λ-Rings and the Representation Theory of the Symmetric Group

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich

308 Donald Knutson Columbia University in the City of New York, New York, NY/USA

Z-Rings and the Representation Theory of the Symmetric Group

Springer-Verlag Berlin. Heidelberg" New York 1973

A M S Subject Classifications (1970): 1 3 A 9 9 , 20-02, 2 0 C 3 0

I S B N 3-540-06184-3 S p r i n g e r - V e r t a g B e r l i n - H e i d e l b e r g - N e w Y o r k I S B N 0-387-06184-3 S p r i n g e r - V e r l a g N e w Y o r k • H e i d e l b e r g • B e r l i n

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer Verlag Berlin . Heidelberg 1973, Library of Congress Catalog Card Number 73-75663. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

CONTENTS

Introduction

Chapter

I:

.

k-Rings

.

i.

The Definition

2.

General

3.

Symmetric

4.

Adams

Chapter

II:

Operations

The

Irreducible

3.

Characters

4.

Permutation

i.

on k - R i n g s

.

46

Ring

Theory

of G r o u p s

of a F i n i t e

Representations

and

Group

Schur's

59

.

60

.

Lemma

76

. .

Representations

81

a n d the B u r n s i d e 104

The Group

III:

15 28

. Algebra

Approach

The Fundamental

The Fundamental Theory

of t h e

2.

Complements

3.

Schur

4.

and D e f i n i t i o n s

Representation

2.

5

.

Representation

Ring

k-Ring

Functions

The

Chapter

of

Constructions

1.

5.

5

Theorem

Symmetric

.115

.

Theorem

.

124

of the R e p r e s e n t a t i o n Group

124

and C o r o l l a r i e s

Functions

137

a n d the F r o b e n i u s

Formula

.

Methods

of C a l c u l a t i o n .

Character 155

Young

Diagrams

~

166

IV

Bibliography

Index

of

Index

.

.

Notation

194

.

198 200

INTRODUCTION

These are notes in the y e a r 1971-72. christened

from a seminar given The object

the F u n d a m e n t a l

of the Symmetric

Group.

isomorphism between

at C o l u m b i a U n i v e r s i t y

is to prove what

is herein

Theorem of the R e p r e s e n t a t i o n

This theorem

states that there is an

a ring c o n s t r u c t e d b y composing

way all the r e p r e s e n t a t i o n

Theory

rings of the symmetric

and the ring of all symmetric p o l y n o m i a l s

in a certain

groups

S , n

in an infinite number

of variables. This

isomorphism has b e e n more or less known

of the subject w i t h F r o b e n i u s reasons

it has b e e n expressed

through its relation linear group.

around 1900, (e.g.,

But for various

in Weyl's Classical

to the r e p r e s e n t a t i o n

The i s o m o r p h i s m

is given

The main technical

in its pure form seems not to

context,

in a b r i e f

in K-Theory,

introductory

in 1956

(~i~

first

in an a l g e b r a i c - g e o m e t r i c

and later used in group theory b y A t i y a h and Tall

The notion of k-ring T h e o r e m of Symmetric

is b u i l t upon the classical

Functions:

where

section.

tool is the notion of k-ring,

introduced b y G r o t h e n d i e c k

Groups)

theory of the general

have appeared until Atiyah's Power Operations a "dual" version

since the origins

(~43).

Fundamental

Theorem:

Let n~l

integer

coefficients

permutation

o6S

and

,

n

having f(X

("f is s y m m e t r i c " . ) F ( a l , a 2 ..... an), coefficients,

(the there

f ( X l , X 2 ..... Xn)

a2

=

XIX 2 + XlX 3 + X2X 3 +

a. l

=

~ jl

proposition.

~rn:Qr ~

to X,

Say we w i s h

the free

~ be c o n s t r u c t e d n

kt(~')=l+~'t'l 1

f(X)=X,

an a u g m e n t a t i o n

the f i r s t m a p is t a k e n

and

and m u l t i p l i c a t i v e ,

an a u g m e n t a t i o n

k-ring h o m o m o r p h i s m

¢:R~)

it is

is trivial.

In a s s i g n i n g ¢(X) ~

kt(kn(f(X)))="An"(kt(f(X))),

I

on one g e n e r a t o r .

adding

Let

an i n d e t e r m i n a t e

Q0 = ~' ~ , as n

Qn = ZZ [~i, ~2 ' "" ' ~n I, w i t h

and for any r > n ,

there

is a k - r i n g

Qn' ~([i ) = [i or O, d e p e n d i n g

on w h e t h e r

i ~ n or not. Let

Q = Lim

Qr"

(Recall

(...,an,an_l, ..,al,a0) Thus

an e l e m e n t

set of v a r i a b l e s polynomial set equal

of

with

if all b u t

is the

set of all s e q u e n c e s

aiEQ.1 and ~ n ~ i , n ( a n + l ) = a n

~ is an i n f i n i t e

~., h a v i n g 1

to zero.)

this

power

the p r o p e r t y

a finite

number

As r e m a r k e d

series that

in the i n f i n i t e

it r e d u c e s

of the v a r i a b l e s

above,

for all n.

(page 22)

to a

~. are l

Q is a l-ring.

25 th Let symmetric and a

n

an = an (?I'~2' .... ~r ) 6 Qr be the n function

of the

6 Q be the a s s o c i a t e d

kn(al)=an , for all n a's n

involving

some

a polynomial and this

Let comments,

of

identity

independent

involving

a I must

generated

contain

each

On the other hand,

f(al,a 2,..) EZ[al,a 2,..] coefficients.

In the next the

to m e n t i o n

section, free

by

ring

form

give

a I.

By the previous

in an infinite Indeed

any s u b - k - r i n g

polynomials,

in the al w i t h

is a s u b - k - r i n g

Three,

on one generator.

r

functions).

kn(f(al,a2,..)),

a polynomial

~ [al,a2,..]

a categorical

already

the u n i v e r s a l

and in C h a p t e r

k-ring

id~tity

an=kn(al ) , and so c o n t a i n

using

of the

as again Hence

Then

©r ) and the

would

A = ~ [ a l , a 2 .... ]-

any e x p r e s s i o n

if n ~ r)

limit.

any p o l y n o m i a l

of s y m m e t r i c

is a p o l y n o m i a l

of indeterminates:

we can e v a l u a t e

wish

inverse

in each

(since

to zero

al($1 '''' [ r ) ' ..,ar(~l,.. ' ~r ) 6

sub-k-ring

A, as a ring,

intensively

is true

out by the t h e o r e m

A c Q be the

Q containing

integer

this

in the

of a l , a 2 , . . , a r, for example,

is ruled

~ [ a l , a 2 , . . ].

(equal by d e f i n i t i o n element

(since

are a l g e b r a i c a l l y

number

~'s

elementary

we w i l l

of

study

H e r e we w i s h

application. A n a t u r a l

Q.

just

operation

26 on the category of ~-rings is a natural transformation from the underlying set identity functor to itself. That is, we have an assignment to each k-ring R, a map

(of sets)

~R:R ~ >

R such that for any map

of ~-rings f:R ---~ S, f~R = ~S f :R--~ S. operations

is defined b y

for multiplication

Proposition: isomorphic

A d d i t i o n of natural

(~R+~R) (r)= ~R(r)+~R(r)

and similarly

and k-operations.

The set of natural operations

is a i-ring,

to the free i-ring on one generator,

Proof: Let ~ be an operation.

and is

A-

Let alEA be the generator of A

and suppose uA(a I) = f(al,a 2 .... ) E A-

For any i-ring R, and rER,

let g:A---~R be the unique i-ring h o m o m o r p h i s m with g(al)=r. Then uR(r) Conversely, A-~A,

= uRg(a I) = g(uA(al))

= g(f(al,a2,..

given any element f(al,a2,..)EA,

taking a I to f(al,a2,..)

)) = f(r,~2(r),..) •

the unique map

extends uniquely to a natural

operation.

I

Hence given a natural operation, in the i-operations.

it is uniquely a polynomial

To check that a given polynomial f(il,i 2,..)

is equal to a given ~, one only need check u(al)=f(al,a 2 .... ). This being a proposed identity in to check in each Qn"

A c Q = Lim Q , it is sufficient n

This can be formally phrased as the

27

Verification uniquely

Principle:

a polynomial

If

in the l - o p e r a t i o n s

particular polynomial

f

(kl

s u f f i c i e n t to c h e c k t h a t of e l e m e n t s

of d e g r e e

I

2 ,l ,..),

and f o r m u l a t e

This process category.

such,

finite groups,or

compact Hausdorff

an o p e r a t i o n

on the c a t e g o r y principle

as f o l l o w s .

Let C be

The

K

0

the c a t e g o r y

s p a c e ~ , w e can d e f i n e f r o m C to the c a t e g o r y

functor C ~>

set of all s u c h o p e r a t i o n s

(l-rings) ~

any of

a K-theory of x - r i n g s . trans-

(Sets)

for a g i v e n C and K f o r m

a n d t h e r e is a n a t u r a l m a p of k - r i n g s

in g e n e r a l

for such.

of

in the K - t h e o r y K 0 is a n a t u r a l

f r o m the c o m p o s i t e

a k-ring/Op(K0~ Of course,

k - r i n g on k g e n e r a t o r s ,

a similar verification

to b e a c o n t r a v a r i a n t ~ n c t o r

to itself.

[l+[2+..+~r

B y a n a l o g y w i t h the c a s e w h e r e C is the c a t e g o r y

~ - m ~ u ~

formation

on a s u m

operations

can be generalized

~u~mmmmm~mmmmm~m~

Given

U = f, it is

E

construct the free

and d i s c u s s k - a r y n a t u r a l

of k-rings,

U is

and for any

to c h e c k t h a t

U = f, o p e r a t i n g

then

i, for all r > 0.

One can similarly any k ~

U is a x - r i n g o p e r a t i o n ,

A~OP(K0)

t h i s m a p n e e d b e n e i t h e r o n e - o n e n o r onto.

.

28

3.

Symmetric Functions

Let that

A b e the f r e e

l - r i n g on one g e n e r a t o r .

A as a r i n g is a p o l y n o m i a l

ov v a r i a b l e s : A was

Recall

this m e a n s

r i n g o v e r ~ in an i n f i n i t e n u m b e r

n [al,a 2 .... ], a n d k (al)=a n-

A = ~

constructed

as a s u b r i n g

of a k-ring

Q = Lim Q , < n

Qn = ~[[i' ~2'''" ~ ' and w e can sum up the r e l a t i o n b e t w e e n

g's by the

a's and the An

element

equation

--

if as a f u n c t i o n of the

of d e g r e e k.

of w e i g h t

Thus

a monomial

Let

of w e i g h t

{al,a2,... ] forms rI aI

r2 a2

a2

r

...an

n

has

n in A. addition,

n is a g a i n of w e i g h t a polynomial basis

n.

of

s i n c e the s u m of two S i n c e the set

A, the set of m o n o m i a l s

rn ..an

integers)

' (all n, all s e q u e n c e s

f o r m an a d d i t i v e b a s i s

of

r l , . . , r n of n o n - n e g a t i v e

A-

Hence rI

consists

_-

aI

r~

An d e n o t e the set of all e l e m e n t s

A is an a b e l i a n g r o u p u n d e r n elements

~ 's, it is

r]

rl+2r2e3r3 +'''+nrn"

isobaric

an tn = ~ (l+~.t)l " n=0 i=l of A is c a l l e d i s o b a r i c

1

homogeneous weight

k t (a I ) = ~

f ( a l , a 2 , . . , a n ) = F ( ~ l , ~2''')

of w e i < ~ t k in t h e a ' s --

the

of all m o n o m i a l s

~ ir. = n. l

The number

of t h e f o r m a I

of such m o n o m i a l s

one b a s i s

r2 a2

of

..an

with

is t h e n u m b e r

of p a r t i t i o n s

rl2r 2 of the n u m b e r n.

Indeed,

to e a c h p a r t i t i o n rI

w e can a s s o c i a t e be denoted

a .

set of all aT,

the m o n o m i a l Thus

aI

r2 a2

~ = 1

rn ...n

of n.

of n,

rn ..an

This monomial will

An is a free a b e l i a n g r o u p w i t h b a s i s

~ a partition

An

rn

the

29

At this p o i n t notation

it is c o n v e n i e n t

on p a r t i t i o n s .

Given

is any s u m n = n l + n 2 + . . . + n k ,

to i n t r o d u c e

a number

n!>0.

some g e n e r a l

n, a p a r t i t i o n

~ of n

If r I of the n's are equal

to l, r 2 are equal to 2, etc., this p a r t i t i o n is d e n o t e d r r r ~= 1 12 2 ...n n T w o p a r t i t i o n s are equal if and only if the corresponding write

rl,r2,..,

the p a r t s

in d e c r e a s i n g

n l ~ n 2 ~ ... >_nk. associated with n squares,

=

Given

~ =

in k rows,

any p a r t i t i o n

b e the lengths

graph

notation

or Y o u n q

(nl,n 2, .... n k)

with

and all of the rows

common

is to

(nl,n 2 .... ,nk) , w i t h

(Ferrar's

the p a r t i t i o n

(6,5,3,3,1,i,i)

Another

order:~=

The diagram

arranged

t h e squares,

are equal.

the i

th

diagram)

consists

of

row c o n t a i n i n g

lined up at the left.

n ! of

Thus

gives

~, w e can d r a w its g r a p h

of the c o l u m n s .

The

sequence

and let l l , 1 2 , . . . , 1 q ( l l , 1 2 , . . . , 1 q)

is /

also a p a r t i t i o n Its d i a g r a m diagonal. The

is o b t a i n e d b y f l i p p i n g Thus

~ E ~(n),

partition

of ~, d e n o t e d

~

the d i a g r a m

for ~ a l o n g

its

for ~ = ( 6 , 5 , 3 , 3 , 1 , I , i ) ,

set of all p a r t i t i o n s

(For a t a b l e For

of n, the c o n j u q a t e

another

common

~'= ( 7 , 4 , 4 , 2 , 2 , 1 ) .

of a n u m b e r n is d e n o t e d

of the size of H(n)

for n=l,2 ..... 200,

notation

.

is ~ ~ n.

see

H(n). [31~

.

30 Now back monomials gives

of

As

Since

the w e i g h t

An>~ik~---m>in+k,

of the p r o d u c t

of two

multiplication

for each n,k.

Hence

in A

h = ~ A n is n a 0

ring. shown

{aT I ~ ~n]. classical bases

An.

i is the sum of the weights,

a map

a graded

to

above, B u t this

theory

are also

The

first

In terms

each

An is a free

abelian

is not the o n l y " n a t u r a l "

of s y m m e t r i c

functions,

group w i t h basis.

several

other

From

"natural"

of t h e s e

is the [,

set of h o m o q e n e o u s

power

we can d e f i n e

l

hl

: ~i

h2

= i~z-j gi ~j

n

This

=

= Zan

k_t(al)

al =

a12-a2

~ ~ . • .. L_ i 11 1 2 ll~- 12L-n

definition

kt(al)

the

indicated.

of the v a r i a b l e s

h

a basis

? in

can also be w r i t t e n

tn

= ~

(l+~it) "

-

~ i (1-~l t)

=

~i t

nnl

•

as follows:

Thus

1 - ~i t

i

=

h nt n

n

We have

sums.

31

Hence

the a's and h ' s

are r e l a t e d b y the i d e n t i t y

~-.~ant

(-i) n h n t n

n

n=0 Equating a

~ /

J

coefficients

of t n g i v e s

H e n c e the set

we

+

n - hlan-I

A, w i t h

each h

.

{ hn

n

i

.

+

n ~0}

of w e i g h t

is

{ h

(-1)nh

n

1

=

0

n>~l

Given

a p o l y n o m i a l b a s i s of r r a p a r t i t i o n ~ = (l 12 2 ° - .)

