CY:X: =fD.

IX: = (a 1 , ••. , a h -], a h -1) ~ n-1.

The tableau which arises from f'k by cancelling the last symbol of the last row will be denoted by i'k; its horizontal and vertical groups, by fIt:, Vi:- We shall prove that

7TEH't pE

Vi:

This shows that (4) is true by the induction hypothesis (apply (5)). To prove (6), we embed Sn-l into Sn as follows. If z is the cancelled symbol, i.e. if

then Sn-l is embedded into Sn as the stabilizer of {z}. The subgroups corresponding to fIt:, VI: under this embedding are denoted by fIt:, Vi:We now decompose Ht: and Vk" into cosets with respect to fIt: and VI: using the symbols in the row and in the column of z:

a fk -

We have

Applications to Combinatorics and Representation Theory 5.2

190

and

so that

(7) c\{:X~=CV-kax~- ~ (ZC;)ey-kax~+ ~ey-kax~(Z'J)- ~ (ZC;)CV-kax~(Z'J)' }

;, }

In order to apply the induction hypothesis to (7) we notice that

(8)

(Multiplication of a summand ±'TT of ey:x~ by (zc i ) or (zr) gives c«zc;)'TT) = c( 'TT( zr) = c( 'TT) - I, since 'TT leaves z fixed. Now use (8) to get (9).) Later on we shall see that furthermore the following holds:

(10) so that we obtain from (7) by an application of (8) and (9)

an equation which implies (6). It remains to prove (10). To do this we bring the element a of f'k in the ith row and jth column into the game:

a fk -

a'"

.. ·c{

It satisfies

ey:=(I-(arj))X and X~=Y(l+(ac;))

191

5.2 Symmetrization of Representations

for suitable X and Y in the subalgebra CSn -

l.

This gives

trace ( (zc i )'VrX~(z'j)) =trace( (zc i )(1- (a'j) )XY(1 + (ac i ))(zrJ )) =trace( [(I + (ac i ))( z'j )(zc i )( 1- (a'j))] XY) =trace([ ((zc i )( c,'j)- (za)( cJj) )(1- (arJ ))] XY) = trace( (( zC i ) - ( za ))W ). Note that W: =(ci'j)(I-(ar»)XY is in CSn -

l;

hence it is obvious that

trace (( Z C i ) W ) = trace ( ( Z a ) W ),

•

so that we are done.

5.2.14 shows thatfDO[a] is obtained by first substituting for the (i, j)-node of [a] the number f D - i+ j, i.e. by forming

and then dividing the product of all these numbers by the product of all the hook lengths in [a]. The resulting quotient is 0 if and only if fD
on~v

if

This shows e.g. that DO[n] always exists (i.e. it is nonzero) and that D[)W] exists if and only if n";;;;'fD. For the dimensions of these particular products we obtain from 5.2.14: D

(i) fDO[n 1 = (f +nn-I ), 5.2.16

(ii)

f DOI ]"] =

(f:).

Both these symmetrized products are of particular importance. DO [n] is called the symmetric part of 0 nD, while D [) [I n] is called the antisymmetric

192

Applications to Combinatorics and Representation Theory 5.2

part of @nD. Their representation spaces are

5.2.17 and 5.2.18

n

E(1") @ V=

(I~ ~ sgn'1T''1T ) @ V. n

respectively. The elements of

5.2.19 B:=
~

a"Pnm('1T)!a"EC} <:End c (

0V).

"ES"

like E(n) and E(I"). are called symmetry operators. The image of such a symmetry operator is called a symmetry class in @"V. By 5.2.9 and 5.2.14. the decomposition of Pnm into its irreducible constituents is

5.2.20

'tIm,nEN P,':'=

~ (nm~~+j)[a].

a'lI

/. J

IJ

We now investigate the case where V is an m-dimensional vector space over C and G is the general linear group, G:=GL(V).

It turns out that if we take D to be the natural representation of G, then all the symmetrized products of D are either zero or irreducible. Here we mean by the natural representation the identity mapping idGL(v), i.e. 5.2.21

idGL(v):GL(V)->GL(V): gl->g.

