The Universal Form of the Littlewood-Richardson Rule

Table of contents :
Introduction
Chapter One - Schur Functors
1. Definitions
2. Universal freeness of skew Schur (co—Schur)
functors
. The filtration of L F e G
3 VP} )
Chapter Two - The Littlewood—Richardson Rule
4. The characteristic zero case
5. Strategy for a characteristic free rule
Chapter Three - A Universal Filtration
6. The main theorem
71 The coefficients occurring in the associated
graded module
8. Further remarks
Bibliography
1 i
10
18
25
3o
35
41
54
64
71

Citation preview

THE UNIVERSAL FORM OF THE LITTLEWOOD-RICHARDSON RULE BOFFI, GIANDOMENICO ProQuest Dissertations and Theses; 1984; ProQuest

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8420758

Boffi, Giandomenico

THE UNIVERSAL FORM OF THE LITTLEWOOD-RICHARDSON RULE

Brandeis University

-

PH.D. 1984

University Miorofilms International 300N.ZeebRoad.AnnArbor.Ml48106

Copyright 1984

by Boffi, Giandomenico All Rights Reserved

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THE UNIVERSAL FORM OF THE LITTLEWOOD-RICHARDSON RUEE

A Dissertation

Presented to The Faculty of the Graduate School ofl.Arts and Sciences Brandeis University

Department.of Mathematics

In Partial Fulfillment of the Requirements of the Degree Doctor of PhilosoPhy

by Giandomenico Boffii

April 1984

.

J

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'This dissertation, directed and approved by the candidate's Committee, has been accepted and approved by the Graduate

Faculty of Brandeis University in partial fulfillment of the

'

'requirements.for the degree of DOCTOR OF PHILOSOPHY

2W 57am

/Graduate School and Science

fArts

Dissertation Committee

Cha

n

’bww

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@Gopyr'i gh'bs. by Giandomenico Boffli

1984'.

r

4

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ACKNOWLEDGEMENT

I would like to thank Professor David A. Buchsbaum for his help and support“ His willingness to freely share,

his time and insights have been invaluable nottonly in the;

preparation of this work, but to the development of my knowledge and appreciation of mathematics in general.

a

.l

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TABLE OF CONTENTS

Introduction

Chapter One - Schur Functors 1. Definitions

10

2. Universal freeness of skew Schur (co—Schur)

3.

functors

18

The filtration of L VP} F e G )

25

Chapter Two - The Littlewood—Richardson Rule

4. The characteristic zero case 5. Strategy for a characteristic free rule

3o 35

Chapter Three - A Universal Filtration 6. The main theorem

41

71 The coefficients occurring in the associated

8. Further remarks

54 64

Bibliography

71

graded module

1

i

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INTRODUCTION

Let V be an n—dimensional vector space over 0 and.

p:

GL(V)-+ GL(W) a rational representation, that:is, a morphism of algebraic groups. After a choice of bases forr

V

and W (dimcw is denoted.by N), P becomes a linear transforh

mation GL(n,®)—a-GL(N,C) satisfying the following property: for all A=(aij) in GL(n,C), the entries of P(A)=(bhk) e GL(N,C) are rational functions in the entries aij’ As thecemmon denominator of the N2 entries bhk turns 0U$Jt0 be a

power of detiA), one restricts one's attention to the case

where the entries of p(A) are polynomials in the entries aij; P is then called a polynomial representation. In order

to study polynOmial representations, it suffices to study

those representations p such that each entry of P(A) is a homogeneous polynomial in the entries aij’ of fixed degree d : one then Speaks of the homogeneous polynomial representation p of degree d. The theory of homogeneous polynOmial representations

of degree d is closely related to the representation theory of S a! the symmetric group in d letters. Consider the d—fdm

tensor product.Vnd; Sd acts on it by

6(vln...nvd) = V34(l)n"'nvad(d) . An element u of End(Vnd) is called bisymmetrio if it commu— l