Then another b a s i s for An as f r e e a b e l i a n

the n a t u r a l w a y to d e s c r i b e

on a k - r i n g R is to s p e c i f y n o t the o p e r a t i o n s r a t h e r the o p e r a t i o n s to the same t h i n g

n

's.

And,

•

{~ ~ n } .

In some cases,

h

.

also forms

n.

w r i t e h~= h l r l h 2 r2 . . . .

group

=

n=0

n

.

By the

formulas

k n, b u t

above,

• since the k n 's are e x p r e s s e d

of course,

c h o i c e of b a s e s

h

the k - s t r u c t u r e

this

in t e r m s of the

the same c o m m e n t w o u l d h o l d

of A , n~l. n

amounts

for any

32 Another basis is given by the monomial symmetric functions. rI r2 Let ~ = (i 2 ...) be a partition of a number n. We give two descriptions

of the monomial

symmetric

function

.



symmetric

functions

provid~ operations,

In this case there is an addition formula

(VI+V 2) =

To prove this,

~. (V 11 (V 2) ~i~2 =~

it is enough,

using the Verification

assume V 1 and V 2 are sums of i st degree elements.

Principle,

powers

of and

just as in the additivity formula for the exterior

(p.6).

Details

are left to the reader.

(Or see

[323, p.91,

Theorem 33). The main subject of this section is the set of operations associated with the power sums s . n Operation,

to

The argument

can then be carried out with the explicit definition the < ~ > ' s , l

also

Y? b y yn(v) = Sn(V),

We define the n

th

for V in a k-ring R.

Adams

48

Proposition :

Let

a,b be

elements

in a k-ring

R, and n,m i n t e g e r s

>~ 1

Then I)

yl(a)

= a

2)

~n (i) = 1

3)

~n(a+b)

4)

'~n(ab) = %+n(a)%+n(b)

5)

~n(km(a))

= km(+~n(a))

6)

Yn(ym(a))

= ynm(a)

= ~n(a)

Thus e a c h yn is a k-ring is a ring h o m o m o r p h i s m Proof:

+ ~n(b)

endomorphism

are each sums of e l e m e n t s of the p o w e r

The A d a m s elements

+~n

of d e g r e e

sums

operations

Principle,

s

n

we can

i.

(p.35)

also serve

assume

Then u s i n g all t h e s e

that a and b

the o r i g i n a l

m

are clear.

to d i s t i n g u i s h

binomial

of k-rings:

Proposition: ~n(a)=a

of R, and the map n ~ >

~-~-~ End R.

By the V e r i f i c a t i o n

definition

= Ym(~'n(a))

Let R be a k-ring

and aER.

Then

a is b i n o m i a l

iff

for all n > l . ~>

Proof: kt(a)

Use the i d e n t i t y =

d___log(kt(a)) dt

(l+t) a iff the r i g h t h a n d

Corollary:

For all m E2Z, ~'n(m)=m.

side

= ~

( - l ) n ~ n + l ( a ) t n.

Then

rl--o

is ~ ( - l ) n a t

n -

a l+t

I

49 Hence, sub-k-ring will

given

R 1 of R b y R 1 =

at least

The

any k-ring

include

Ix I ~n(x)=x,

the unit

first p r o p o s i t i o n

in v e r i f y i n g

that v a r i o u s

we n e e d

a definition:

element

r£R,

R, we can p i c k

pre-k-rings

include

This

binomial subring

a copy of ~.

a converse,

which

are in fact

a ring R is t o r s i o n - f r e e

and any integer

a maximal

all n ~ i}.

i, so will

above has

out

will be useful

k-rings.

First

if for any n O n z e r o

n ~ i, nr = r+...+r

(n summands)

is

Let o p e r a t i o n s

yn

also nonzero.

Theorem:

Let R be

a torsion-free

pre-k-ring. oo

be d e f i n e d by,

for a£R,

(So in particular, Suppose

yn(1)=l,

all a,b6R

pre-~-rinq

we h a v e

n,m.

the theorem,

~n:R~>R,

we m a k e

a general

for

definition:

unit,

n ~ i, s a t i s f y i n g

for all a,b£R,

is a p r e - ~ - r i n g

satisfying

Yn(ym(a))=ynm(a)

and Yn(ym(a))=ynm(a)

ring R w i t h

yn(a+b)=yn(a)+yn(b), also

all a,b,n.)

Then R is a k-ring.

R is a c o m m u t a t i v e

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

n

'~'l(a)=a, "~n(a+b)=yn(a)+yn(b),

yn(ab)=yn(a)yn(b),

and integers

TO p r o v e

d ~ =~--~-l°gkt(a) = ~ ( - l ) n ~ n + l ( a ) t n=0

for all a,b6R

yn(1)=l,

together

yl(a)=a,

and integers

A with

and

n. A

~n(ab)=yn(a)~n(b),

and integers

n,mZ

a

i.

and

50

Given

any c o m m u t a t i v e

ring

R w i t h unit,

countable

sequences{(rl,r2,r3,..)

structure

by defining

For each yn(

integer

R

w

R w is g i v e n

I r i £ R ].

and m u l t i p l i c a t i o n

n ~ i, we d e f i n e

(rl,r2 .... )) =

Proposition:

addition

an o p e r a t i o n

a ring

coordinatewise.

yn:RW~R

w, b y

(rn,r2n,r3 n .... ).

is a Y-ring. if the m a p

If R is a p r e - Y - r i n g ,

Y-ring

if and only

Y(r)

(yl(r),y2(r) .... ), is a h o m o m o r p h i s m

=

let R w be the set of

then R is a

Y:R---~ R w, d e f i n e d b y of '~-rings.

D Proof:

Clear.

Note

that

since

1

is the i d e n t i t y map,

the m a p

Y:R ~

R ~ is

one-one.

Let R be

a torslon-free

kn : R - - ~ R, n>~O, so t h a t E.g.,

following

p.

pre-~-ring.

Sul]pose t h e r e

are o p e r a t i o n s

co

~d

logkt(x)

= ~ - ( - 1 ) n y n + l ( x ) t n, a l l x£R. n=l

, we can c a l c u l a t e

l(x) 1

,

I~ y2

n. k n(x)

= de

yl

(x)

(x)

yl(xI

n(x)

T h e n we s u p p o s e

that

b y the d e t e r m i n a n t so t h a t kn(x) that this

for each n > i, and each x 6 R the e l e m e n t

on the right h a n d

is d e f i n e d .

division

side

is d i v i s i b l e

(The t o r s i o n - f r e e

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

provision

if defined.)

Note

b y n:

defined in R,

guarentees

that

if

51

R contains possible

a field of c h a r a c t e r i s t i c

so the kn's

As usual,

automatically

the p o w e r

series

division

is always

exist.

k t =~knt n gives

R----~I+R[[t]] + and we can d e f i n e diagram

zero,

a map

L making

a map the

following

commute: 1+R[[t~ 3+ R //kt/~

L is

defined

by:

d -~log(t+alt+a2t

B(l+alt+a2t2+...) 2

Proposition:

~L

+...)

i)

=

(-1)nrn+l t

L(x"+"y)

2) L("I")

= (rl,r

= L(x)

2 ....

) if

n

+ L(y)

= 1

3) L(x .....y) = L(x)L(y) 4) L("Yn"(x))

= ynL(x)

5) If R is t o r s i o n - f r e e , 6)

If R c o n t a i n s onto,

Proof:

(Recall

operations there.)

a field of c h a r a c t e r i s t i c

and h e n c e

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

first that the

"..."

in l + R [ [ t ~ + - thus

I) and 2) are easy.

easy observation

i),

notation

so the i n j e c t i v i t y

that the k e r n e l

refers

L is

to the k-ring

to the A d a m s

T h e n to c h e c k i, g i v i n g

zero,

of Y-rings.

,,yn,, refers

to take x and y to b e of d e g r e e F o r 5), we can use

L is one-one.

3),4)

again

operations

it is s u f f i c i e n t

an e a s y v e r i f i c a t i o n .

of L f o l l o w s

of L is trivial.

from the

For 6) the

52

inverse map L -I: RW--~ l + R [ [ t ~ + is easily c ~ I c u l a t e d n

-i

as

D

((bl,b 2 .... )) = exp(-g(t)) , where g ( t ) = ~ (-l)nbn tn" n+l

Corollary: zero

If R is an a~gebra over a field k of c h a r a c t e r i s t i c

(e.g., k = Q

Proof:

), then so is l + R [ [ t ~ +.

R w is c e r t a i n l y

isomorphic

a k-algebra,

and under the hypothesis,

is

to l + R [ [ t ~ +.

The Proof of the original R has k-operations, R is a k-ring

theorem is n o w accomplished.

and hence Y-operations,

iff R is a Y-ring.

A useful

If

and is torsion-free, restatement

of the

theorem is the following proposition.

Proposition:

Let R b e a t o r s i o n - f r e e

Suppose there

is a ring homomorphism.

There

and S b e any ring.

is given a map of sets ~:S--~> l + R [ ~ t ~ +. Then

is a ring h o m o m o r p h i s m

k-structure

ring,

iff the c o m p o s i t e map S --~ I + R [ [ t ~ + - ~ R If S is a pre-k-ring,

~ preserves

the

iff the c o m p o s i t e map L~ does.

is an interesting

a p p l i c a t i o n of this p r o p o s i t i o n

algebraic g e o m e t r y of varieties

over finite

fields

to the

(which the

n o n - g e o m e t e e r m a y ignore since it will not b e relevant to the sequel.)

w

Let k be a finite

field and S the G r o t h e n d i e c k

ring of

53

varieties

d e f i n e d o v e r k, w h e r e

of v a r i e t i e s and ~ : S ~ >

and p r o d u c t

~x(t).

S --~ X j

on X w i t h c o o r d i n a t e s

on e a c h v a r i e t y ,

proposition

implies

that

One m i g h t h o p e t h a t seems not

so.

the n - f o l d

for d e f i n i t i o n s . )

L ( ~ x ( t ) ) n is the n u m b e r of field k'

for p r o d u c t s ,

~ is a ring h o m o m o r p h i s m

the r e s t a t e d

R ---~ l + ~ [ t ] ] +.

{ is also a m a p of k - r i n g s .

symmetric power

, where

is the sum of the r a t i o n a l

and s i m i l a r l y

The o b v i o u s

of k

S i n c e the n u m b e r of r a t i o n a l p o i n t s

on a d i s j o i n t u n i o n of two v a r i e t i e s points

~44J

in the e x t e n s i o n

o v e r k is n.

Let R = Z,

to e a c h v a r i e t y X its

(See S w i n n e r t o n - D y e r

T h e n in the c o m p o s i t e m a p

the d e g r e e of k'

f r o m the d i s j o i n t u n i o n

f r o m the c a r t e s i a n p r o d u c t .

l + ~ [ [ t ] ] + b e the a s s i g n m e n t

zeta-function

points

sum c o m e s

k-structure

Alas,

it

to p u t on R w o u l d be to take

of a v a r i e t y X to b e h

(X), and d e f i n e n

the o t h e r o p e r a t i o n s 1-space A

1

is of d e g r e e

in a k-ring,

degree

in t h i s d e f i n i t i o n , We

accordingly.

s h o u l d remark,

i, b u t

U n d e r this d e f i n i t i o n

1 1 2 its s q u a r e A X A = A is not,

is m u l t i p l i c a t i v e .

powers,

seems n e a r l y

as p l a u s i b l e .

t h a t t h i s c a t e g o r y of v a r i e t i e s

is a g o o d e x a m p l e of a c a t e g o r y w h i c h has symmetric

but whose Grothendieck

and

H e n c e R is not a k - r i n g

and no o t h e r d e f i ~ i t i o n

however,

_affine

sums, ring

products,

is not

over k and

a k-ring.

54

W e n o w u s e the t h e o r e m to c o n s t r u c t k-rings. K(S)

Let S b e

c l a s s of

a set, K a field of c h a r a c t e r i s t i c

the set of all m a p s

two m a p s

a general

is d e f i n e d

as usual,

identity.

Suppose there

n = l , 2 .....

satisfying i)

from S to K.

o

The

m a k i n g K(S)

sum and p r o d u c t a commutative

is g i v e n on S a set of m a p s

is the

zero,

and of

ring with

~ :S---~S, n

identity map

i ii)

o ~ = n m

We t h e n d e f i n e o p e r a t i o n s 9n(f(s))

= f(On(S)).

possible

in K(S)

nm 9n:K(S)~)

K(S)

As in the theorem,

for e v e r y

by,

for f : S ~ )

K,

and sES,

s i n c e d i v i s i o n b y n:

is

i n t e g e r n, we can d e f i n e o p e r a t i o n s

in by

kn

_

1 n:

det

91

1

0

,2

~I

2

~n and w e can c o n c l u d e to the o p e r a t i o n s ~-ring,

k

t h a t the n

th

~n are the A d a m s o p e r a t i o n s c o r r e s p o n d i n g since K(S)

is c l e a r l y

a

it is a l s o a k-ring.

one is g i v e n

n

0

41

By the theorem,

This construction

of G.

/

a

n

power

applies

a g r o u p G,

is the m a p in G.

on G w i t h v a l u e s

in p a r t i c u l a r

and S is the set of c o n j u g a c y c l a s s e s

i n d u c e d on S b y

K(S) in ]~,

to the c a s e w h e n

the o p e r a t i o n of t a k i n g the

is t h e n c a l l e d the ~__~rin~ of c e n t r a l

functions

55

The context Let

L l + R [ [ t ] ] +~ P R

map

- that

R be

of u n i v e r s a l

a commutative

the moment)

and

of elements

of R.

different

ring

let W R b e

ring

Consider

classically

As

the map

rings

with

unit

the

a set,

structure

Witt

set

of

WR = R

w

appears

- which

we now

(assumed all

another

describe.

torsion-free

w-tuples

, but

in

we will

for

(Wl,W2,~.-) put

a rather

on W R.

M:WR----~ R ~ d e f i n e d

by n/d

M ( ( w ! , w 2 .... )) = Thus

(rl,r 2 .... )

where

r

= n

~dw dh

d

can

identify

rI = w 1 2 r2 = w I

+

2w 2

3 r3 = wI =

r4

+

3w 3

4

wI

+

2w22

+ 4w 4

etc. If R is t o r s i o n - f r e e ,

M

is

a one-one

map

and we

W R

with

its

image

Proposition: there

M(WR)

WR(M)

c R

.

is c l o s e d

are p o l y n o m i a l s

F.,

under

G. w i t h

sum

and product

integer

coefficients

l

F n depends

[wl I i d i v i d e s

on t w o

n])

sets

of variables:

Indeed

such

j

M ( ( W l , w 2 .... ) ) + M ( ( w l , w 2 .... ) )= M ( ( F I ( W l , W l (here

in R e.

that

e

)'F2(wI'w2'wI'w~)

[wi I i d i v i d e s

n]

.... ))

and

and

M ( (Wl,W 2, .. )) -M ( (wl,w ~, .. ))=M((G 1 (Wl,W i) ,G 2 (Wl, w2,w{, w ~, .. )) (where

similarly

Hence cation

using

in WR,

G

n

is

these

a function polynomials

W R becomes

of

the

two

to d e f i n e

a commutative

ring

sets

of v a r i a b l e s ) .

addition with

and multipli-

identity

- the

56 (Universal) ring of Witt vectors of R.

For the proof of the proposition,

we must construct the

polynomials F., G., which can be accomplished by just proving l ] the special case for the ring R = Z. polynomials,

For, once we have the

they define operations in the set W R, for any ring R,

(torsion-free or not).

The ring axioms, associativity ~£c., will

follow for an arbitrary ring because they are true over Z, hence are polynomial identities valid in all rings.

Actually,

to carry out the proof, we will just assume that

R is torsion-free

(an obvious property of Z).

WRM---~R w is injective.