The corresponding symmetrized inner product is denoted by

:

5.2.22

a>

In order to prove that < is either zero or irreducible, we consider besides the subalgebra B (cf. 5.2.19) of Ende< @IIV) the subalgebra C generated by 5.2.23

II I~[GL(V)]:= @idGL(v)[GL(V)].

5.2 Symmetrization of Representations

193

i.e., we consider 5.2.24

C::::: (I:'[GL(V)]).

In the approach we are adopting here the following theorem is crucial: 5.2.25

THEOREM.

(i) C::::Endcs(lZl"V). (ii) 0" V is a ~ompletely reducible GL(V)-module. Proof We consider representing matrices for the elements of the subalgebras Band C of End c (0"V) which correspond to the basis given in 4.3.4:

" (a) Denote the matrix representing 'IT E S" under P,,'" by 11\"'( 'IT):::: (a",,,,( 'IT )). We obtain from 4.3.6 for its coefficients (I )

if

denotes the usual Kronecker symbol. (b) An element yEEnd c (0"V) is contained in End cs (0"V) if and only if for its representing matrix n l)

the following holds:

i.e.

By (I) this is equivalent to 'tf'IT,

cp, I/;

a",,,, ( y )::::a",o"-',,,,o,,-'( y).

This shows that y is contained in End cs (0"V) if and only if the coefficient function (cp, I/; )t->a",,,,( y) is constant on e~ch orbit of S" on mn X mn under the action

Applications to Combinatorics and Representation Theory 5.2

194

We notice that this permutation representation of Sn on mn X mn is just 02Pnm, so that the number of orbits is equal to the multiplicity (0 2Pnm,[n]), which is

The last equation is obtained from 5.2.16(i). This yields for the (>dimension of End cs J0 nv)

If we put

0

0

0 0

Ecplf:=

0

0

1

0

0

cP

0

0

0

o If

and (denoting the orbit of (ep, 1/;) by [ep, 1/;]) E[cp,If]:=

~
EH ,

then if 'J is a transversal of the orbits, we have a C-basis of End cs J0 nV):

(c) In order to prove statement (i), we notice that by 5.2.1 C c: End cs n(0 nV). If End cs (0 nV)* denotes the vector space dual to End cs (0 nV), then C=r!::E~dcs(0nV) if and only if there exists a nonzero linear iunctionaljEEnd cs (0 nV)* which restricts to the zero functional on C. Denoting by E[cp,If]* n the elements of the dual basis, then jE End cs J0 nV)* is of the form

5.2 Symmetrization of Representations

195

An application of f to g0 ... 0 gEI';:[GL(V)] yields (writing g: = (d ,k ( g» in matrix form):

This shows that the corresponding polynomial

has the property that for each non-singular mXm matrix (aid, the result of making the substitutions X ik I->a ik in Pf is zero. Therefore, Pf ( a,d' det( aid = 0 for all m X m matrices (aik)' Since C is infinite, Pf must be the zero polynomial. But for distinct (cp, 1/;) E5, the monomials II ;xJ;(j) are distinct, so b(J;) =0 for all (cp, 1/;). Thus there is no nonzero linear functionalfin Endcd0nV) which restricts to the zero functional on C. This proves C~Endcd0nV). Assertion (ii) follows from (i), 5.2.24, and the semisimplicity of C. n • THEOREM. If a=( a\, ... , ah)~ n, and m is a natural number with a; ";;;;;m, then (a) is an irreducible representation ofGL(m,C) over C.