I

i

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tes with the endomorphism identified by 0’, for all 6 in S a. In other words u belongs to the centralizen-of the im-

age of CS

in End(Vnd) (68d denotes the group algebra offsd

d

over 0). As cs d is semisimple, so is its image in End(Vnd). It then follows from the Commutation Theorem, that.the a1;

gebra 3(a) of bisymmetric endomorphisms is semisimple too (3(a) is in fact.the subalgebra of End(Vnd) generated by all tEd for teEnd(V)). Hence every finite dimensional B(d)-

module decomposes as a direct sum of simple B(d)—submodmms. As it is possible to show thatievery homogeneous polynomial representation 9 of degree d factors uniquely as

GL(n,C) ——> B(d) t

t->

13nd

———> GL(N,C) H

g('tnd)

y

where g is a suitable homomorphism of the algebra B(d) into

M(N,C) (= NxN matrices over G), one concludes that every na

tional representation of GL(n,C) is completely reducible, i. e., GL(n,C) is reductive. (For more details cf. ED-C] , p.14 and ff}, whose presentation we have closely followed.

Also cf. [3,1] and [P]; notice that.Schur calls "order" what we call the degree of the representation.) Furthermore one proves that every simple B(d)amodule

is isomorphic to one of the form

c'VEd, where c generates

a minimal left ideal of the image of CS

d

in End(Vnd). Thus

one is led to classify all minimal left;ideals of CSd,wmmfll is what Young did in his celebrated memoirs. For details on

Young's work we refer the reader to [vdw], §14.7o In the following, we just state the main results.

I

I. To every partition x of weightid, associate the tag

I

l

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leau Tb.’ that.is, the diagram ofifh filled row-wisezwith'un indices l,2,...,d in this order. Let.ax and b} bezthe ele— ments of CS (1

a

defined by ax: Z(-l)-c’ and bx=Zt , where 6’

runs over all the elements of 8d which preserve the rows of

TA and t'runs over all permutations in Sd which preserve the columns of TA. Let cA be the product;aaba, and LAY be

the subspace of Vmi defined by

cA-Vnd. The spaces

{LAVI A has d boxes, its firstgrow has sn.boxesz form a complete set of distinct irreducible homogeneous polynomial representations of degree d.

II. Let SK denote the character of LAY’ and let’X'be the character of the Sd-simple module associated to;%. Then

for every A in GL(n,C), one has

’X.

57x00 =

,

j:

(i) AVnV—>1\,VEAV—9L

EB:I

V—>0

,

i. e., in the Grothendieck ring, one has

L EEF‘v = A3VmA2V - A4a =

f

A3v v

A4v 2 Av

I

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°

7

Similarly for the seconliambelli fornula. For quite a while now, it has been a problem of con-

siderable interest to see how far one can go in generatflfing the classical theory over G'to any commutative ring R ( in

fact 2 ). The reason of this fact lies in the develOpment of modular representations (cf. for instance [C—L]) and of algebraic geometry over fields of positive characteristic

(of. for instance [Hi]), not to mention the work spurned in

several directions by LasCoux'[La].' The.first step to achieve a generalization is to find a suitable definition for 1&3; Fla free'module of rank'n

over R. This'can be done ad abundantiam. One can define sex

eral (indecompOSable; but not irreducible) Schur functors;

distinct over 2, that yield the same irreducible representa tion when tensored by Q. We define two such free modules in

Chapter One, denoted by LAF and KWF. The second step is to see whether the classical formur

las for Schur functions (formulas true over 2) still have some meaning for our integral representations, For instance,

say for LRF'

consider IHEFFH The sequence

0 ——. A4FnF ——.~/\3Fn/\2F —->L F _.o EEF is not exact:over 2. However, it is possible to find an ex—

act sequence 0 ”Ab-F —>A5F 9 (A4FEF) —>A3FEA2F —> L

Efl

F —-2 0

so that in the Grothendieck ring one has

I’EEF‘F = A3FnA2F - Mm - ASF + ASF

'

l

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,

A3F

Mr

F

AZF

’

and the first Giambelli formula still holds! In fact.Akin

and Buchsbaum [AeB] have shown that it is always possible to generalize the first Giambelli formula in this sense, no

matter what A one takes in IKF' The problem is still open for the second Giambelli formula.