Then the map

Also, the map l+R[[t]] + is injective.

The proposition follows by observing that these two inclusions give the same subset of RW:

Indeed, we can define f : W R ~ > l + R [ [ t ] ] +

by, for w = (Wl,W 2, ...)£W R, f(w)

Then L~f(w) -

=~(i d

d log ( ~ ( i dt d

1 d\ l-Wd(-t)d ]

= ~{d

= ~

Wd(-t)d-i

1

-

Wd (- t )

C¸-d

= ~ d

( - l ) ~ d ~ d wdn/d(-t)n I

(-l)

n

- Wd(-t) d)

n-i r

tn-i n

d)

)

Wd(-t) d t

)(i + Wd(-t) d + w d 2 j

= ~n(-l)n+/d~hd wdn/d I t n-I

(-t)

2d

+.

57

Hence,

b y the d e f i n i t i o n

M is one-one, via

so is f, so

f as a subset

it is n e c e s s a r y which

of the map L, L f(w) (for R t o r s i o n - f r e e )

of l+R[[t]] +.

once we w r i t e

(rl,r2,...).

Since

W R can be c o n s i d e r e d

To see that W R is all of l+R[[t]] +,

o n l y to s h o w that

is o b v i o u s

=

f is an i n v e r t i b l e

out the d e f i n i t i o n

map,

a fact

of f in m o r e

detail: Given w =

(Wl,W2,W3, ...)

f(w) = ~ ( i d =

£ W R,

- Wd(-t) d)

(l+Wlt) (l-w2 t2) (l+w3t3) (l-w4 t4) ...

= 1

+

+

(wl)t

(-w2)t 2

(w5-w4wl-w2w3)t5

The c o e f f i c i e n t sum is over

+

of t n is

w ..w nI n2 nk

(-w4+wlw3)t 4

where

the

of n into k d i s t i n c t

if l + a l t + a 2 t 2 + . . .

solve

+

( - w 6 + w 5 w l + w 4 w 2 - w 3 w 2 w l)t 6 +

n l > n 2 > .. > n k

Hence

the image of f, we m u s t

(w3-wlw2)t3

( - 1 ) n ~( - l ) k w < -

all p a r t i t i o n s

and all i n t e g e r s k.

+

+

El+R[ [t~ + is to be

the e q u a t i o n s

aI = w I a 2 = -w 2 a 3 = w3-wlw 2 etc. Since

in the n

coefficient w's

th

equation

w

n

occurs

+ l, the e q u a t i o n s

in terms of the a's.

in o n l y one term w i t h the

can b e

solved

inductively

for the

parts, in

58 The

first

few terms

are

w I = a1 w 2 = -a 2 w 3 = a 3 + ala 2 2 w 4 = -a 4 + a3a I + a 2 a I 2 w 5 = a 5 - a4a I - a3a 2 - ala 2

2 3 + aI a3 + aI a2

etc. H e n c e W R is i s o m o r p h i c

as a ring to l+R[[t]] +,

(It s h o u l d be m e n t i o n e d have

set up the i s o m o r p h i s m

(Wl,W 2 .... ) - ~ 2 ~ i d to take the to s a t i s f y

- wdtd ) .

(l+at) (l+bt)=(l+abt) Indeed our~-~ d

cited

degree - so

above.

( g~i

~4~, I~7

+ as

forces

them

of first d e g r e e Our c h o i c e

elements

authors

elements

in l+R[[ t ]I +

of f is g o v e r n e d b y

should m u l t i p l y

is to t h e i r ~--~ (l-wdtd) d n is to k_t(x)

properties

to the p - v e c t o r s

several

(l-at) (l-bt)=(l+(-a) (-b)t)=(l+abt).

(l-Wd(-t)d)

kt(x)=~kn(x)t

Additional relation

This

(1-at) (l-bt)=(1-abt). first

that

WR~)I+R[[t]]

"multiplication"

our rule that

expression

here

for any ring R.

=

( ~ h n ( x ) t n ) -I.

of this c o n s t r u c t i o n of W i t t can b e

as the

found

)

of W R and its in the r e f e r e n c e s

CHAPTER

The

II

following

:

notation

All

groups

are

All

vector

spaces

denotes If V

the

and W

linear

maps

S

Denote

these

o(1)=2,

relations

cons~[sts this

and

this

chapter:

numbers,

act

is t h e

are

over

= n,

picking ),

the

z*

is t h e

~ unless

complex

is

its

vector

space

a basis

the

group

numbers.

conjugate.

of Hom(V,V)

o(j)=j,

"column

vectors"

group

j>2.

7(n)=l. n

= i,

elements

Let

Then

S

and

1,7,T

2

otherwise

of

consisting

of V gives

of

invertible

specified.

left multiplication

1 , 2 , 3 ..... n.

~

z6C,

subset

~ Gl(n,~

on V by

symmetric

2 ~ =i,

V

AutV

for

Hom(V,W)

If d i m V

, are

of V

and

over

.

~

objects

of the

order.

~

to

the

o(2)=1,

i = i ; 2 .... n - 1 the

over

elements

OF G R O U P S

throughout

finite-dimensional

of groups

products,

denotes

n

are

complex

Aut V

THEORY

finite

elements.

matrices

so t h e

used

Aut

isomorphism

take

be

f r o m V t o W.

an

We

will

spaces,

invertible

Tensor

REPRESENTATION

are vector

of

n~fn

THE

n

rather

than

of permutations Let

oES

T be

the

n

be

oT = T

n-i

,o,o~,o7

2

~.

the

n-cycle

is g e n e r a t e d

by

(f(g(v))=(fg)

"row vectors".

of n objects. transposition T(i)=i+l o,

we

,

T subject

In p a r t i c u l a r ,

which

(v)

usually

to

S3 take

in

)

60

i.

The R e p r e s e n t a t i o n

A reprgsentation

~ :G ~ >

Rin 9 of a F i n i t e

Gzoup

of a g r o u p G, of d e g r e e ~,

is a h o m o m o r p h i s m

GI(n,C) .

For example,

if G = S 3, we can a s s i g n

-1

f:

9 :

-1

p Thus

~

for x , y ~ S 3, the m a t r i x p r o d u c t

of

~ (x) and

~(y)

is equal

to

(xy).

While take

this

is a nice c o n c r e t e

a coordinate-free

approach

Let V b e a v e c t o r

space,

in V is a h o m o m o r p h i s m is the d i m e n s i o n an i s o m o r p h i s m the c o n c r e t e

While the p a i r

of V.

of

and G a group.

f:G ~Aut (Of course,

to

V.

A represent gtion of G

The d e ~ r e e

picking

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

a basis

of V gives

so this does g e n e r a l i z e

above.)

speaking,

(v, ~ ), we w i l l o f t e n

is c l e a r

we p r e f e r

to the problem.

A u t V w i t h Gl(n,~),

definition

strictly

definition,

from the context.

the r e p r e s e n t s t i o n speak of

of G is g i v e n b y

"the r e p r e s e n t a t i o n

In a s i m i l a r

way,

given

V"

if

p : G _ - ~ k-u-t j

V,

61

and g ~ G, we should and write g both

refer

to the a s s o c i a t e d

~(g) (v) for v 6V.

for the element

In other

common

of G and

of G

defined

Qr a c o m p l e x

the m o r e g e n e r a l arbitrary is said

Given G if g w ~ W

notion

of invariant

Zf gv=v

elements

vector

spaces

be

The set of all representation

for all in V.

is then exactly

a linear representation,

(to d i s t i n g u i s h

~GI(n,R), a

a subspace

where

(left)

it from

R is an

G-module

and

W of V is invariant

An element

g EG.

v ~ V is invariant

We w r i t e

V G is a s u b s p a c e

linear

maps

of G b y defining,

from V 1 for ~

is a linear

map of f(gv)=gf(v).

V 1 to V 2.

to V 2 , H o m ( V I , V 2) gives :V 1 -~) V 2 and g ~ G,

9Y(~j~)(~iy;jfor

Hom(Vl,V2)~ ~

, or

is invariant.

for any v e V # and g ~ G , of G - ~ o d u l e s

under

V G for the set

o~ V and

f: (VI, ~ ) ....) (V2, ~ )

the set of maps

V 1 --~ V 2 is the map

gv for F ( g ) (v).

of G on V.

f : V l - - ~ > V 2 satisfying,

Let HomG(VI,V2)

g~:

F:G

and w ~W.

A map of G-modules

called

to write

it from a p e r m u t a t i o n

V is called

of G on V,

for all g ~ G

and to write

representation

ring).

on V as /(g)

simpler

(V, p ) is also

of a map

an action

an action

a fixed point,

for F ( g ) ,

(to d i s t i n g u i s h

commutative

to give

it is u s u a l l y

terminology,

representation below)

But

operation

all v ~ V I.

H o m G ( V I , V 2)

a

62 A map of G-modules

f (VI, ~ ) --~ ( V 2 , ~ )

there is a map of G-modules composites (Vl~t)

f': ( V 2 , ~ )

if

(VI, ~j) so that the

ff' and f'f are the identity maps on V 2 and V I.

and

(V2j ~

) are isomorphic

there exists such an isomorphism. (V2, ~ ) ,

-~

is an isomorphism

(also called equivalent)

Note that, given

V 1 and V 2 can be isomorphic

if

(V I, ~,) and

(indeed identical)

as

vector spaces without being isom~Dphic as G-modules.

W e are interested really only in isomorphism classes of representations. of basis

Given a representation F

:G -m Aut V, each choice

for V gives a matrix representation of G, G --~ GI(n,C),

and all these are isomorphic. more natural than others,

Proposition:

Let

But some choices of basis of V are

as the following proposition shows.

~ : G ---~ Aut V be a representation of G.

there is an inner product on V, call it each g 6 G ,

and all Vl,V 2 in V, ¢ i , v 2 >

,

Then

such that for

= ~gvl,gv2>

.

Hence

every representation of G is isomorphic to a representation b y unitary matrices.

Proof:

Pick any basis e I .... e

n

of V and let

(-,-) be the

usual inner product with respect to this basis: (~a.e., z

z

~b.e.) 3

conjugation.

]

3

= ~a.b.*, ~

i

Now define,

the asterisk denoting complex

z

for Vl,V 2

in V,

63

_

This

1 S ( g v l , gV2 ) IGi g~G

is c l e a r l y b i l i n e a r ,

the u s u a l

inner p r o d u c t

VlhV2>

skew-symmetric is.

1

=

~

J GI

hg,

the m i d d l e

g ~G

lhgVl,hgv 2)

~



in m a n y ways.

For n=0,

The t r i v i a l

the

vector

A I , . . . , A n. and

~iAi

map with

by

n-dimensional

representation

the m a p S ~ n

n= i Gi

G---->S

Gl(n,C)

G ~>GI(n,C).

left m u l t i p l i c a t i o n ,

degree

, called

Generalizing

of the n e l e m e n t s

representation.

n with basis the b a s i s

of a g r o u p n

.

Given

given

elements

Let V

labeled

elements:

given

~6

Sn

a linear

the r e g u l a r

this process,

we can c o m p o s e

the g r o u p G acts

representation

representation

w e can start w i t h

G, and any g r o u p h o m o m o r p h i s m

maps

a linear r e p r e s e n t a t i o n

of H.

is b y this

to get a linear

In p a r t i c u l a r ,

giving

G, o_~f d e g r e e ~,

such,

above

of a g r o u p gives

the

e( ~'~iAi ) = ~iA~(i).

a homomorphism

representation

unit m a t r i x

is the zero r e p r e s e n t a t i o n :

Sn acts on V b y p e r m u t i n g

A permutation definition

of the n y n

of all p e r m u t a t i o n s

n

space of d i m e n s i o n

~ V,

representation

dimV=0.

the g r o u p S

A I , A 2 ..... A n has a c a n o n i c a l be

this

A u t V, w h e r e

trivially,

all the

if G is abelian.

o__ffd e g r e e ~ of a g r o u p G is the a s s i g n m e n t to each e l e m e n t

diagonalize

H ----2G.

on itself

of G of

of G.

any

linear

representatioz

Then composition

In p a r t i c u l a r ,

if H is a

of

65

subgroup

of G, each representation H of H, Res G ~

a representation

Conversely, cosets

if H is a subgroup

representation

representation generalized, trivial

of G.

V.

of G, G acts on the set of left

left multiplication,

of G, so b y the above,

Using a ~ e r m i n o l o g y

one-dimensional

~:G---~ A u t V

let H be the kernel

G/H~)

Aut V.

normal

subgroup

representation

of ~'

giving a

a linear

which will

we say that this representation

A representation not,

:H - - ~ A u t

of H in G, ~ g H ~ g 6 GI b y

permutation

~ :G---) Aut V gives by restriction

later be

of G is induced b y the

of H.

is faithful Then ~

gives

if

is one-one.

a faithful

representation

In this case, ~ will be said to be associated

G, the subgroup

H.

In particular,

if ~

K e r ( ~ ) is contained

with the

is induced b y a subgroup

in H and equals

If

H of

H if and only

if

H is normal.

Thus

already

for the group S 3, we have

(where we just indicate ~(~) a consequence

l)

and

~(t)

the following

since

representations

the other matrices

of these):

The trivial

representation

of degree

n:

the n • n

unit m a t r i x

are

66

2)

The canonical

representation:

p (~)= !

3)

o

0

1

r(~)=

The regular

representation:

the o r d e r

9, ~ ' .

0

0

1

0

0

0 i

0

0

0

0

1

0

0

0

0

0

1 0

0 1

0 0

0 0

0 0

The r e p r e s e n t a t i o n

I

o

o

0

1

0

,A2:

Let the e l e m e n t s

induced by

Cf (H) = H

1 0 0 0

Let V b e the v e c t o r

g

=

This

in

~H

0

0 1 0 0

The

=~T-~H : ~ / :]f~I, a n d

=

~

H

H

~ H

"3CH =

=

io0) 1 0

0 0 0 1

space with basis

1

0 1

0 0 0 0

the s u b g r o u p H : I I,@~of S 3.

T (v ~H) =

Then

taken

OOOOoo

=

x(H)

I°l

, then

of S 3 b e

o

~(~)

c o s e t s o f H in S 3 are H = S H - - ~( ~

H =

,A3=

~-~ ~'~ ~ ~, t h e n

~0 a (~,)' =

4)

o

Let A I =

and

~

is isomorphi:c to the c a n o n i c a l

=

d i m V,

of W,

is the

the K r o n e c k o r p r o d u c t .

be the t r i v i a l

the d e t e r m i n a n t

V and W

The a s s o c i a t e d

Note

V.

o den°tes

representations

g(vl, % ... ,\vi)=gVl,% .... '\gv i.

g 6G,

in

of g in Aut V ~ } W w i t h

g @ G acts b y g ( v d ~ w ) = g v ~ g w .

e x t e r i o r p o w e r of the v e c t o r

~nv

of G in Aut V

zero m~trices.

The p r o d u c t V o W of the two vector

matrix

is a

and B the m ~ < m m a t r i x

to w I ..... win, the m a t r i x

to the c o m b i n e d b a s i s

and m k n

v I ..... V n , W 1 ..... w m

~ I v = V.

We

representation

of the m a t r i x

assigned

take ~ 0 V

to

If n= d i m V, assigning

to e a c h

to g in the r e p r e s e n t a t i o n

is the z e r o r e p r e s e n t a t i o n .

V , W we have a l r e a d y d e s c r i b e d

as a G - m o d u l e . G-module,

In p a r t i c u l a r ,

Hom(V,W)

is the dual

how H o m ( V , W )

if W

is

is the t r i v i a l

V of the

69

representation ( g ~ ) (v) =

V.

G thus acts

~(g-lv).

the c o r r e s p o n d i n g

dual b a s i s

a c t i n g on ~ is the

Let

inverse

complex number

Then

of ~.

transpose

Let

check

the c o n j u g a t e s

of V*,

the t r a n s i t i o n

on the c h o i c e

Proposition:

Proof: G-invariant

Proposition:

of V, and

assigned

to g

to g in V.