5.2.26

Proof Recall that B is defined to be (Pnm[Snl). Now, (a) is afforded by e®nV, where e is a primitive idempotent in EcxB. (We may therefore assume that (a) is afforded by c,\{cx')Ccx@nv.) If (a) were reducible, then 5.2.25(ii) would yield a nontrivial GL(V)-decomposition

Let W be a GL(V)-complement to e@nV in @nV, and let ei be the projection of 0 n Vonto V; with kernel WEB Tj, i=l=j. Then e, and e 2 belong to the centralizer End c (l8l n V) of C. Since B has the double-centralizer property, we get e=e, +e2' contradicting the primitivity of e. • Theorem 5.2.26 gives many irreducible representations of GL(V). But every group with a finite-dimensional complex representation has a homomorphic image inside some general linear group, so the concept of symmetrization enables us to produce many new representations from a given one.

196

Applications to Combinatorics and Representation Theory 5.2

A well-known theorem of Burnside states that if D is a faithful representation of a finite group G, then 0 oD: = I, 0 I D, ... , 0 sD (s being the number of different values X D takes) between them contain all the irreducible representations of G as composition factors. An interactive computer program which does symmetrization and subtracts known irreducible characters from new ones can therefore be used to great effect on the problem of determining the ordinary character table of a given finite group, once a faithful character has been found. In Chapter 8 we shall investigate the representation theory of general linear groups more fully, proving that id GL( V) 0 [a] is irreducible whenever char F=O, and showing how to find an irreducible composition factor when charF=p.

The applications of symmetrization DO[a] of representations D of a group G by representations [a] of symmetric groups give rise to the question whether it might be possible and even useful occasionally to symmetrize D by an ordinary irreducible representation D i of a subgroup P of Sn' That such a symmetrization 5.2.27 is well defined is trivial from the definition of DO [a], since we merely applied Schur's lemma in order to define DO[a], but we never used the fact that the permutation group acting on I8i n V was the full symmetric group. 5.2.27 is nonzero if and only if the centrally primitive idempotent E i , which corresponds to D j , i.e. 5.2.28

E j.'--

1 ~ i'i( -I) -IPI '" ~ 7T 7T, '1TEP

satisfies n

5.2.29

Ej

0 Vr"{O},

which holds if and only if D i is contained in the restriction of Pnm : 5.2.30 Or, equivalently, it holds if and only if there exists an a ~ n such that 5.2.31

197

5.2 Symmetrization of Representations

A closer examination of the homogeneous component of type D i is

E®V=E( EBE ®V)

5.2.32

I

I

a,n

a

"

EB fD'([a] t P, Di)er 0 a,,,

V,

er

where again (see Chapter 3) denotes the minimal idempotent corresponding to the first standard tableau of shape [a]. 5.2.32 can be rephrased in the following way: 5.2.33 a,n

This equation shows in particular that the decomposition 0 n D = ~a D 0 [a]) of 0"D into symmetrized products DO [a] with ordinary irreducible representations of S" is at least as fine as the decomposition into symmetrized productsDDD i with ordinary irreducible representations of a subgroup P of S". Nevertheless it is occasionally useful to symmetrize by representations of subgroups P~Sn' A surprising example is a proof of the Amitsur-Levitzki theorem on the standard polynomial identity:

r(

5.2.34 AMITSUR-LEVITZKI THEOREM. If M I , ... , M 2n are n X n-matrices over C, then the following standard polynomial identity holds: [M 1 ,···, M 2n ] : =

~ sgn 1T M 1T (I)'

••••

M 1T (2n) =0.

'lTES:1n

Proof

(i) We first prove that the mapping ~

2,,+1

2n+1

sgn1T(e;1T(1.··2n+I)1T- I ): 0 cn-. 0

'lTES:1n+l

which maps the tensor v l 0 ... 0v 2n + 1 = ... 0v i 0 " . onto

~

sgn1T('"

0V1T(2n+I

.. I)'IT-1(i)0 . . . )

'lTES:1n+l

is the zero mapping. In order to prove this, we notice first that

en

Applications to Combinatorics and Representation Theory 5.2

198

is the zero "representation" by 5.2.15. Hence 5.2.33 together with

(see 2.5.7) yields that both

are zero. Expressed in terms of the corresponding centrally primitive idempotents E(~+ I, I")' this reads as follows: 2n+ I

E(~+l,I")

0 Cn={O}.

Hence also 2n+ 1

(E(~+I,I")-E(~+l,l"») 0 Cn={O}.