The purpose of this thesis is to show how one can gen? eralize to integral representations another classical result about the multiplication of Schur functions. If X and

p are partitions, then she“ must be an integral linear com?

bination of Schur functions, say

SAS}4 2

g C(Mffiv) SP

,

where u ranges over all partitions having a numbeniof boxes

equal to the sum of the boxes of N andib. In fact;

all

c(A,p;P) are non—negative, and there is a rule describing them, known as the Littlewood—Richardson rule. The parallel statement for irreducible representations is a rule de-

scribing c(h,p;P) in

LAV n LPV = Z C(7\,}A;P) LPV p

(GL(n,C) reductive implies that any tensor producu.of Schur functors must be completely reducible). Is there any Littlewood—Richardson rule for integral

representations? The answer is yes, in that it is possible to find a filtration for LXFEL P F such that its associated

graded module is isomorphic to Z c(A,p;p)LuF ,with c(A,p;v) p

described by the same rule which works over 0. (An analo-

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gous result.probably holds for KAEEKHF') Furthermore, the combinatorial description of c(A,p;v) leads to a description of the associated graded module as Z: LPF n CP , where

‘CP denotes a suitable universally

free module of rank c(A,u;u) over R.

Here is what can be found.in the following chapters.

Chapter One discusses the definition of Schur functors over any ring R and explains why we can just work over Z.

Chapter Two sketches the proof of the Littlewood-Richardson rule for Schur functors over 0, and outlines our strategy over Z. Chapter Three contains the main results and some

applications (both in mathematics and physics).

F

I

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CHAPTER ONE SCHUR FUNGTORS

In this chapter we survey the work of Akin, Buchsbaum and Weyman about a Schur and a co-Schur functor over h,'flat

are universally free. In particular, we describe the.fil— tration they put on the Schur functor L%(F9G).

In SOme case,

for details we refer the reader to either [A—B—w] or [13]. 1. Definitions

It is a well known fact that several results about the Schur functions can be extended to the so—called skew Schur functions, i. e., Schur functions defined by means ofexskew shape. Hence in this section we give the definition odmn‘

functor associated to a skew shape, or to a more general configuration. We start with some preliminaries.

A graded R—algebra is a graded Remodule together with a homogeneous multiplication

and a unit

A =

e A

120 1

,

A g A-—ELeA

u: R—e»A , such that the following diagrams

commute:

Ann—mi Am lam l

I l m

Am —m—. A (associativity)

Rm Elia“ m ‘//1;

(r) .

A (identity element)

Given two graded R—algebras A and B, the graded R-

module AnB becomes a graded R-algebra by the obvious unit u

and by

mAnB: (AnB)n(AnB)-—>AEB

defined as the composite

10

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11 m.

Am‘BnAm'B 3113i AmAaB flAnB

,

where T is the "twisting morphism" AnB‘—9BnA that sends xmy

to (-l)ijynx, for all xeAi, yeB.. J A graded R—algebra is commutative if the diagram

T

AHA

>AEA

/m

m \

is commutative.

A A graded R—co—algebra is a graded R—module A, together with a homogeneous diagonal morphisni

co—unit

A: An—+A g A

and a

E: A-—+R, such that all the above diagrams (r) wflh

reversed arrows commute (we Speak of co—associativity and

co—identity element). Similarly, by means of T,

one defines

the tensor product of co—algebras and the property of

co—

commutativity.

A graded R—Hopf algebra is a graded Remodule A that is

at the same time a graded R-algebra (A,m,u) and a graded R—

co—algebra (A,A,£), and satisfies the following conditions: I

(i) m and u are maps of R—co-algebras

(ii)4A and 8 are maps of R-algebras. A Hopf algebra is commutative if it is commutative both as algebra and co—algebra. We usually deal with Hopf

algebras that are commutattye, connected (AO=R), and free of finite type (every Ai is a finitely generated free

R-

module). Examples of Hopf algebras are AF, SF and DF (F a free R—module); recall that in SF and DF we only give even de-

grees, that is, SkF and DkF denote the elements of'degree

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12

2k. The diagonal map of AF (reSp., SF) is obtained by ap— plying the functor /\(resp., S) to the map F —>FoF , x F—v , and composing with the isomorphism

(x,x)

A(FeF) E AFnAF

‘(resp., S(FoF) g SFnSF) .