Pick a basis

is an n x n m a t r i x w i t h

of the entries.

&'~4~,

(Of c o u r s e

V / X ~ > V * does not depend,

obtained

9*(g)

denoted

of V.

By the p r e v i o u s inner product.

V, V

proposition,

V*,

e

Aut V,

and

one m u s t

up to i s o m o r p h i s m

is i s o m o r p h i c

matrices.

is i d e n t i c a l

to

to V*.

we can a s s u m e V has a

P i c k an o r t h o n o r m a l

basis

But a u n i t a r y m a t r i x

of V.

Then

is one w h o s e

its c o n j u g a t e .

Let 0 be the zero r e p r e s e n t a t i o n dimensional

the m a t r i x

a basis

of that a s s i g n e d

For any r e p r e s e n t a t i o n

transpose

we take

g e G,

of basis.)

G acts on V via u n i t a r y inverse

v 6 V,

be the n w n m a t r i x

the c o n j u g a t e r e p r e s e n t a t i o n that

~6V,

representation.

~*(g)

and th~s r * : G --~ Aut V is a n o t h e r called

Then

for each g &G, r (g)

entries.

r (g) b y t a k i n g

for

of matrices,

C :G --~ Aut V be a l i n e a r

v I ..... v n of V.

from

In terms

on ~ by,

of G and 1 the t r i v i a l

one-

representation.

Let U,V,

and W be G - m o d u l e s .

Then

t h e r e are

isomorphisms:

70

i)

(U(gV)~jW

ii)

VC~U ~

iii)

O~V

iv)

U-V

vi)

~ I~U ~

U~V

ix)

U

xi)

~

U~V d e g r e e V .

we can define

(since

have

of H gives

if G is the trivial

Conversely,

~H~V,

We

to see that

involution

augmentation

G acts

of H.

(i.e., ~ : G ~ Ind(V)

the s u b s p a c e H acts

groups

Ind:R(G) ----> R(H).

are

by g(h)=hg,

Aut ~4

is an a n t i - h o m o m o r p h i s m : defined

generated by,

= hlh~gv we

action

Let C H be the vector

This

is then

involved,

of H as follows:

of H.

on Ind(V)

v) = h h l g ~ v

Ind(V)

of G makes

as the vector

by all e l e m e n t s for h16 H,

write

~

a

space

of the form

hl(h~v)

= hl(h~gv), sometimes

of G,

=

(hlh)~ v-

this makes Ind H for G

sense.)

74 It is e a s y to see that gives

an a d d i t i v e

Ind(V)

m a p R(G) ~

= Ind(V).

= Ind(V)~)Ind(W)

R(H) .

To see this,

since H acts b y m a t r i c e s whence

Ind(V~W)

with

note

Slightly

less

so that

trivially,

that as an H-module,

real entries.

Hence

Ind

CH = ~£*,

Ind(V*)

=

(Ind(V))*,

the c o n c l u s i o n .

Ind

increases

degree(IndH(v))

These

Theorem:

operations

Let G c H.

dim~Oms(V,

2)

Ind(V~Res

In o t h e r words, spaces

Proof:

if n =

IHI/~G~

, then

= n d e g r e e V.

i)

product

the a u g m e n t a t i o n :

L e t V be a r e p r e s e n t a t i o n Res W) = d i m ~ o m H ( I n d

W)

=

There

of G, W of H.

V, W)

(Ind V ~ W

I) Res and Ind are a d j o i n t R(G)

Rec iproc ity:

Ind and Res are r e l a t e d by F r o b e n i u s

and R(H),

and

is a n a t u r a l

operators

2) I n d ( R ( G ~ C

imbedding

R(H)

on the is an

V---~ ~ H ~ V ,

inner

ideal.

v ~) l~v.

C3

Consider

the d i a g r a m

of v e c t o r f

V inclusion Ind V

Given

a G-map

diagram

determined

-'> Res W

1

: ~{~rg

f:V---~ Res W,

commute:

spaces

I

isomorphism

of v e c t o r

spaces

is a u n i q u e H - m a p

g, m a k i n g

this

--~

g there

take g ( h ~ v )

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

W

= hf(v).

Given

an H - m a p

of g to the s u b s p a c e

g,

f is u n i q u e l y

V of Ind V.

75 The isomorphism h~(v~gw)

~

Ind(V~Res W) ~ >

(h~v)~hw.

Ind(V)~W

is defined by

The inverse map is ( h ~ v ) ~ u

Each is an H-module homomorphism.

~> h ~ ( v ~ h - ! u ) .

76

2.

Irreducible

Representations

L e t V be a r e p r e s e n t a t i o n if t h e r e

is a s u b s p a c e W ~ V ,

w i t h W ~ CF(G)

preserving the involution, augmentation.

is a homomorphism of ~ -rings

conjugation,

It is a one-one map,

inner product,

identifying R(G) with

image, which we call the character ring of G. irreducible representations,

and its

The images of the

called the irreducible characters,

form an orthonormal basis of CF(G).

Proof:

The fact that the map preserves

the involution~

the conjugate,

sum, product,

and the augmentation

i, 0,

follow

immediately from the lemma above. To show that the map preserves sufficient

the b-ring structure,

to show that it preserves

it is

the Adams operations.

Thus,

85

given a G-module V, if we take the element ~ k ( v ) & via the universal polynomial ~ k ( v ) = Q k ( then compute ~k(~/~V) .

~k(v ),

R(G), computed

k2v ..... kkv),

~IV,

and

the result must be the same as computing

Hence we mu~t~ show, for every g6 G , ~ k ( v

) (g)=~k(~v)(g).

Let g 6 G and assume a basis for V is picked so that the matrix for g is diagonal

(which is possible by

).

gl O}

g =

g2

0 gn

Then, using the lemma above: 1 + TrCg)T + Tr~ ~ 2 g ) T 2 +

=

Det II + gT)

~+gl T Detl

: \ Thus Tr~ i g )

~

l+g2T

~

(l+giT)

.th is the i elementary symmetric function of the gi s.

We now compute =

Qk I Tr ~g,

=

gl

k

Tr /~2g ..... Tr ~ k g )

k + g2

+

+ gn

k

(by definition of Qk )

= Tr(q k)

= ?~v(g k)

= VkNv~g) Hence the map R(G) ---9CF(G) preserves the

~ -rinq structure°

86 N e x t we show that will

have

two o t h e r

the set of

the m a p p r e s e r v e s

consequences:

irreducible

characters

F i r s t we n e e d a fact: 1

S

I GI Proof:

Then

=

lq

set

and

in CF(G).

V of G,

linear m a p d e f i n e d b y

g~G

gJ

Hence

so v G •

is equal

Image~J) d

Image{J) ~

-

to J for all g 6 G.

It is a s i m p l e

and e v e r y gd G acts

trivially

V G.

But also,

H e n c e V G = Image(J)

~ Tr (g) geG N o w take any two r e p r e s e n t a t i o n s d i m V G = Tr(J)

is an o r t h o n o r m a l

This

dim V G

that J is i d e m p o t e n t

image of J.

m a p on V G,

the m a p m u s t be one-one,

any r e p r e s e n t a t i o n

=

be the

the c o m p o s i t i o n

consequence the

~v(g )

inner product.

g (

Let J : V ~ ) V

J

given

the

1 ~GI

J is the

identity

so b y the lemma,

QED. V and W of G, w i t h

characters~v,

1 gEG 1

~

]IV, (g) 7 < w ( g )

iG% g ~ G

Pl

geG

1

~_

on

~TCgw(g

)

(GI g c G i ~i'/Hom~v, I Gl g C G =

dim C HomG(V,W)

=

~v,w)

wl ~g)

(using

the fact above)

~W"

87 Finally, we show that the set of irreducible characters span CFIG).

Let f~-CFIG) be any central function.

an irreducible representation of degree n.

Let

~:G~)Aut

Define ~f =

~

V be f(g) ~ (g),

gc~G a map from V to V. identity map, where

Then

I

~ f = ~I, a constant times the

k--G/V,

g

-i

UgCV

some hEG. has

g-lh-ivhg=V,

for

some

gEG.

The

Hence

the

same

(finite:)

so g

-1U g = V .

number

I

109

Corollary: subgroups

Rank(B(G))

= k = the n u m b e r of c o n j u g a c y c l a s s e s

of G.

As in the

first s e c t i o n

of this chapter,

representation

of G, we can a s s o c i a t e

G.

a homomorphism

This gives

but

for G = S 3, t h e r e

Consider

is not u s u a l l y o n e - o n e .

of e l e m e n t s .

Hence,

as groups,

= ~ 3, and so the m a p h e r e c a n ' t b e o n e - o n e .

representation

of G,

the c h a r a c t e r

of the

linear representation.

Proposition: associated

> R(G).

the c o m p o s i t e m a p B(G)~--> R(G)----~>CF(G), a s s i g n i n g

to each p e r m u t a t i o n associated

B(G)

of

are four c o n j u g a c y c l a s s e s of s u b g r o u p s

only three conjugacy classes

B ( S 3) = Z~ 4 b u t R(S3)

to e a c h p e r m u t a t i o n

a linear representation

of k - r i n g s

It s h o u l d b e n o t e d that this m a p E.g.,

of

Let S b e

a G-set

and X the c h a r a c t e r

linear representation.

n u m b e r of p o i n t s integer-valued

of S left

central

Then

fixed b y g.

function

for gEG,

of the

x(g)

is the

In p a r t i c u l a r ,

on G, t a k i n g

X is an

only nonnegative

values. Proof:

Immediate

from the c o n s t r u c t i o n

of the c h a r a c t e r

representation, Matrices

matri~s

l

s u c h as o c c u r

the o b v i o u s b a s i s

in such r e p r e s e n t a t i o n s

of the v e c t o r

space

e a c h r o w and c o l u m n c o n t a i n

that entry being

of a

"i"

involved)

(choosing

are p e r m u t a t i o n

o n l y one n o n - z e r o

entry,

110

Hence the map B(G) valued characters

~ R(G)

of R(G),

has image at most the integer-

and so isn't onto in general.

it isn't even onto the set of integer-valued example,

the group discussed

So far, permutation

The first new ingredient the composite characters

B(G)~

of the theory

is the analog

R(G)~>CF(G)

do not suffice

characters

(see,

for

on p. 103).

all of this is a straightforward representations

In fact

generaliTation

for linear representations.

of character

is not one-one,

to distinguish

to

unequal

theory.

Since

usual

elements

of B(G).

So the notion must be extended.

Definition: ~:

A super central

(conjugacy

classes

The set of super central ~I([U~)+~2([U~),

Given

a G-set

In shorthand

on G

of subgroups functions

is a map of sets

of G)

is a ring

(~1~2) ([U~)=~I([U])~2([U~)

S, the super character

representation H of G, ~s(H)

function

is the function

notation,

~s(H)

= I HS I

(by (~i+~2) ([U]) = denoted

SCF(G).

of this permutation

%~S' given by,

= the number of elements

~

for each subgroup

Of ~sES I h(s)=s,

all h£H~.

111

Note that ~SI+S2(U) Hence the map B ( G ) P

= ~S I(U)+~S2(U ) and ~SIS2(U)=~S l(U)~S2 (U).

CSF(G)

is a ring homomorphism.

One interpretation of ~s(U) Hom

G

(G/U,S).

~S (T) =

The notation generalizes

I HomG(T,S)

i

all UCG.

Two G-sets S and T are isomorphic

associated super characters Proof:

to apply to any G-set T,

Of course it is easy to compute ~s(T),

given the values of ~s(U),

Theorem:

is the number of elements in

One direction

if and only if the

~S and ~T are equal.

is of course trivial.

To prove the other,

it

is convenient to set up first a partial ordering on the set of simple G-sets, G/U < G/V

We write U


Aut V,

homomorphism)

(over C:)

our a p p r o a c h

theory.

There,

and

one

t h a t it is s e m i - s i m p l e : is a s u b m o d u l e

Theorem,

the M a s c h k e

a special case

is a sum of m a t r i x

t h e o r y of a sum of m a t r i x

this produces

the m a i n

facts

of G. of view,

a representation

(group h o m o m o r p h i s m )

is the same as a r e p r e s e n t a t i o n

of C[GT,

a v o i d e d this a p p r o a c h f o r m i n g R(G),

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

of course,

is that any s e m i s i m p l e ~ - a l g e b r a

all g.

V. h a v e the same l

and s u b m o d u l e N I, t h e r e

T h e n one i n v o k e s W e d d e r b u r n ' s

algebras. algebras

and ~ E n d

p is a l s o onto,

For ~ [G7 this

~-c g = 0, g

i m p l i e s t h a t c =0, g

a p p r o a c h to r e p r e s e n t a t i o n

for any ~ [ G ~ - m o d u l e M,

Then

the c o n n e c t i o n b e t w e e n

s t a r t s w i t h the a l g e b r a C[G~

Theorem.

T a k e y = le.

~[G~---~> End Vo

The r e a s o n

(aside from O c c ~ ' s

razor)

w e w a n t to t a k e t e n s o r p r o d u c t s

(~ - a l g e b r a t h a t we h a v e is t h a t

in

and e x t e r i o r

p o w e r s or r e p r e s e n t a t i o n s . Consider f,g,h:V ~gV

a vector inducing

f(vl)^...^f(Vn)

s p a c e V and l i n e a r t r a n s f o r m a t i o n s fn ,g n ,h n : Anv---~ Anv

and s i m i l a r l y

n for g , hn).

(by fn (v 1 ^...^Vn) Suppose

f = gh,

=

the

117

composite A

n

is not

of maps.

Then

"additive".

fn = g n h n .

Even

if f=g+h,

it can still h a p p e n

that

shows).

of groups,

groups

Thus

a map

doesn't

induce

the e x t e r i o r

Given

representations

there

is no n a t u r a l Another

[GI~ = C[G2~. the t h e o r e m

their

For example,

dihedral

4-group

b y the t h e o r e m groups,

their g r o u p

and the g r o u p

algebras

as k-rings,

they d i f f e r

is that

irreps G

to ~

group

of

? End V

a pair

(E.g.,

groups

algebras this

are

tables

though

is true

Again

of these

to

R(D4)=R(H) .

out on p. 95).

for

Or take the

u n i t s H.

isomorphic

In fact as rings

(as p o i n t e d

is, u s i n g

two a b e l i a n

of o r d e r 4.)

character

are each

C[G3

Hence group

of

algebras

of a b e l i a n

of q u a t e r n i o n

and the k n o w n

C G ~ G ~ ~> ~ • E n d ( ~ 2) .

~

isomorphic

rings m a y differ.

above,

a map

End(V~W).

isomorphic

and the c y c l i c

g r o u p D4,

induces

if G is abelian,

isomorphic

representation

any e x a m p l e

~ End Anv.

~I~

fact that

of the same order have

the K l e i n

map

of the p r o b l e m

above and the

all v£V,

are not the only p r o b l e m .

GI,G 2 may yield

all o n e - d i m e n s i o n a l , groups

~[

But

End V, ~_~--> End W of a C - a l g e b r a

C-algebra

groups

V,

of ~ - a l g e b r a s ,

powers

~i ~ >

indication

nonisomorphic

a map

f(v)=g(v)+h(v),

(as almost

G--TAut

a m a p of C - a l g e b r a s ,

Of c o u r s e

i.e.,

fn ~ gn + h n

G - - 7 Aut Anv, b u t

A n is a functor.

i.e.,

But

118

Let G b e

a group.

to a sum of m a t r i x

elements

By the t h e o r e m

rings,

e. 6 ~ [G],

above,

is i s o m o r p h i c

-[-7- End V i, we can ask for t h o s e i6IrrepG

i6IrrepG,

satisfying

e.=l on V.,

1

and v., ]

C[G]

1

j~i.