But the corresponding irreducible characters r(n+I,I")~ of A 2n + 1 differ on the two classes (2n+ 1)'"' of the (2n+ I)-cycles only (recall 2.5.12, 13), so that E(~+ I, I") - E(~+ I, I") is a multiple of ~

p-

pE(2n+ 1) +

~

T

= a multiple of

TE(2n+ 1)-

This yields our first statement. (ii) We prove now that

for arbitrary n X n matrices M; over C. trace[M I ,···,

M 2n + I] =

trace ~ sgn7T' M,,(I)'

= ~sgn7TtraceM"(I)"

.. M"(2n+ I)

" ·M,,(2n+l)

" = ~ sgn 17' trace (

"

2n+1

#

idEnd(v)

)-

(M 1 , ... , M 2n + I; 17'(1 ... 2n + 1)17' -I)

199

5.2 Symmetrization of Representations

(d. the proof of 4.3.9). Hence

trace[ M 1 , ... , M 2n + I] = trace ( 2n; 1 id End ( V)

) -

~sgn17(e; 17(1·· '2n+l)17- 1 )),

X ((M 1 , .. .,M2n + j ; 1)0

7T

and the right-hand side is 0 by (i). (iii) We are now in a position to prove the statement. In order to do this we first remark that

if S2n denotes the stabilizer of {2n + l} in us

S2n+ I'

This together with (li) gives

O=trace[M 1 , .. ·, M 2n + 1 ] ~

sgn17·traceM7T (I)·· ·M7T (2n+j)

7TE~n+l

=

~ sgna aE~n

~

sgnT·traceMaT(l)·· ·MaT (2n+l)

TE(l]···2n+l)

=(2n+l)traceM2n + 1 • ~ sgnaMa(l)···Ma(2n) aE~n

= (2n + 1) trace M2n + I' [M p

... ,

M 2n ]·

Since this holds for arbitrary n X n matrices M 2n + j ' the statement [M j , . . . , M 2n l=O must be true. • Besides 5.2.33 there are many other useful equations- for example the following one: If D is an ordinary representation of a finite group G, then for any af- m, {3f- n

5.2.35

( D o[ a]) 0 ( D 0 [{3 ]) = D 0 ([ a] [{3 ]).

(This sometimes holds for ordinary representations D of infinite groups also, e.g. if G: = GL(k, C), in which case in the proof the characters have to be replaced by S-functions or a different approach adopted.) Proof For the character of the right-hand side on g E G we have

200

Applications to Combinatorics and Representation Theory 5.2

But the product

is the value of a class function fg on Sm+n' so that the right-hand side of 5.2.36 is an inner product of class functions, to which we can apply Frobenius's reciprocity theorem, obtaining: X DD([aH,B])( g) = (X[aH,BJ, fg)

=(X[a]#[,BJ, fg JSmEBSn) m~n! ~ ~a(p-l)~,B(a-l)Q

xD(gJrJ(p\D(gkr'(")

J.k-I

'T1'=po

=XDD[a J(g h DD [,Bl( g) =X(DD[a])0(DDfP])( g).

•

An immediate corollary is the following determinantal form: 5.2.37 It reduces the decomposition of symmetrized products D CJ [ a] to the decomposi tion of "completely symmetric" representations D CJ [r] together with decompositions of inner tensor products. For inner tensor products D 0 D' of ordinary representations of a finite group G we find (if y f- n) ~

5.2.38 (D0D')CJ[y]=

([a]0[.8],[y])(DCJ[a])0(D'CJ[.8]).

a,p. n

Proof If gEG and (p, a) ESnEBSn, we write

Then the character of the right-hand side of 5.2.38 on g EGis equal to

~ ([a]0[.8]'[Y])~ n,

<>,/3

~

ta(p-l)t,B(a-1)fg(p,a)

(p,o)ESnEBSn

= ~ ~ ( ~ ([ a] 0[P]' [y ])x[a)#[PJ(p, a) )n XD(gkr,(p\D'(gkr'("). n!