DF is defined to be the graded dual Hopf algebra of"SF*. In general, given the Hopf algebra A, A*gr is defined (as a

module) by

(it"igr)i = HomR(Ai,R) ; it becomes a Hopf algebra by setting '

m *gr = af

and

A

A *gr = m* - Hence DnfmFanykgr ; if one identifies SFale A

with the polynomial functions from F to R, DF is identified with the partial differential Operators on polynomial func—

tions from F to R. (Cf. LAéB-W], Chapter I.)

When charR=O, DF is isomorphic to SF as an algebra. But when charR#0, this is not so; in particular, DF is not noetherian,

even if it is finitely generated in each degree.

This leads to the following general notions.

Let TR(F1,...,Fn) be a functor defined for all commu— tative rings R and all n—tuples of free Remodules Fl,...,Fn;

TR(F1,...,Fn) is called a universally free functor_if (a) TR(F1,...,Fn) 1s a free R-module (b) whenever oar—es is a ring morphism, TS(S§F1,...,

SfiFn) is naturally equivalent to SfiTR(Fl""’Fn)' For instance, the k—th exterior power of a free mod-

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l3 ule is a universally free functor because AFF is fnee oven~

R, and the Semodules A#(S§F) and SgfiFF are Lisomorphic. Also Sk and Dk are universally free functors. Moreover, as already observed, they are naturally equivalent when charR =0.

Let Ti and TE be two universally free functors. We say that Tfi is a Z—form of Tfi if T6 is naturally equivalent to To

C

The idea behind the notion of universal freeness is to give constructions over 2 that can be carried over to any

ring. The idea behind the notion o£.Z—forms is to distin— guish distinct characteristic free constructions that specialize to the same classical object. For example, in classical invariant theory only SFEDF is needed; but in order

to generalize the Schur functors one has to distinguish the role of DE from that of SF.

Proposition 1.1 ([B],Pr0position 1.2, Chapter Two). If the functor TR is the cokernel of two functors satisfying prop-.

erty (b) above, then T

R

satisfies it too. In particular, if

TR satisfies (a) and is the ookernel of two universally flaw functors, then T

R

is universally free.

We now turn to the definition of two Z-forms of the classical Schur functor. ' t A Le

I

d eno b e

any Sx t matrix ' .. (a13)’ where aije{0,l}.

l

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14 b.filaij (the sum along a row

and

1:] a.i=Zlai

Also let

and along a column,‘ resp.). We are going to define a mor—

dA: AA—> SK , where

phism of functors

as

8.l AA

= A

SK

’

KIOOEA

=—-Sblnooon

and X denotes the transpose of A.

,

Sbt

(Notice that we are. going

to use only formal properties of the Hopf algebras involved.) dA is described as follows. Diagonalize AA to get:

a

a

a

a

a

a

(A 113. . .n/\ 11;):¢zi(/\ 21m. . .n/\ 2t)n.. .n(/\ Sln. . .m/\ St), which is isomorphic (by re—arranging along the columns of

A) to

a

a a a a a (A llnA 21E. . .n/\ 81)m(/\ 12n/\ 223:. . .n/\ 82m. . .

a

a

a

.. .n(/\ 3't 2th. . .n/\ S1I) .

As ai . e{o, 1}, this is equal to (S

nSa

all

mumsa

21

)11:(Sa

$1

nsa

12

mumsa

)n...

22

'

s2 ...n(S

nS

alt azt

n...nS

ast

).

Now we use multiplication in S to go to S

bl

ES

b2

n...RS

bt

=S~

A

I

a The whole thing amounts to first antisymnetrizing A r and

spreading it over the r—th row, and next; taking symmetric products along the columns.