These

elements

are u n i q u e l y

0 on

1

characterized

by

the p r o p e r t i e s i)

e i y = y.e.

all 76(12 [G],

all i

(centrality)

1

ii)

e.-e. = 0 l 3

i#j

(orthogonality)

e.. e. = e~ 1

ill)

Theorem:

1

(idempotence)

1

~e. = i i i

Let X

i

(completeness,

.th • b e the i irrep of G.

positivity)

Then

i e.

=

1

Proof:

deg X

J

/Gi

ki(g),g

g£G

Let Y. E ~ [G] d e n o t e l

the e l e m e n t

For each G-module, p:G.--> Aut V,

Y.

g i v e n b y the a b o v e

acts on V by,

for v£V,

l

i Yi(v)

=

deg X

~

k l(g)*

pg(V)

IGI The t r a c e of this e n d o m o r p h i s m deg X T r y (Yi) = IG 1

can be c a l c u l a t e d

i

= degree

~_" xl(g) * Tr(Qg)

X

i

(kl,P)

as

formula.

119 Furthermore

the map Y1. : V ~ >

deg X

PhYi (v) =

V commutes

i

"

i GI

- ~ - x l (g) *Oh gEG

with

Ph for each h6G:

(pg(V))

i deg X

~

IGI

k I (h-lq) *p

q~G

(v)

(here taking

q = hg)

(here taking

q = gh)

q

and i YiPh(V)

deg X IGl

=

~ x!(g).p (Ph(V)) g gEG i

deg;Gl ~

~

x i ( q h - l ) p q (v)

q6G Since

X

i

is a c e n t r a l

H~nce, is a scalar

by

function,

Schur's

multiple

Lemma,

of the

these

two sums

are equal.

if V is irreducible,

identity:

Y. = rI.

Y~:V----~V 1

Applying

the

1

calculation

of the trace,

i V = X , r=l and Y.

Tr(Yi)

is the

' = r-deg V = deg x i °(XX,V). i If V ~ k , r=0.

identity.

Hence

If Y.

1

is the p r o j e c t i o n V of G, p r o j e c t s this

is what

operator

constructed

which a p p l i e d

onto y.i (V)cV,

Y., 1

operators them

t o any r e p r e s e n t a t i o n

i

the X - i s o t y p i c a l

the e. 's o b v i o u s l y l

The o p e r a t o r s Symmetrizing

1

do,

for i£Irrep

the two m u s t be

G,

are c a l l e d

(after the Rev.

in the algebras

component.

C[Sn]

Alfred - see

Since

identical.

Young

Y o u n g who

~39%~7J

first

120 One a p p l i c a t i o n of the

symmetrizer,

of these Y0'

operators

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

symmetric

group

S . n

symmetric

power

of V has two e q u i v a l e n t

i)

Symmnv

Let V be

The n a t u r a l subspace

identification

of V ®n,

Isomorphism

Symmnv

Theorem

we have domain(Y0)

Image(Yo) = V ®n,

We next p r o v e later.

Given

a Young

for the a s s o c i a t e d

Lemma:

Let Y b e

G-modules the sum, Proof:

as l i n e a r

This

is u n i q u e l y

constructed

operator

is a simple

corollary

a sum of i s o t y p i c a l

th

exterior

out of an irrep of

w h i c h will be u s e f u l Y,

lets w r i t e

of Y : V

~V,

and

Y(V) operator

transformation

transformation,of

the n

and

of V ®n.

proposition

symmetrizing

as

By the F i r s t

Similarly

o n t o map V ~ >

V , W the l i n e a r

fixed b y

~6Sn]

= Domain(Yo)/Kernal(Yo)

g i v e n b y the action

a Young

left

from the fact that,

image of Y0"

symmetrizing

YV for the e n d o m o r p h i s m

the n - f o l d

factors).

[ (x-~x) I x6V ®n,

and q u o t i e n t

a technical

Then

of V ®n b y W, w h e r e W is the

Ker Y0=W.

as s u b s p a c e

of a

(n factors)

of these comes

is the

case

definitions:

(permuting

n

and any other type of p o w e r

S , appears both n

YV'

W =

space.

of V ® . . . ® V

of S

S y m m n v = the q u o t i e n t subspace

power,

any v e c t o r

= the s u b s p a c e the action

ii)

is in the s i m p l e s t

of G.

Given

YV~w:V@W----~V@W

Yv:V

-~V and Yw:W.

is ~ W

.

of the fact that e v e r y G - m o d u l e

components.

721

Lemma:

Given

a G-map T:V~>

linear transformation

T'

W of G - m o d u l e s ,

so t h a t the

V

there

is a u n i q u e

following commutes:

~W

YW

T'

Y(V)

Proposition: natural

.....~> Y(W)

Let HT-G and V b e an H - m o d u l e .

Then there

is a

isomorphism

(Ind~ V) G

vH--~

Proof:

R e c a l l V H d e n o t e s the e l e m e n t s

h6H,

thus the i m a g e of the Y o u n g

unit

representation

Consider

operator

Similarly

1 H of H.

of V left fixed b y

each

Y associated with

the

for the s u p e r s c r i p t G.

the c o m p o s i t e m a p VH

where

m

the

the second

2 V

~

first m a p

Y

Ind~ V

is i n c l u s i o n

is V--~ C[G]

®

V,

. dGv .G ~> (In H )

of an i s o t y p i c a l

v ~

l®v.

s u b s p a c e and

This composite map

is an

[H] isomorphism:

Indeed,

b y the f i r s t

lemma,

the f o r m a t i o n of i n d u c e d r e p r e s e n t a t i o n s , w e can r e s t r i c t

our a t t e n t i o n

V is n o t the u n i t (Ind V,

IG) I G =

it is a d d i t i v e

)i G =

of H, V H = 0 and also

of

in V,

to the c a s e of i r r e d u c i b l e V.

representation

(V,Res~l G~

and the a d d i t i v i t y

so

If

(Ind V) G =

(V, IH)! G = 0 1 G = 0 and any m a p

122 f r o m 0 to 0 is an i s o m D r ~ h i s m . (by F r o b e n i

reciprocity)

so the c o m p o s i t e

is a m a p

s h o w t h a t the m a p

In c a s e V = 1H, V

(IndV) G =

1 H - - - ~ l G.

isn't zero.

(i G + Hence

This

= V,

and

(other)) G = 1 G and it is s u f f i c i e n t to

follows easily by unravelling

the d e f i n i t i o n s

of i n d u c e d r e p r e s e n t a t i o n

this case.

final d e t a i l s

The

H

and Y o u n g o p e r a t o r

are left to the r e a d e r

in

as an

a exercise.

Here of g r o u p

is a final h i s t o r i c a l note. representation

algebras,

but with

one c o m p u t e s

theory were

done,

the s o c a l l e d g r o u p

its g r o u p d e t e r m i n a n t ,

variables

X , g£G. g

consider

the m a t r i x

representation

The o r i g i n a l

of G

® is the d e t e r m i n a n t

Given

the

assigned

®,

"generic

not w i t h

determinant. a polynomial element

this element

~ g£G

f r o m our p o i n t of view,

to the i r r e d u c i b l e

factor occuring

the n u m b e r of t i m e s

conjecture

t h a t an o r i g i n a l

a g r o u p G,

function

Xgg

in

in C[G~"

of t h a t space).

Looking

w e see t h a t the i r r e d u c i b l e

of ® c o r r e s p o n d

interesting

Given

The p r o b l e m w a s to r e l a t e

the s t r u c t u r e of G to the p r o b l e m of f a c t o r i n g ®. this

semisimple

in the r e g u l a r

(in t h e o b v i o u s c h o i c e of b a s i s of t h i s m a t r i x .

investigations

representations as its degree.

of G,

at

factors each

It is

i m p e t u s to the t h e o r y was D e d e k i n d ' s

( c o m m u n i c a t e d to F r o b e n i u s

in 1896)

that the n u m b e r of

123

linear

factors

commutator

of ® is equal

subgroup

For a h i s t o r y see H a w k i n s

([22]).

(a s i m p l e

to the index fact

in G of its

for us - see p.92).

of the early w o r k

in r e p r e s e n t a t i o n

theory

CHAPTER

III

The Fundamental

i.

of the

The

of k-rings certain rings

of all

Before

n letters,

the

say t h e

such that

asserts

X-ring

symmetric

o is a c y c l e

of t h e

the

constructed

about

combining

this

object,

symmetric

letters

o =

group

if t h e r e

- indeed

under

breaks

a number

of o r b i t s ,

=

structure

(k I ..... k n) w h e r e

different

structures:

=

(i,i,i)

=

(2,1)

~ =

(3)

and a

the

example,

corresponds corresponds

corresponds

Let

~ be

to the

o(m)=m, that

associated

(23) (i),

to the

elements

set

{I ..... n]

is a c y c l e .

e = o =

~ =

~ES n

o=(il,.,iq) (ji,.,jr)..

of S 3 h a v e

and

one

partition

element

OT =

{I .... n}

element

of the cycles

elements

of S n-

for m n o t

every

of w h i c h

the elements

to the

an e l e m e n t

of o, t h e

sizes

of

{i I .... i q ] C

cycles:

action each

to recall

of p e r m u t a t i o n s

n

of d i s j o i n t

t h e kl are t h e

For

S

Observe

of o is the

decomposition. cycle

I,

it is n e c e s s a r y

is a s u b s e t

" .. .,lq). (ll,

..(k I ..... k s )

The cycle

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

all t h e r e p r e s e n t a t i o n

..... O ( i q ) = i I a n d

as a p r o d u c t

into

there

in C h a p t e r

1 , 2 , 3 ..... n.

can be written

up

Theory

groups.

of length ~

Write

that

A discussed

by

~(ii)=i2,~(i2)=i3

ij .

THEOREM

Group

we construct

a few facts

FUNDAMENTAL

of t h e R e p r e s e n t a t i o n

in q u e s t i o n

between

k-ring

THE

Theorem

Symmetric

theorem

:

~

in t h e

one of three

(i) (2)(3)

(12) (3), 2

(123)

=

(13) (2) and

T 2=

(132)

125

Proposition: h a v e the

Proof:

Two elements

same c y c l e

Suppose

as a p r o d u c t

of S

are c o n j u g a t e

n

structure.

~ and J' h a v e the

of d i s j o i n t

a b o v e the other,

if and o n l y if t h e y

cycles

so t h a t c y c l e s

same c y c l e

in the

structure.

Write

following pattern:

of equal

each

one

length correspond:

(i I ..... iq) (Jl ..... Jr ) "'" (kl ..... ks)

(ll,

Let

T 6S

n

...

')...(k ,i ) (3' 1 ..... 3r

b e the c o r r e s p o n d e n c e

I ....

g i v e n b y the v e r t i c a l

T (il)-' --±i' '. .--' T (jl)-'' --31, ...etc. ' . . T. (lq)--l~ ~' are c o n j u g a t e .

Corollary:

Then

The c o n v e r s e p r o p o s i t i o n

The n u m b e r

of c o n j u g a c y

the n u m b e r of i r r e d u c i b l e the n u m b e r of p a r t i t i o n s

classes

representations

for the c o n j u g a c y c l a s s of S

-i

~ T=u so o and

|

is clear.

of S n,

and h e n c e

also

of S , is e q u a l to n

a

n

[~

, for ~ a p a r t i t i o n

of e l e m e n t s

The o t h e r c a l c u l a t i o n we w i l l n e e d (which n u m b e r w i l l

T

lines:

of n°

H e n c e f o r t h we a d o p t the n o t a t i o n

of [ ~

'k 's)

of

cycle

s t r u c t u r e ~.

is the n u m b e r

also be denoted by

[~.).

of n,

of e l e m e n t s

126

Proposition:

Suppose ~ =

(i~2 ~ . . . n ¥)

is a p a r t i t i o n

of n.

Then

n' t

Proof:

Given

the c y c l e

(-) (-) ... (-) ( - , - )

s t r u c t u r e ~:

(-,-)

let us c o u n t the n u m b e r n: w a y s to w r i t e one-cycles

can b e

rearranged

arise.

.........

to fill

(-,-,

. . . . -)

in the b l a n k s .

1,2 ..... n.

However

in ~: w a y s g i v i n g the

d i v i d e b y ~:

the ~: ..... y: t e r m s

have

of w a y s

in the n u m b e r s

of S , so we m u s t n

l e n g t h q,

... ( - , - )

to r e m o v e

Furthermore

are

the f i r s t same e l e m e n t

the d u p l i c a t i o n .

Similarly

in e a c h of the c y c l e s of

any one of the q e l e m e n t s can be l i s t e d

to d i v i d e b y q for e a c h c y c l e

There

of l e n g t h q.

first

- so we

H e n c e the

f a c t o r l ~ 2 ~ . . . n Y.

We now construct n=0,1,2,.., recall,

let R(Sn)

R(S0)=~).

X-structure

on e a c h R(Sn)

as follows. s u b g r o u p of S m

First n+m

of 1,2 ..... m.

is a p a i r i n g

observe that n

ring of Sn

(where,

f o r g e t the m u l t i p l i c a t i o n

and c o n s i d e r

by taking S

as p e r m u t a t i o n s

For e a c h i n t e g e r

b e the r e p r e s e n t a t i o n

F o r the while,

The o u t e r p r o d u c t

S

the k - r i n g R(S).

it as just an a b e l i a n group.

R(Sn)XR(Sm ) ~

R(Sn+m)

S ~ S can b e c o n s i d e r e d n m

to b e p e r m u t a t i o n s

of n + l , n + 2 ..... m,

and

and S

n+m

defined

as a

of 1,2 ..... n,

as p e r m u t a t i o n s

In d i v i d i n g up the n + m s y m b o l s p e r m u t e d b y S

n+m

127

into

one

set of n a n d a n o t h e r

the division

can be

of c o n s t r u c t i n g of S

done

set of m,

there

in ~ n ~ m ) w a y s , b u t

an i n j e c t i o n

S ~ n

S

m

is s o m e

any two

--> S

n+m

ambiguity

such

-

ways

give c o n j u g a t e subgroups

n+m Given

~n ~ ~ m

now

elements

£ R(Sn~

Sm)

~ n 6 R ( S n ), ~ m £ R ( S m ) , c o n s i d e r

(see d e f i n i t i o n ,

p.

).

the

The

element

outer

product

~n ~m o f ~n and ~m i s t h e e l e m e n t o f R(Sn+ m) g i v e n b y

~n~m (Note t h a t

S n+m = Ind S ~ S n m

since

any two of our ways

Sn+ m are conjugate, the way

chosen.

is j u s t

(n,~)~?

In t e r m s described and ~

m

:S

X

21

,X

41

F o r n=0, ~+~+..+~

32

,X

221,~nda (using

R(Sn)m

(n s u m m a n d s ) ,

n

~ ~

m

two

S < S n m

into

~ n ~ m £ R ( S n + m ) is i n d e p e n d e n t

the map

Given

R(Sm)---~

and ditto

R ( S n + m)

for m = 0 .

the product

representations

:S ~ S - - 7 A u t ( V ~ g W ) n m

of

~

n

can be :S--~ n

Aut(V)

and

~_~ (V~gW)) . -S U S n m

the X

of i m b e d d i n g

representations,

C[Sn+m]

an e x e r c l s e , ,X

element

of a c t u a l

~ Aut

respectively then

)

-----2A u t ( W ) ,

~ :S • n m n+m

As

the

as f o l l o w s .

m

(~n ~ ~ m )

311

reader denote

the notation

is i n v i t e d characters

to show that

2

•

~,~ of S 2 , S 3 , S 5 , S 5 , S 5 , a n u

of t h e c h a r a c t e r

2 21 41 32 221 311_~II~ X X = X + X + X + X

if k

tables

S5

128

Definition: R(S) =

- the sum as abelian for all n , m ~ 0 ,

Proposition:

Proof:

~=0

groups.