(p,o)

k

a./3

Now [y 1 of Sn can be considered as a representation of diag(S/B Sn)' in which case we have

[ a] # [P] J diag( Sn EB Sn) =[a] 0 [P] = ~ ([ a] 0 [.8] , [ y ]) [ y ], y

201

5.2 Symmetrization of Representations

or, equivalently,

[y ] t Sn EB Sn == ~ ([ a] ® [ P], [y ])[ a] # [ P]. a.f3

In addition, the product fi p, (J), defined above is, for each g E G, the value of a class functionfg on SnEBSn' so that the character of the right-hand side of 5.2.38, evaluated at g, is equal to

•

==X(D0D')[J[Y)(g ).

Analogously one obtains

(D#D')[][y]==

5.2.39

~

([a]®[p],[y])(D[][a])#(D'[][,B]).

a.f3

If we restrict attention to symmetric groups, we find the following:

5.2.40

THEOREM OF ASSOCIATES (IN CASE OF SYMMETRIZATION).

If

af-

n,

,Bf- n, then

if n is odd if n is even.

Proof

•

202

5.3

Applications to Combinatorics and Representation Theory 5.3

Permutrization of Representations

The preceding section was devoted to the symmetrizations D8[a]. These representations yield the decomposition n

o D=

~ r(D8[a))

of 0 n D. The basic equation for the proof of this was n

n

so that Schur's lemma could be applied to the matrices 0 n [P(g) under the assumption that Pnrn is completely reducible into absolutely irreducible representations. Hence an application of Schur's lemma to iPl,;"(?T) yields analogously (cf. 5.2.9): 5.3.1 THEOREM. Let G denote a finite group, and D an ordinary representation of G on the vector space V, while n E r\I. If Di denotes an ordinary irreducible representation of dimension f of G, then the homogeneous component of type Di of 0 n V (as a left G-module) affords, if it is nonzero, f i times a representation of Sn' We denote this representation by D 6. n Di and call it the nth permutrization of D by Di • Its dimension is equal to the multiplicity (0 nD, DJ of D i in 0 nD, and it satisfies, for each gEG, ?TESn • and with respect to a suitable basis of 0 n V, I

n

@[P(g)·/pnrn(?T)=

+ (Oi(g)X[P6. i

n

[P,(?T»·

(We allow D 6. n D i to denote the zero" representation" of Sn if D i is not an irreducible constituent of 0 n D, so that the corresponding homogeneous component of type D i is the zero subspace. Thus D 6. n D i is defined for each i, but the corresponding summand in the above equation may have to be neglected.)

To 5.2.10 there corresponds now the decomposition 5.3.2

Notice that this holds for each m-dimensional representation D of any finite group G. From the equation in 5.3.1 we get that the character of D 6. n D i satisfies n

5.3.3

VgEG, ?TESn

II k=l

X D(gkr,(7T)= ~ ~'(g )xD6"D'(?T),

5.3 Permutrization of Representations

203

where ~' is the character of Dj , so that we obtain by an application of the orthogonality relations 5.3.4 Comparing this with 5.2.13, we get the following very close connection between symmetrization and permutrization which is in a sense a duality: 5.3.5 DUALITY THEOREM. For each ordinary representation D of a finite group G, for each n EI\I, each a~ n and each ordinary irreducible representation D j of G we have the following equality of multiplicities: ( D 0 [ a], D j ) = ( D 8" D j , [ a] ).

This multiplicity is just the multiplicity of Dj#[a] in the representation of G X S" afforded by 0" v.