(Notice the analogy with the dc

finition of the "Young synunetrizen" c7L in theorem I of the

Intrdduction. )

Similarly, one defines a morphism

r

dA: DA—>/\K , where '

I

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15 b

A

A

aS

3.1

-b

sponds to the product. bka)‘ in theorem I of the Introduc—

tion.) Clearly, if one dualizes 'dA(F): AAF—fiSIF , one d'(F*): D~F*—>/\ 13‘”E .

precisely gets

K

A

A

Definition 1.2. Let F be a free R-module; the Schur functor of F with

respectuto any matrix A, denoted by LAF’ is

im(dA(F)). The co-Schur functor of F with respect; to A, de— noted by KAF’ is

Let

lN‘7°

im(dA(F)) .

denote the set of all finite sequences of

non-negative integers ("improper partitions"). E. g.

7\=

:

(Al,7\2,...) with his-.0 for almost all indices 1. If MN“, we define KEEN” (the "dual" or "conjugate" of 3.) by

X3 = number of all indices 1 such that Aizj . Clearly Alzkza... ; moreover, the terms of A are precisely those of A, written in decreasing order. If we define

AetN”

to be a partition whenever A1275)... , then ~ is an involu— tion on the set of partitions.

Given a partition h, iM = (Eli is called its weiat, and the number of its non-zero terms is called its lenah,

UM. If l7tl=n, we also say that 7i. is a partition of n. Re;— mark that N is a weight preserving involution. A partition X is sometimes expressed by means of a diagram of boxes

(the "Ferrers—Sylvester" diagram; of.

[8y] on the origin of

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16

this double name), as in the example below:

ifx: (6,4,4,3,l) , then N

Hence )x. just counts the boxes in each column, and its dia—

gram is the mirror image of that of A, with respect; to its left -t o—ri ght diagonal :

K:

x

/\:

\

If >t is a partition such that €()»)=s and e(X)=t, consider the sxt matrix A(>\.)=(aij), where

1

section of the i—th row and the j-th column

.

a, , = 13

if the diagram of A. has a box at; the inter

0

otherwise

.

co— Then LMMF (resp . , KA(7\.) F ) is called the Schur ( res p . , __

Schur) functor of shape 7x, denoted by LXF (resp., KLFL then LAIF=AtF and KXFr—D'bF'

If X=(‘t),

.

With

If )v=(l,...,l)

,

, _ 1:

IM=t, then LXF=StF and LXF—AF .

Given two partitions X and p, we write pg)» if‘pigki for all i. We can thus consider the “skew" (or "distorted")

A.

shape

5/ /

(the shaded region has been erased). .

,-

m

The corresponding element (kl—’41, 5142,...) of- IN

by A/p ("A mody")- It is easy to check that

17?

u

is denoted

and tit/P:

have the same diagram.

u

d

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17.."

Let 8(7\.)=s and E(X)=t as above, and consider the sxt

matrix A(A/p)=(aij), where 1

if the diagram of 7V“ has a box at the inte;

aij=

section ofthe i-t'h row and the j-th column 0

otherwise

.

Then LAM/ME (resp., KAm/NF) 18 called the skew Schur

(resp., co-Schur) functor of shape h/E, denoted by LMAF (resp., KNHF). and

33- g. = if 7\= (6,4,4,3,l)

P= (3,3,2,l), A044) is

the matrix 0 O O l l l O O O l O O

o o 1 1 o o O l l O O O

l O O O O 0

.

Notice that (hl—y1,7\2-t42,...) is not necessarily a pa; tition, and even if it coincides with a partitionv, LHF is {c

quite differ-en same as

o

L

:EF‘

t

from Lz/V‘F

Fc_,FmS2FnF

:

LEBFC-esnSZF

-

_

is not the

.

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1 ‘3 .LC)

2. Universal freeness of skew Schur (co—Schur) functors

1 ' L M" and K ’Vi" sec t'10n w e discuss the fact tlratf In this

are universally free funotors (hence, also LR and K1. are so). We are not going to give a proof (which can be found

in all details both in [A—B—W],II.2 and in [B],§§3—4, Chapter Two). We just notice that by virtue of Proposition 1.1, one has to show that LN? F (resp., KA/FF) is free. and that L My

( res P . 9

K “4“) is the cokernel of two universall y free

functors. This can be done by means of a suitable descrip-

tion of LMPF (resp., KA/VF) by generators and relations, together with the "straightening law".