R(Sn )

The outer p r o d u c t s , R ( S n ) ~ R ( S m ) ~ >

induce a m u l t i p l i c a t i o n

R(S)

axioms

commutativity

for multiplication,

Commutativity

follows

follows

~

S

n+m

ring w i t h identity.

are a s s o c i a t i v i t y

and

and the d i s t r i b u t i v e

law.

from the fact that R ( S n ~ S m) is isomorphic

to R ( S m X Sn ) in the obvious way, S ~ S n m

on R(S).

is a graded c o m m u t a t i v e

The only n o n - o b v i o u s

R~Sn+m ~

and the two subgroups

and S ~ S c- S are conjugate. m n n+m

from the fact that the operation

Distributivity

of inducing

representations

is additive. To show associativity,

it is enough to show, given elements

~ n £ R ( S n ), ~ m £ R ( S m ), and ~p£R(Sp) S - d n+m+p in Sn~ Smx S

= (~n~m) ~p

We will

just p r o v e

Writing

this p r o p o s e d

S I d n+m+p n S xS n+m p

that

p

the left-hand

(~n~ ~m ~ ~p)

equality,

=

~n (~m~p)

the other b e i n g

equality out in more detail,

S _ n+m ( Inds x S n m

(~n ~ m

) x ~p)

=

similar.

we get

S Inds~s~sn+m+P ( ~ n m p

~m~p)

129

Let o6S (p. ~

n+m+p

and apply the formula for induced characters

) to both sides. (n+m+p):

i

n: m: p:

[o]

The right hand side yields

~i×~2~3 ~ S~Sm~S p



space.

WimV~V~q..

~)V

...C~'~Vn)=O(w)C~v o

Let W(V)

be the s u b s p a c e W(V)

=

(WdgV C~n)

F o r example,

is the n - f o l d

alternating

representation W(V)

The c o n s t r u c t i o n Given

a m a p T:V I - ~

iwd~ T n : W d ~ V ~1 n action

The g r o u p

symmetric of S

n

W(V),

-7 W 4 ~ V 2~ ] n

acts

of V) by,

for

fixed u n d e r

one-dimensional

power

of V.

(the n o n t r i v i a l

this

representation

If W is the one-dimensional

power.

for fixed W,

This m a p

is f u n c t o r i a l

[W~VI6~n ) Sn

W(T):

W(VI) ~ W ( V 2 ) .

in V.

V2C~] n and h e n c e

is c o m p a t i b l e

: (W~JV 1~] n.) Sn~ 2

as we m i g h t write,

then

w i t h the

a map

IW ~9 TCW n

or,

n

n

V 2, w e h a v e T C ~ n : V l ~ n ~

of S , so g i v e s n

S

be

n

-i GO v -i ~j ...C~)v (i) o (2) ~-l(n)

is the e x t e r i o r

V ~

in the

Aut W of S

(n c o p i e s

if W is the t r i v i a l

of S , W(V) n

representation),

S ~> n

of ZU_~V ~-gn of v e c t o r s S

action:

A involved

(W~V2~n)

Sn

131

S u p p o s e that V 1 = V 2, and t h a t the t r a n s f o r m a t i o n diagonalizable

and has

eigenvalues

t h a t T r a c e ( T ) = t l + . . . + t k. monomials

t I ..... t k,

The e i g e n v a l u e s

in t I .... ,tk and s i m i l a r l y

(k=dim V)

T is so

of T ~gn are n t h - d e g r e e

for i w ~ f ~ / n .

Recall

the

elementary Lemma:

Let S:U----)U b e

a linear transformation.

subspace UI~ U satisfies the r e s t r i c t i o n a c t i n g on U. Hence

of d e g r e e n, w i t h

following

of W(T)

are m o n o m i a l s

reason:

to a c h a n g e

be unaffected by Finally,

of b a s i s

o f W(T)

of the s y m b o l s

of V.

But

Then,

iI ik tI ... t k has d e g r e e n and so c a n n o t t h a n n of the v a r i a b l e s a monomial would

t I ..... tk.

o c c u r in W(T)

of p o w e r s

of the

symmetric

a l = t i + . . + t k ..... a k = t l . . t k is i n d e p e n d e n t s u p p o s e k > n.

the a n s w e r w o u l d

change.

in t e r m s of e l e m e n t a r y

Indeed,

t I ..... t k

since the g i v e n d a t a

are c o o r d i n a t e - f r e e ,

such a c o o r d i n a t e

This

t I ..... t k for the

we c l a i m t h a t the e x p r e s s i o n

f u n c t i o n s W(T)

same p a t t e r n

of S

of d e g r e e n in

in t I ..... tko

in the v a r i a b l e s

any p e r m u t a t i o n

and the c a l c u l a t i o n

as k ~n.

of

is a h o m o g e n e o u s p o l y n o m i a l

integer coefficients,

is s y m m e t r i c

corresponds

the e i g e n v a l u e s

B

and so the t r a c e of W(T)

polynomial

Then

some

of S to U 1 are a s u b s e t of the e i g e n v a l u e s

the e i g e n v a l u e s

t I ..... ~ ,

S(U I) c~ U I.

Suppose

symmetric

functions

of k = d i m V,

still,

as long

any m o n o m i a l

involve nontrivially more

S i n c e W(T)

is s y m m e t r i c ,

such

if and o n l y if a m o n o m i a l w i t h

i I ..... ik o c c u r s

in W(T)

involving

just

the

132

the variables t I ..... tn. of monomial

symmetric

Hence the expression of W(T)

functions

is independent of k ~ n, and so

likewise for the elementary symmetric Thus,

function ®(W)

Lemma:

functions.

after all this work, we have a map:

representation

in terms

given a

S --9 w in R(S ), there is an associated n n

symmetric

£ A . n

®(Wl(*~W2) = ®(WI)

@(W2)

*

Proof: (Wl~9 W 2) Since S

n

=

(

(Wl~

in) ~

)

(W2~) V ~n)

for any V.

acts independently on the two factors

(WI~ v~n) S n ~ Hence,

v

( W 2 ~ V ~ n ) Sn

=

(

(WlfmvC~n) ~

(W2~Vd~n)) S n

for a linear transformation T,

Tr(WI+~JW2(T)

)

=

Tr(WI(T)(~gW2(T))

and this latter is

equal to Tr(WI(T))+Tr(W2(T ))

Corollary:

Lemma:

I

I

® gives a well-defined map R(Sn) --> A n .

® is multiplicative.

I.e.,

product of &n£R(Sn ) and ~m£R(Sm), product of symmetric

functions

if ~n+m E R(Sn+m)

is the outer

then ®(~n+m)=®(~n)G(~m ), where the

is taken in A = ~ A

n

.

133

Proof:

We can assume ~

~n:Sn ~

and ~

n

are actual

m

representations:

Aut W n, ~m:Sm---~ Aut Wm, ~ n + m : S n + m - o

V be any vector

Aut Wn+m.

Let

space.

W n + m ~ V ~w'n+m

=

= Ind

Ind

(Wn~ W m ) ~

~h+m

S ~ S

•

n

(Wn~WmC~)(ReSs

m

V ~n+m)

(by Froben ius

)

n+m C~n ~

Reciprocity)

v@Qm

= Ind

( Wn~Wm~)V

)

= Ind

((WnC~V CWn) ~_P(Wm~)v~)m))

Hence S v~m) i n+m

S (Wn+mC~ v~n+m)

n+m

=

v~n)

:

This

isomorphism

S

Wn+m(V)

if T is any linear it ind~ces

ind((Wn6~V~n

v~m)

Ca (Wm

= Wn(V)~_JWm(V )

operator

an equality

of a tensor p r o d u c t

n

) ~(Wm~

with

Wn+m(T)

S

m

is functorial

eigenvalues = Wn(T)~Wm(T)

is the p r o d u c t

t I ..... tq, q "

give a map of rings ® : R ( S ) ~ > A .

so

n+m,

Since the trace

of the traces,

this gives

I

®(Wn+ m) = ®(W n)®(W m) .

Hence the maps ® : R ( S n ) ~ >

in V,

An, n=0,1,2 .....

can be added up to

134 Lemma: Proof:

®:R(Sn)--->An One b a s i s

is onto,

for A

is

for each n~0.

{h

I ~ ~n].

Such an h

n S n is ~ ( I n d s k ~ s k ~

..~Sk

~

i) = ® ( t h e

outer product

= h k l h k ..h k 2 n

of the unit

n representations

Corollary: Proof:

®:R(S

of Skl ..... S kn )"

n

)~:)A

is o n e - o n e

n

An onto h o m o m o r p h i s m

the same

finite

rank m u s t

An i m m e d i a t e

corollary

(it is c o n v e n i e n t

between

two free a b e l i a n

of the

Indeed,

are c l e a r l y

integer-valued,

of

m

fact that

of the g r o u p s

functions.

groups

also be one-one.

for each n, R(Sn)

to think of ® as the i d e n t i t y map),

fact that the c h a r a c t e r s

basis

for all n~0.

the characters

S

-i

®

n

is the

are all i n t e g e r - v a l u e d S n

(hkl ""

and they give

hkn) = IndSkl''Sknl

an integral

for R(Sn).

Thus we h a v e rings.

Since

that ® : R ( S ) r - - ~ A

A is also a k-ring,

a corresponding additional

shown

k-structure

structure,

we have

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

the i s o m o r p h i s m

on R(S),

and t a k i n g

the m a i n

= A

result:

of

® induces

R(S)

with

this

n

135 Theorem:

The Fundamental

of the Symmetric

Theorem

Group).

of the Representation

The map ® : R ( S ) ~ > A

Theory

is an isomorphism

of k-rings.

The induced integers

k,n~l

k-structure

and ~6R(Sk),

these operations

plethysm).

applied

k-rings,

R(Sn),

on R(S),

to a representation the induced

guess

&:S k

2V

to

Classically

were referred

from the k-structure

n~0, which

is to describe

It is a reasonable

constructing

for example,

an element hn(~)E(Rnk).

(to be distinguished

One problem

explicitly.

assigns,

on A, so b y extension,

to as outer p l e t h y s m of the individual

on R(S)

is called

inner

the outer plethysm that the operation is performed

representation

h

n

by first

of the wreath product

S n [ S k ] ~ ? V ®n, and, using the n a t u r a l i n c l u s i o n Sn[Sk]CSnk, inducing verify

to a representation

this explicitly.

reasonable

algorithms

Few calculations

The point pass

Another

have b e e n made

and forth,

ring R(Sn)

on one hand.

and the group

(=the group

of k-operations

But we have been unable

outstanding

for computing

of the fundamental

freely b a c k

representation

of Snk"

outer

problem

(or inner)

is to find plethysm.

( see Littlewood ~ 2 7 ~ 8 ~

theorem

is that

of symmetric

functions

"of weight n")

).

it allows us to

for each integer n, between (=the ring of characters

to

the

of R(Sn)) of weight n

on the other.

On

136

each

s i d e of t h e

one c a n

do,

involved, relate this

equality

a number

and

these.

chapter.

some

there

of o b v i o u s

"canonical"

This project

are

a number

bases

for the

elements,

is c a r r i e d

of c a l c u l a t i o n s abelian

group

and the g a m e

is to

out

in t h e

rest

of

137 2.

Complements

Let n ~ l

and C o r o l l a r i e s

be

the i s o m o r p h i s m Given

a basis

a fixed i n t e g e r of a b e l i a n

[h

I ~n}

groups

for the g r o u p

for A , and v i c e - v e r s a . n are a b a s i s

and let ®:R(Sn)----~A n be given

R(Sn),

the s y m m e t r i c

and we h a v e

n

Theorem.

its image u n d e r ® is a b a s i s

F o r example, of A

in the F u n d a m e n t a l

functions

already noted

that,

if

is the p a r t i t i o n ®-l(h

where

~=(k I ..... kn) of n, S ) = Ind n S k 7< .. x S k 1 1 n

1 is the p r o d u c t

1 ~ ... ~ l

representations

of Skl .... Skn"

Littlewood

) this

~29~

it is n e c e s s a r y

associated identify

representation

h

is the

representation

R(Sn)

with

A

n

functions,

[a I ~ ~ n ] ,

S

(aT)

=

Inds

w h e r e ~=(k I ..... kn) , and representations and w r i t e

on the

"a °' not only

But

of A

n x

n

the n o t a t i o n

R(Sn)

by products

• .. × S k

~ .

When adhere

is the

it is e a s i e r

to

for b o t h objects.

of e l e m e n t a r y

) is t h e r e p r e s e n t a t i o n

(alt)

kl

n

(alt)

is the p r o d u c t

groups

and ~

in g e n e r a l

®-l(a

of

and A , we will n

function

v i a ® and use h

-i ®

between

of S . n

one-dimensional

is often d e n o t e d

symmetric

In the c a s e of the b a s i s symmetric

(Following

to d i s t i n g u i s h

to this n o t a t i o n :

of the t r i v i a l

of the a l t e r n a t i n g

Sk. As a b o v e , we w i l l 1 for the s y m m e t r i c f u n c t i o n a

be but

sloppy also

138

for the representation ®-l(a ), again treating ® as an identity map.

So far, only the abelian group structure of R(S n) has appeared. But R(Sn)

is also a k-ring with a dot product.

To avoid confusion

with the outer product defined above R(S n) ~

R(S m) - ~ f,g

R(Sn+m)

I~/ fg

the usual representation-theoretic

R(S n) ~ will b e called, denoted f , g ~ >

n,m~ 0

R(S n) ~ _

product

J~ R(Sn)

in this chapter,

n~ 0

the inner product,

and

f*g.

The scalar product

R(S n) X will be called,

R(S n) ~

~

Z

in this chapter,

n~O the dot product,

and

denoted f,gw%~o)f.g . It is essential to keep these three products

straight,

and

we will adhere rigorously to these terms throughout this chapter.

As we have seen, the Fundamental Theorem corresponds outer p r o d u c t

to

since ®:R(Sn) ~

the

usual

product

of

An is an isomorphism,

symmetric

functions.

the But

we can induce an inner

product and a dot produc t on An from that on R(Sn).__

A bit later

139

we will g i v e e x p l i c i t

Using

the inner product,

en of the group all ~ 6 R ( S n ) , refer

R(Sn),

eniS

9(a )=h

for these.

the e l e m e n t

o.-~an*O.

an involution:

also to the i n d u c e d

the m a p of g r a d e d by

formulas

groups

, all ~

n,

Since

e2=l. n

involution

given

( t lq Y )

a £R(S n) g i v e s n an*an=hn,

We l e t

on A . n

by ~=t~ n.

all n~0

the

Let

symbol e

for

n

0:A - - ~ A be

also b y

characterization is given on p. 181

Another

and h n ~ O = o

e can also

and again,

a map

Note

be

specified

e(h )=a

.

e is not

a ring h o m o m o r p h i s m .

The m a i n Theorem:

result

The e l e m e n t

character

corresponds

Proof:

L

section

£ R(Sn)

is the f o l l o w i n g

defined(as

theorem•

previously)by

the

I:0 u n d e r ® to the p o w e r

F i r s t we

show the t h e o r e m

n into one part: only n e c e s s a r y

~ =

n

Then

to show that,

~

[(n)~

0 is an n - c y c l e in A, a n d t h e

(n).

of L k . . . . 1

outer p r o d u c t

Since

in this

- n,

or not.

immediate

sum f u n c t i o n

s

for the t r i v i a l

partition

of

since ® is m u l t i p l i c a t i v e ,

if ~ =

(k I .... kn),

it is

L~ is the

,L k n

Ln(O) s

n

= n or 0, d e p e n d i n g

is the

object

is

symmetric

to

show t h a t

on w h e t h e r

function

n n ~i+~2 +

®(Ln)=S n.