This shows that we can define for D and each n E 1\1 a matrix S( D, n), the rows of which correspond to the partitions a of n (in lexicographic order, say) and the columns of which correspond to the ordinary irreducible representations D k of G as follows: 5.3.6 The duality theorem says that SeD, n) contains in its kth column the multiplicities 5.3.7 so that its columns yield the decompositions of the permutrizations D 8" D k while its rows yield the decompositions of the symmetrizations DO[a]. A numerical example is [14] [2.1'] [22] [3.1] [4]

o o o

(~

1

2

1

o

1

1

1)

o o

[I J] [2.1] [3]

The multiplicities sak allow the following symmetric version of the equations in 5.3.1: n

5.3.8

@[}(g).I?"rn(7T)=

+ +Sak([}k(g)X[a](7T)). a k

204

Applications to Combinatorics and Representation Theory 5.3

Let us return to formula 5.3.4, which yields the character of D 8 n D i • We would like to point to a few consequences of the fact that by 5.3.4 each expression of the form

is the value of a character of a symmetric group. Since the character table of Sn is a matrix over l, we obtain as the first result 5.3.9 THEOREM. Let G denote a finite group with ordinary representations D and D'. Then for any a k EN o, I ".;;k"';;n, we have

A particular case is (put a n := I, a]: = a2: = ... : = a n -]: = 0) 5.3.10

I~I ~t(g-I)xD(gn)El. g

But this number is the multiplicity of ~i in the class function X~ defined by 5.3.11 so that we obtain 5.3.12 COROLLARY. For each ordinary representation D of a finite group G, 5.3.11 defines a generalized character, for the multiplicity of ~i in X~ is equal to

Let us denote by D] the identity representation of G and compare the particular multiplicity 5.3.13

5.3.14

XD,6nD'(l)=Wi ~~i(g)". g

205

5.3 Permutrization of Representations

This leads us to a characterization of ordinary irreducible characters of G where the averages on the right-hand sides of 5.3.13, 14 coincide: 5.3.15 THEOREM. If ~i is an ordinary irreducible character of a finite group G and nEf'\I, then

~ ~~;(gn)= ~ ~~i(gr g

g

if and only if Di 8 n D j is of the form

where the multiplicity bl vanishes if n is even. If on the other hand D; 8 n D, is of this particular form b\ [n ] + b2[ In], then still (i. e. whether n is even or odd, no matter If bl =1= 0)

~ ~t(gn)= I~I ~~i(gr

(mod 2).

g

Proof

(i) A comparison of 5.3.13 and 5.3.14 shows that the averages coincide if and only if (1 ... n) is contained in the kernel of D i 8 n D J • But we know from 2.1.13 that [n] and W] (for each n Ef'\I) together with the representation [2 2] of S4 are the only ordinary irreducible representations of symmetric groups which are not faithful. Since [2 2] does not contain (1234) in its kernel (for ~(22)«(1234))=0), the first part of the statement is true. (ii) If D i 8 n D J =b'l[n]+blW], then by 5.3.14

while by 5.3.13 and 2.4.8,

I~I ~~;(gn) =X D ,6

n

D 1 ((1

.. ·n)) =b i, + (-1)n-J bl =b i, +bl (mod2) .

• This is the first case where we meet a congruence modulo a prime number. We would like to show now that our result, that each expression of the form

206

Applications to Combinatorics and Representation Theory 5.3

is the value of a character of a symmetric group, opens an easy way to a systematic use of congruences. (This can of course be done in other ways, too, since characters are sums of roots of unity and the binomial theorem and number-theoretic arguments may be applied, but such arguments need case-by-case proofs.) Let p denote a prime number. If 17 E Sn' then we know that this element is of the form 17=pa, where p denotes the p-regular component of 17, i.e. p does not divide the order of p, while the order of a is a power of p and pa = ap. We know from general representation theory that for each character X of Sn we have 5.3.16

X ( 17 )

= X ( p ) (mod p ).

Hence for example 5.3.17

A particular case is the following generalization of a part of 5.3.15:

Wr 2: r'( gP) =Wr 2: t( g)P (mod p).

5.3.18

g

g

In terms of multiplicities 5.3.17 reads as follows: 5.3.19

where (XDV denotes the character of (iYD, and X~ is the class function defined in 5.3.11. Let us abbreviate this by 5.3.20

Since X D6pD '((l ., .p)) is linear in D, and
THEOREM.

For each generalized character