Let us suppose for the moment that K and H have only two rows, say }\=(7\l,?\2) and H=(Hl,y2). We denote by DA/i" the morphism

AAAIPIFaAsF’

a- -v >\--| v A|F+F®A2Vz F

..)\ ti".2: _2

'Y—l

!

("'1 V—H,'-y1+|

Where/\l 1 F

AZFZ F—eAlraAZ‘AZF isthe

composite map N't‘fl"

/\

I-

)\-

-

A-

Ar.

‘V

)\-

FaAW‘VF 33—.» A'P'Fa/Q’Fe/V”2 F 1813A IAFoA’P’F.

Let us go back to the original 7\ and Vi, and define

i A

i =(Ai’AiflI-l)

3

P

. =(Vi’Hi+l)

1-

=

l,ooo,€(7\)—l

It makes sense to define CIA/H as the morphism

z lu...nlnD i

i

islanaul

.

A 4A

It is not difficult to see (by co—associativiity, and the

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n

fact that AZFAFnFfis is zero) that

im(D7\/F) g

ker(d>‘/V(F)) (cf. [B],§3, Chapter Two). But more is true.

Theorem 2.1 ([23], Theorem 4.7(b), Chapter Two). L

x4. F i S

the cokernel of

”w .

The proof of this theorem requires the explicit de~

scription of a basis for LX4AF’ which we now give below. Let S be any totally ordered set and think of the diagram

of h/p, denoted by ARA" as the set of all pairs (i,ji) Where

i

=

l,aoo,e(>\l)

and

jj.

=

“1+19000,A i

I

Definition 2.2. A tableau of shape h/p with values in S is

a function

T: AAA—>3

(i. e., a way of filling the boxes

of Mi with elements of s). A tableau T is row-standard if T(i,j)\.)=s, €(p)=x~, and.

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call a the shape (91+fll,...,Vr+hl,hl,...,k S ), i. e.:

:37 A h

Also, let G = R ; . Assuming that GP

beB(o’/'v)

Then mum) induces a map $0.510): LPF ——> NIP/Mp . Proof. Let

xeimfljp), where UP is the composite map: P.

p. -e

mi:

:2.

”I

A F®,,,®/\HF®/\' Fc/x'“ FcA‘*‘Fc... l

it

PM

AF®...sA

V;

l®...®l®A®i®l®...

t:

“:u‘b

F®/\F®/\F eA 1

“m

F®/\

Fe...

I®...clelemc|c...

' i AV'Fc... @AP"'F® AP; F c /\Pi+ ‘F®/\V*2F®...

for any i, and any t such that tspi+l. By Theorem 2.1, it

suffices to show that w(w,b)(x)eli'llp .

Let us say that

x} = Up (aln. ..nai_lnaimai+lm.. .nar+s) = D

= g aln. . .nai_ln(ai)uwin(ai)ut.ai+lnai+2n. . . , ~

1

g (ai)owin(ai)ott =A(ai) (Note that we write

,

. i

where P-

, With A: Api+tF —>-/\'F n AtF .

p=(Pl,V2,...,pr+s). )

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Then ig(v,b)(x) = d6,(cpp(0’)(xnb)) is equal to

Z L 2. atWtakbe 0‘

”45‘ ($30“

_

where

AbiJJeaiJJ/tbw"easAeS)3

ea. Mb Mm) JJJJAE chitJim U"

(i)

{51.513

Prtrj

3

, as usual.

A... A b

b

b . stands for

Consider the composite map Dt: 0'- 2

0" "t

0%“:

(1-

6

A'LFG&)®...®A"'LF®&)®A (F%)®A'" (FemeA'Hreem... 1/ m...eieAeime... 0';

5" ‘t

t.

0’;

5;.

KeenaneA “(Fem/x Gem “(Fame/x ‘*‘ (Pecos/x ”Game... 1 i®...®l€9l@/m®l®... 56

”H

“in.

511+!

0’ A‘(Fe®®...®/\ tree» 9A Home A (Fame/x

(Notice that tsVi

di+1') Then

.

(1°,t = 0 , again by

Let 2 be

Theorem 2.1.

Z

+l$

\

twee...

(upheaeam Ab;would} (90min): “GOM‘L’Me...)