140

This will be n=l,

Ll=~l,

For n > l ,

accomplished

we use the N e w t o n

the t h e o r e m

the c h a r a c t e r

~£S

is true

associated

to s

®

-i

structure

®-l(srhn_r)

(o) =

Lr*~n_r

[o]

Given

the formula

dl£SrX

Sn_r,

" + Slhn-i

n

on R(S

n

In the case is trivial.

= nh n We will

1 6 r{ n-l.

(l~2$...nY).

(o)

in S

(Lr~ i) (Ol)

n

for induced

characters

(Lr~l) (ql) will be

of p.

zero u n l e s s

). o I is of the

form ~i

=

(

).(

an r-cycle c o n t a i n i n g the n u m b e r s i, ...,r in some order If o I is of this

form,

evaluate

).

= Lr*~n_r,

OlESrXSn_ r o ~ 1

(applying

.

(Srhn_r)

cycle

r

+

for Sl,S 2 .... Sn_ I.

have

n

on n.

formula

+ Sn-2h2

First we e v a l u a t e Let

induction

® ( ~ l ) = h l , and hl=S I, so the statement

s n + Sn-lhl Suppose

by

).(

) .....

(

)

other cycles i n v o l v i n g the numbers from r+l to n

(Lr~< I)(dl)=r.

141

Hence we want to count the ratio

iii[u[ilii ini:Snr ~

r

~ of / 01 in S n which a re ~ conjugate in S n to

(Srhn_r) =

. r . (this ratio)

G~ven that o has cycle structure is [l~2~..r~.]

(i~2 ~ . .r Z. .) , the denominator

(using the notation of p.

).

The numerator of

the ratio is then the number of ways of taking one r-cycle on the letters 1,2 ..... r times the number of permutations (l~2~..r ~-I..) on the remaining letters r+l .... . n. numerator is [r].[l~2 ~...r Z-I..~.

< rrl~. )

(

i

(n-r) '. I~:2~,...rZ-I(z_I)

n

of type

Hence the

The ratio is then

:...

)

i ~ : 2 ~ : . . . r ~ • ...

~ence~,~r~nr>f(u)

is additive

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

The map U i ~ Q H o m G ( U i,U) - ~

(i.e.,

and the i s o m o r p h i s m

in U,

when

and the

U=U.,

some

identity

i).

U

follows

is true w h e n

Indeed,

this

is just

1

a restatement

of Schur's

Let n o w V b e on ~

a vector

by permutation

V

~n

~

lemma.

~

space

of factors.

(W~Hom

~-n But HornS

(W ,V ~ n )

=

and U = V ~i)n.

Then t h i s

S

Let G=S

identity

n

act

becomes

(W , ~ n ) )

n (Hom(W

, ~ n))

Sn =

(wdual S n ) n

S n = wdual (V)

n the last e q u a l i t y associated

being the definition

to a r e p r e s e n t a t i o n

dual

W~

.

of the operation

wdual(

)

153

N o w n o t e t w o things:

W

is an i r r e p

and t h e y b o t h h a v e t h e same d e g r e e . An i r r e l e v a n t

fact is t h a t

if and o n l y if W d u a l

(This is t r u e

is,

for any group.

for S , the i r r e p s W and W n ~

dual

are

isomorphic. ) Hence: V C~n

wdual~(~Dw~(v)

= ~i-n

This

is a f u n c t o r i a l

endomorphism

of V w i t h

t r a c e of b o t h

of b i n o m i a l

in V,

eigenvalues

so r e p l a c i n g V b y

t l , t 2 .... tn,

an

and t a k i n g the

sides we get

(tl+t2+''")n

Schur

isomorphism

=

~ H ~ n

functions

. {~] (t I .... t n)

interpreted

coefficients.

i n t e g e r n and let V b e

as o p e r a t i o n s g i v e

Namely,

a vector

let ~ b e

a generalization

any p a r t i t i o n

of any

s p a c e of d i m e n s i o n m.

Write (~> w h e r e ~'

=

dimension

is the p a r t i t i o n

If ~=(n), usual binomial

(~I

im) n)

, =

conjugate

=

to ~.

ira3 : dim ~in~v~: dim A n v the

coefficient.

The last p r o p o s i t i o n binomial

[~](V)

gives

an i d e n t i t y

coefficients mn

=

S ~n

H~

() m

for t h e s e g e n e r a l i z e d

154

(~1

In terms of k-rings, operations

[~/~ £ A to the e l e m e n t

We c l o s e this

identify group

(i.e.,

we have extends

t h e s e two

product,

a dot product,

b y a ). n

There

basis

{h ,~I-n},

{s , ~ n } .

of p a r t i t i o n s

natural bases

{,~n},

Given

R(Sn) _ ~ > A

object

and an i n v o l u t i o n

are s e v e r a l

(binomial)

isomorphism

The r e s u l t i n g

the n u m b e r

of a p p l y i n g

This

integer matrix combinatorial

if the s theory

for c a l c u l a t i n g of the r e s u l t i n g

of the

numbers.

some of this theory.

For

which We

is a free a b e l i a n an inner

(as free a b e l i a n group):

integer matrix

and a r a t i o n a l (or one b a s i s matrix between

(or l/n:

times

an

and a large p a r t of the

symmetric

these matrices

~.

(inner m u l t i p l y i n g

{ [~}, ~I-n}

are chosen)

n

of n, w i t h

any two of t h e s e b a s e s

is an i n v e r t i b l e

k-ring

R(S)~>A).

and the set of s ) one can ask for the t r a n s i t i o n them.

the

w h a t we n o w have.

an i s o m o r p h i s m to a ring

sets.

of rank ~(n),

{a , ~ n } ,

m in the

section b y r e v i e w i n g

each i n t e g e r n ~0, is n a t u r a l

is the result

group consists

explicitly

In the n e x t

of m e t h o d s

and i n t e r p r e t a t i o n s

two s e c t i o n s w e e x p l o r e

155

Schur Functions

The m a i n functions

and the F r o b e n i u s C h a r a c t e r

object

of this

section

are the i m a g e s u n d e r

representations

the c o m p u t a t i o n

is to p r o v e

t h a t the S c h u r

the i s o m o r p h i s m ® of the i r r e d u c i b l e

of the s y m m e t r i c

m e t h o d of p r o o f g i v e s

Formula

groups.

It t u r n s

out t h a t the

a set of f o r m u l a s w h i c h w i l l b e b a s i c

of the c o m b i n a t o r i a l

t h e o r y of the

in

symmetric

group.

The p r o o f c o m e s

f r o m the

simple observation

in this c o n t e x t b y P h i l i p H a l l g r o u p of f i n i t e must be unique

rank,

with

[20~

) that in a free a b e l i a n

a dot p r o d u c t ,

(if it e x i s t s

at all)

This c o m e s

bases

the t r a n s i t i o n m a t r i x

be orthogonal with matrices which

integer

from the fact t h a t

entries.

are s i g n e d p e r m u t a t i o n

e a c h row and e a c h c o l u m n

entry,

that entry being ~

R(Sn)

= An has

irreducible

an o r t h o n o r m a l b a s i s

- at least,

sign and order. are given,

(pointed out

u n i q u e up to

if two o r t h o n o r m a l

from one to the o t h e r m u s t

The only o r t h o g o n a l

matrices contain

- i.e.,

square matrices

e x a c t l y one n o n - z e r o

i.

a dot p r o d u c t ,

representations.

Hence

and an o r t h o n o r m a l b a s i s , any o r t h o n o r m a l b a s i s

m u s t c o r r e s p o n d up to sign and o r d e r w i t h the irreps. l e m m a is the

following.

integral

the

in A

The k e y

n

in

156

Lemma:

Given

qk'

rk £An

[rk

I k~n]

under

any e x p r e s s i o n

(notation

facts

as in

are b o t h b a s e s

the dot p r o d u c t

The p r o o f about

product.

of this

satisfying, i)

I ~7

there

with

rk(y)

), the sets

[qk I k~n]

to each

and

other

n

lemma

requires

us

first

in free abelian

a free abelian

Thus,

-- ~. qk(x) XPn

of An and are dual

in A

dot p r o d u c t s

Let F be

hn(XY)

group

some

groups•

of finite

is a function

to recall

(-,-)

rank k w i t h

defined

a dot

on F ~ F

for all x,y, z6F

(x,x)

ii)

£

(x,x)~ 0

and e q u a l i t y

iii)

(x,y)

iv)

(x,y+z)

=

(x,y)

+

(x,z)

(x+y,z)

=

(x,z)

+

(y,z)

This gives

a map

:

holds

only

x.u~

(x,-)

if x=0

(y,x)

~ : F - - ~ Hom(F,~),

which

is n e c e s s a r i l y

one-one. Suppose, sense that,

now, if [rk]

det(

(rk,r~))

know

that

R(S

n

is any b a s i s

of F,

=~i.

(This is true

) has

an o r t h o n o r m a l

representations.) the map

that the dot p r o d u c t

An e q u i v a l e n t

8:F~-->Hom(F,Z)

given

is also normal

in the

the k × k d e t e r m i n a n t

in F = R(S n) = An since we basis,

definition

above,

the i r r e d u c i b l e of n o r m a l i t y

is onto

(hence

is that

an isomorphism).

157

Another

equivalent

is a u n i q u e (ri,s9)

statement

dual b a s i s

= 5ij

{sk}

(Kroneckor

Let us p r o v e to a g i v e n b a s i s

is that,

of F , dual

in the

implies

To find the

s k, one m u s t

for i n t e g e r s

equation

,r

by r kI

1

=

=

By Cramer's

all(rkl,r

this

the i n v e r s e map

There

,rk2)

+ a12(rki,rk2)

+

+

set of e q u a t i o n s

this

...

...

is s o l v a b l e

over Z since

in ~ .

n o w in the c a s e of F = A

n

, we w a n t to look at

~-I:Hom(F,Z)----->F.

is a n a t u r a l

for any free a b e l i a n is a n a t u r a l

Dotting

equations

kl

d e t ( ( r k . , r k ))= ~ 1 is i n v e r t i b l e i ]

Specifically

aij.

solve

of a dual

.... we get

) + al2(r

all(rki,rkl)

rule,

that

k2

~i

0

sense

[rl} of F, there

the e x i s t e n c e

like s k = a l l r k l + a 1 2 r k 2 + .... in s u c c e s s i o n

a basis

delta).

that n o r m a l i t y {rk}.

given

isomorphism

group

isomorphism

of finite

Hom(Hom(F,ZS),F)= rank F,

Hom(Hom(F,~),G)

F~F.

Indeed,

and any g r o u p G,

= F~G.

This m a p

is

there

158

given

by ~: F ~ G

by,

--~ H o m (Hom (F, ~) ,G)

for ~fiOgi

9( ~ Both

sides

to b e

show

entail dual

fic~}gi) (9)

are

~

-i

that

that

to one

Hence

{qk I k I-n] another.

and For,

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

~-I =

[rk

simply

I kL--n]

chasing

B -I is o n t o

Furthermore,

for r6A

map

given

n

by

r = ~ - i B ( r ) = B-l(q)

are , the

just

so this enough

with

(r,rk)s k.

of l i n e a r

qk,rk£An

are b o t h

bases

isomorphism

), ~-l(q)

definition

dotting = ~

the

,

is e a s i l y

=

shows

algebra

must

a n d are

back,

(.]~rk(~sk) (q) = that

of them, of

shown

in g e n e r a l .

a matter

~ q~rk kl n

so for q E H o m ( A n , 2 Z

there

i 6 G

for F = ~ , t h i s

It is n o w

any e x p a n s i o n

Since

A --7 ~ n

S~(fi)g

in F and

A . n

= ~_ q ( r k) s k E An. k A . n

=

additive

£ A ~ n

~-i = ~ q k i ~ r k ,

span

and ~£Hom(F,Z)

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

Thus to

6 F~G

the

they

~ gives

r: q =

(r,-).

Let

r = sk,.

sl m u s t

form

q=8(r)

a basis. = the

Thus Then

k s k, = z~ ( s k , , r k i s k. S i n c e t h e s k are a b a s i s , (sk,,rk) = 1 or O, k a c c o r d i n g as to w h e t h e r k'=k or not. H e n c e the set of r k is a dual basis

to the

set of s k.

159 This argument dual basis r k ,

is reversible:

given any basis qk' and

o f An, ~-1 can be e x p a n d e d as

~_ qk@grk. k~-n

To identify ~ - I £ A ~ A , it would be sufficient n -~ n

everything with the rational find the element

~-i = k~--~nq ~

~

rk

-1

numbers

("introduce

in ( A / ~ Q ) ~ ) (Anl(kl"

. .,k~) . . i f . k l = k i ,.

k i = k [ , b u t ki+l>k'i+l

and (k I ..... Xn)>2(k 1 ..... kn) if kn=kn,..,ki+l-ki+ I -' Thus the first

is lexicographic b y largest part first,

second is lexicographic b y smallest part last carefully, kI ~ k2 ~

,but kik i and kj>kj+ I.

R.. associates 31

to k the partition

=(k!,k2,..,ki+l,ki+l,..,kj-l,kj+l,..kn

is removed the i

th

from a small part, We say,

Young Raising

That

is, one

, and added to a larger k ,k' of n, that

from k b y applying

the class

of convex

[i .... ,n] to itself.

is less than or equal a partition

).

a finite

series of

Operators.

: Consider integers

the j

th

for two partitions

k ~ y k' if k' is obtained

The Y o u n g

k =

to 2f(i),

functions

(Convex means for each

from the set of

that f(i+l)+f(i-l)

i = 2 ..... n-l.)

(k I ..... k ) of n, consider n

the convex

Given

function

189

fk given b y fk(i) function

f from

= k I + 12 +...+ k i.

{i ..... n~ to itself

the form fk for k =

(f(1),

two sets are in one-one define

functions,

Proof:

i.e.,

a convex

f(n)=n

f(2)-f(1) ..... n-f(n-l)).

k,k'

is of So the

and it makes

sense to

of n that I ~D k' if fk ~ fk'

fl(q) i fl,(q)

implies ~ ~ ' .

that if ~ *~=~', of raising n,n-l,.,

to get

distinct.

Let ~ =

from w h i c h the number

tracted

mutation,

a series

are now all

from the first number, Consider Suppose

rows.

Then ~*G = R

to a partition

the number

the number

it is larger Let p,q,

Let ~' be a followed qP

(~*~').

Proceeding

from which

But this can only happen so ~*~I = ~"

a series

of the form ~*al, where,

from which the number

is less than the number

for all i.

o(n)

Let i6{i .... n}.

(i,i+l).

is to show

we can assume ~*a is obtained b y applying

of raising operators ~*~i'

The numbers

of these

likewise

A d d in the numbers

i+l is to be subtracted.

be the indices

in this fashion,

(41 , .... ~n ) .

i is to be subtracted.

from w h i c h

b y the t r a n s p o s i t i o n

computing

etc.

essentially

from ~ b y applying

one now subtracts

from the second,

respectively,

remains

(~l+n,~2+n-l,..,~n).

To get ~*~,

than the number

What

then ~' is obtained

operators.

as

for all q=l,,,n.

It is easy to see that ~ y and ~ D are equivalent,

that ~ ~ y ~

g(n-l)

satisfying

correspondence

for two partitions

Conversely,

in

i is to be sub-

i+l is to be subtracted,

if o I is the identity perf

190 Another property of the natural ordering is that for ~,k 6 H(n), w ~ k if and only if, taking conjugate partitions, ~' < ~'. One might conjecture

that the natural ordering is also

obtained as the intersection of the two linear orderings ~i,~2 above.

But this is false,

(6,6,2,2)

as (7,4,4,1)

is greater than

in each of these, but the two are incomparable under

the natural order. This partial ordering on partitions has been investigated by, among others, Doubilet Inversion Theorem on it Liebler and Vitale

([14]) who has proved a Mobius

and also b y Brylawski

([9]) and

([25]) .

Young originally gave the algorithm in terms of his Raising operator,

and conversely the expression of the h

of the { ~ ' s

b y means of a lowering operator.

T = with

s in terms

(See

Now that we have the transition matrices R = {k} = ~ r k

!

[39]).

(rk~),

h ~, it is a simple matter to compute the matrix

(tk~),

tk~{~}.

To find tk~, dot this equation

[~}: = (,

[~)

~tk~({~},{~})

= t~

(using the orthonormality of the [~}'s)

191

But

{~] = ~ r w

since the h the m a t r i x

h

so

's and 's

Theorem.

the transitions

matrix U =

= T - R =

(and easy to prove) is a symmetric

Recall

gives

known

h ~ is just the product

and

{~} = ~ r h

R transp°se-

R.

of

so the

It is a general

fact

of a matrix b y its transpose

matrix W =

the conjugate

a *{k] = ~ w k ~ a n * a n

Hence wk~ = rk, ~. calculation

Hence tk =r k, so

= ~ u k

h p,

for all k,~.

a *{k] = [k'} n Thus

~k

Hence in the expression

is the transition

a *a =h n ~ ~

= r

T is also triangular.

that the product

matrix.

we have uk =u k

Recall

= ~ U k

= ~ t k ~ { W } ,

(uk~)

(<X>,h)

of R, a fact traditionally

In particular

N o w the transition

r

are dual bases .

T is the transpose

as Kostka's

Next

(, {~]) = ~ w

Similarly the Schur

that passing

turns

the partial

(rk~)

is triangular

f

(wkw),

partition, ~

from a partition

ordering implies

upside that

down. (wk~)

so this

in terms

a ~.

and an*h =a w and

so {k'] = ~

'

= an*,

functions

{k} = ~ w k

wkwh w.

same sort of

of the f .

to its conjugate Hence

is also.

just

the fact that

192

Notice k£H(n),

in p a r t i c u l a r

(ak,,h X) = i.

still g e t s

this b e c o m e s

=

This

is

in

(hk,ak,)=l

This

that

shape

n

(hk,ak,)=l

] - the Y o u n g

"positive

[~}.

1 ,

with

a , n

k' b y k , ~' b y ~,

Hence

there

correspondence

, (by u s i n g

is then

the M a c k e y

a standard

Theorem,

are b o t h

between

actual

exists say)

k X -

is first and

representations,

one and only one irrep

labeled

in

The m a i n p r o b l e m irrep

of S

n

arises

in

k. of A l f r e d Y o u n g

(for w h i c h

see L 3 9 J

c o m e s up in his c o n s t r u c t i o n symmetrizers.

is c o n s t r u c t e d

symmetric

Indeed,

) that the c o r r e s p o n d e n c e

since h k and ak,

for a u n i q u e

k~n,

r

is to show that e v e r y

In the a p p r o a c h

in ~[S

Relabel

{k} of S . n

that t h e y h a v e

irrep

of course,

fact that

{~'].

~Z~

entries

the inner p r o d u c t

the n a t u r a l

Ii2 J ~9J

fact i m p l i e s

this w a y

[k} +

k of n and irreps

then to o b s e r v e

common.

r

R one

([k], {k}) = I. QED.

(e.g.,

to c o m p u t e

this

ak, =

~ W>k

for each

the m a t r i x

diagonal

Taking

[~}

{k'} +

phenomenon:

inverting

matrix with

fact again gives

partitions proof

r

a~ = A

and the result

(hk,ak,)

Proof:

a triangular

thus h k = {k} + ~ "

the f o l l o w i n g

group

on the

For each Y o u n g

an e l e m e n t P(k)

of

), the

idempotents

tableau

of

£ ~[S n] - the

rows of k" and an e l e m e n t

193

N(k)

6 ~ IS ~ - the n

of k"

"negative

and then one shows

is an o r t h o g o n a l

symmetric

that

{

group

on the c o l u m n s

--~ P(k)N(k) all Y o u n g t a b l e a u x of shape k

I ki- n}

set in ~ IS I. n

The c o m b i n a t o r i a l

aspect

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

theory

the s y m m e t r i c

group

is the s t u d y of t h e s e

transition

(rk~),

etc.

An e n t i r e l y

approach

(tk~),

subject

(in, e.g.,

symmetric = ~

functions

Algorithm. properties

and the

R .

This

are derived.

representation

to us, m a y

bodily

elements

approach,

certain

in that context.

one has v e r y n a t u r a l

not h a v e o c c u r r e d

, wk to us,

It seems

has yet to b e w r i t t e n .

, etc.,

Knuth

thus

and their of

shortening

which now appear natural On the other hand,

interpretations

in

of the

w h i c h we c o u l d derive,

synthesis

(w~k),

the w h o l e n o t i o n

had not the c o m b i n a t o r i a l

the final

W =

functions

ignored,

notions

to the

R for w h i c h

is done b y the i n g e n i o u s

In this

However,

rk

the m a t r i x

Schur

matrices,

[ 3 8 J ) starts w i t h

of the m a t r i x

can be c o m p l e t e l y

look ad hoc

that approach,

found them.

symmetry

Then R is u s e d to d e f i n e

the e x p o s i t i o n .

matrix

['437 , and Rota

w kh k and c o n s t r u c t s

W = R transp°se"

group

Stanley

different

of

but might

approach

first

of the two a p p r o a c h e s

194

BIBLIOGRAPHY

Eli

Adams, J.F., ~ -Rinqs and lecture, 1961; Adams,

~-Operations,

J.F., Lectures on Lie Groups,

(unpublished

1969, Benjamin

Atiyah, M., Power O~erations in K-Theory, Quart. J. Math., (2) 17 (1966), 165-93. (Also reprinted in Atiyah, M., K-Theory, 1967, Benjamin [4~

Atiyah, M., and D.O.TalI, Group Representations, -Rings, and the J-homomorphism Topology, 8, 1969, 253-97

[9

Bergman, G.M., Ring Schemes: The Witt Scheme, Chapter 26 in D. Mumford Lectures on Curves on an Algebraic Surface, Princeton, 1966

L63

Berthelot, P., Generalities sur les ~-Anneaux. Expose V in the Seminaire de Geometrie Algebrique, Springer-Verlag Lecture Notes in Mathematics 225, 1972

[~

Bir~hoff,G., and S. MaeLane, A Surve Z of Modern Alqebra, 3 Edition, 1965, MacMillan Co. Boerner,

LgJ

H., Representations

of Groups,

1970, North-Holland

Brylawski, T., The Lattice of Integer Partition~s, of North Carolina Dept. of Math. report, 1972

university

Burroughs, J., Operations i__qnGrothendieck Rings and Grou~ RePresentations, Math. Dept. Preprint 228, State University of New York at Albany

LIO

Cartier, P., Groupes formels associes aux anneaux de Witt generalises, C.R. Acad. Sc. Paris, t.265, 1967, A-49-52

Coleman, A.J., Induced Representations with Applications to S and Gl(n), Queen's Papers in pure and Applied Math No.4, Queen's University, Kingston, Ontario, 1966

El%

Doubilet, P., Symmetric Functions through the Theorff o__ff Distribution and O c c u r , (No.VII of G.C. Rota's On the Foundations o__ffCombinatorial Theory) (to appear) Doubilet, P., An Inversion Formula (mimeographed notes, 1972)

involving

Partitions,

195

[19

Dress, A., R__epresentations of Finite Gro_~, Part i, The Burnside Ring, (mimeographed notes, Bielefeld, 1971)

[36]

Foulkes, H.O., O__nnRedfield's Grou~ Reduction F_unctions, Canadian J. Math, 15, 1963, 272-84 Frame, J.S., G.de B.Robinson, and R.M.Thrall, The Hook Lengths o f ~ , Canadian J. Math., 6, 1954, 316-325 n

L183

Grothendieck, A., La Theorie des Classes de Chern, Bull. Soc.Math.France, 86,1958,137-54 Grothendieck,A., Classes de Faisceaux et Theoreme de Riemann-Roch, (O, Appendix, in Seminaire de Geometrie Algebrique, Springer-Verlag Lecture Notes in Mathematics No. 225, 1972) Hall, P., The Algebra of Partitions, Proc. 4thcanad.Math. Congress, Banff, 1957 (1959) 147-159

[20

Harary, F., and E. Palmer, The Enumeration Methods of Redfield, Am.Journal Math., 89, 1967, 373-384 Hawkins, T., The Origins of the Theory of Group Characters, Archive for the History of the Exact Sciences,VI__II,2,1971, 142-70 Koerber,A., ReDresentatio~ o_f Permutation Groups~, 197~, Springer-Verlag Lecture Notes in Mathematics, No. 240

[20

[20

Lang, S., Algebra, 1965, Addison-Wesley Liebler,R.A. and M.R.Vitale, Ordering the Partition Characters of the Symmetric Group (to appear) Littlewood, D., A Universit~Algebra,

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Littlewood, D., Plethysm and the Inner Product of S-Functions, J.London Math. Soc., 3_22,1957, 18-22

L20

Littlewood, D., The Inner P!ethysm of S-Functions, Canadian J. Math., i~, 1958, 1-16

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Littlewood, D., The Theory of Group Characters, 2nd Ed.,1958, Oxford

196

[30]

1 31]

MacMahon, P.A., Combinatory Analysis, Vol. I,II, Cambridge, 1915, Reprinted Chelsea, 1960 MacMahon, P.A., table of the number of partitions of n, n ~ 200, in G.H.Hardy and S.Ramanujan, ~ Formulae in Combinatory ~ , Proc.London Math. Soc.,17,1918, p.l14 Milnor, J., Lectures on Characteristic Classes, notes, Princeton, 1957)

(mimeographed

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1948, Edinburgh

1885, Cambridge

Serre, J.-P., Representations Lineares des Groupes Finis, 1967, Hermann Snapper, E., ~ Characters and Nonnegative Integral Matrices, J. Algebra, 19, 520~35, 1971 Stanley, R.P., Theory and A pplicatio n of Plane Partitions, Studies in Applied Math., 5__O0,1971, 167-88 Swinnerton-Dyer, H.P.F., ~plications of Algebraic Geometry to Number Theorz, Proc. Symp. Pure Math, A.M.S., 1969, Number Theory Institute

197

~

Weyl,

H.,

The Classical

~6~

van der Waerden,

~7~

Young, A., Q u a n t i t i v e S u b s t i t u t i o n a l A n a l y s i s , I - VII, ( p u b l i s h e d at v a r i o u s t i m e s 1 9 0 1 - 1 9 3 4 in Proc. L o n d o n M a t h . Soc. - s e e R u t h e r f o r d ~39~ )

B.L.,

Group_s, Modern

1949,

Algebra,

Princeton Ungar,

1950

198 INDEX

a

2

OF N O T A T I O N

55

W R

n

K(F)

6

S

A V

6

R(G)

70

kt(a)

8

CF(G)

81

P n ( S l .... s n ;c~n ~' "''~ n )

K, i

89

12

L.

90

n

59

n

1

[rq

96

P n d ( Sl .... Snd)

12

1 + A[[t]] +

15

A

25

B(G)

107

a

28

S C F (G)

ii0

28

~I~[G]

115

v) 29

R(S)

128 130

h n

TT'

(conjugate

of

99

~Wn

29

®

If(n}

29

S

17 1

137

*S

2

h

30



32

s

35

139

A(X)

39

145

{k]

43

~/n

47

H

151

R

5O

X

151

S

.S

rrl

~2

~Prr

laij Ik

137 137

147

t99

153

T =

(tk~)

190

162

U =

(uk~)

191

182

W =

(Wk~)

191

o*k

184

P(k),N(k)

192

R.

189

ITI

f

R

=

.

13

(rk~)

the n u m b e r in a set T

of e l e m e n t s passim

lO0

INDEX

Adams O p e r a t o r s

47

Ferrar's graph

algebraic g e o m e t r y

52

finite degree (of an element in a k-ring) 8

b i n o m i a l coefficient, generalized b i n o m i a l type

153

forgotten symmetric

29

function 162

9

Brauer's Theorem

i01

B u r n s i d e Ring

107

Frobenius Character Formula

163

Frobenius R e c i p r o c i t y C a u c h y ' s Lemma

40

central

81

function

centralizer

105

character

84

c h a r a c t e r ring

84

F u n d a m e n t a l Theorem, T h e o r y of S n

Rep. 135

F u n d a m e n t a l Theorem, symmetric functions G-module -

c h a r a c t e r table

74

- -, map of

2 61 61

90 - - -, i s o m o r p h i s m of

c h a r a c t e r of p r o d u c t of two g r o u p s G H 97 c h a r a c t e r s of S n' integrality

G-set; -

G-map

- -, sum of

104 106

134 - - -, p r o d u c t of

characters,computation

62

106

101 - - -, symmetric p o w e r 106

conjectures

i00,i13,135

conjugate partition

group a l g e b r a

115

group d e t e r m i n a n t

122

29

cycle

124

cycle index

146

group reduction formula 146 H o m o g e n e o u s p o w e r sum cycle structure

124

dot p r o d u c t

138

in R(S)=A

30

hook n u m b e r

172

immanent

147

201

indecomposible

76

induced character formula inner

automorphism

inner p r o d u c t (=A) irreducible

96 105

in R(S) 138

character

84

k-ring, n a t u r a l o p e r a t i o n on - - -, p r o d u c t

28

isotypical

78

- - -, t e n s o r p r o d u c t of two

21

-

,

lattice permutation

monomial

isotypical

component

Jacobi-Trudi

Equation

K-Theory Knuth

Theorem

k-ring ---, -

-

binomial

- -, c a t e g o r y

of

- -,of central functions

---,

definition

---,

finitary

-

-

184 27

Algorithm

Kostka's

79

21 15

-

Theorem

Maschke's isobaric

of two

special

-

Mackey's

irrep ( = i r r e d u c i b l e representation)

25

99

Theorem

76

group

103

natural ordering partitions

on 187

Newton' s F o r m u l a s normal

167

dot p r o d u c t

35 156

193

normalizer

105

191

orbit

104

5

orthogonality

9

outer product

relations

91 127

20

partition

54

partition, natural o r d e r i n g on set of

187

13

permutation

109

8

plethysm, outer

29

matrix

inner

and 135

- -, free on one generater

24

power

sums

35

- -, m a p

20

pre-k-ring

7

of

202

pre-~-ring

49

Schurfunctions

43

Y-ring

49

Schur's Lemma

77

semidirect p r o d u c t

98

regular addition of squares

177

representation,linear conjugat~

of

semi-simple

116

simple G-set

105

60 69 Splitting Principle

decomposible

76

degree of

60

dual of

68

18

standard Young tableau 168 super central function ii0

exterior power faithful

super c h a r a c t e r

ii0

super c h a r a c t e r table

113

68 65 symmetric

induced

73

inner p r o d u c t

72

irreducible

76

permutation

64

-

function

- -, e l e m e n t a r y

- - -, forgotten

-

-

p r o d u c t of

-

76

- -, regular

64

-

-

-

-, Sum of

-

S-functions

2

- -, hom. p o w e r sum

30

- -, m o n o m i a l

32

- -, p o w e r sum

35

- - -, Schur

43

symmetric p o w e r

46

torsion free ring

49

64

trace

83

67

transitive

105

70

t r i a n g u l a r i t y of transition m a t r i c e s

189

64 of S

- -, a l t e r n a t i n g

representation

162

68

- - -, trivial representation canonical

28

68

- - -, reducible

-

- -, Fund. T h e o r e m

2

ring

n'

44

203

Verification Waring

Principle

formula

Wedderburn's

27 35

Theorem

i16

Witt vectors

56

wreath

98

product

Young

diagram

Young

Raising

29 Operator

189

Young Symmetrizing Operator

119

Young

tableau

168

zeta-function

53