PN‘WP’i

and take Dt(z); one has:

{5s ( E(alhblengambme(oQWJAbie(aft;bAa;JlAbi+JoaiJJAbiJze...‘)) r""’ I

i

+ Z F’n"'a(5l

Z .

w u(-nt 3(4)” 72(5- [aJABJemaLJALiJoJI ”(Frit— 'A(b‘)[$d;-V;-t+f‘ J

oét

_".

JJ

MAL”) “in“we ”beam/rt) e

where X (as) T

v +1:

VVJAvk-j

@[M'“VJ: Ma.) . With A: A

p. +t-j

F—+A

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3

FPDAFJ

51 t-L

(II--"’i t+j

M1»), with A. A6" o-—>A and i5: (MIN P. V jwmmm

C1®A u

Hencetph», b)(x) is equal to:

If; —t(

_3[: 3 2 (43'1“ .--. 0 E

“72“.“... 4. Warm“, 1.453,” _V_at" Sn;”HBMQHJ

Mai»“AQMAQJVHNb

(3‘

I3: (3'

denote the element

For every fixed j and T, let 2.

J :7

Pi‘wm J

(4)

Z (emits .au._»5 Ali-QJMVWH Jb‘yflda-Pi-“ie

d[Z

|

(5

f’rWF‘i

8 (0:37.IAQ‘M Ur3,1t .A Jam 8 oitlAbimG m]

,

Following almost verbatim the argument given for z; a in

the proof of Proposition 6.1, one obtains I

I Z

=

Z

T

where

c'(

dG[EZ_

k

c(kl,ooo,kr)

1,000,

k

)E Z , and

r

-. (kl’coo,Kr)

I

Z

(kl, ...,kr )

’

is equal to

Q.Ab®.. 8.0n_ Ab;W8KGJ1PHJAL'E @LQ31A QI+‘AUE+‘® Qi+1AL,‘:+2®.u]

’

f’m’Fl

with b'j the appropriate element, of usual type.

(90/, b)(x) is thus equal to a linear combination of terms like 2

(k1,...,k )

. As jPi, and

df w‘3( vl,...,pll,u+t a,P11’G+JP i+2 ,...)>v

,

it follows that Lp(P,b)(x)eMp, as required. (Notice that

u' may be an improper partition,

(kl""’kr) e MP

it is not immediate that z'

6. 3 shows that acting on z( Proposition 6.1,

kl. . . . ,kr)

in which case

. However, Remark

as in the proof of

our claim eventually follows.)

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Theorem 6.7. LA? n L F

i‘

has a filtration whose associated

graded module is isomorphic to a direct sum of Schur func— tors.

Proof. For every P, order the finite set B(d/P) in some way.

Consider the finer filtration given by

Mp’b = Z im(Lp(v'.b')) where either v'>V,

,

or P'=|J and

b' F

a: L

(5.1) FOL( 4,2) FoL (4,1,1)F

I‘L

fl:

-> 0

F

:EP

-+ 0

F—>L F—>0 EH

’

so that the total complex:

O—>LF-—>L FeLF—>L F—>LF-»o B: 1333 g: EEP EH H *‘- Pieri formula LDJFébL F

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is exact, and in the Grothendieck ring one has: L

L

But

ES!

:83

L

=

F

F + L

:53

fij

F — L

F = L

‘33

m

FnL F + L

FmF

E

B”

F

.

, because there is a short

exact sequence: 0 —+ L

:EP

F —+'L

EF

F'QF‘—»-L

EF

F —+ O

.

Hence the Giamhelli formula still holds.

In general,

it is clear that applying repeatedly the

Littlewood-Richardson rule to the products in the expansicn

det(L

of

a. 1

b

:3

F is equal to det(L a. F). What is not clear, how

L a ...a l

F) , one proves that in the Grothendieck ring

r]

bl...br

l l] bj

ever, is the existence of an actual resolution of L [al."arf bl.nbn

corresponding to that equality. One hopes that examples like the one above may suggest a way of constructing such a resolution.

It is no accident that a physicist, Takahiko Yamanou-

chi (active in Japan in the Thirties and the Forties), appears in our story. The representations of the symmetric group play an important role in physics. It was precisely

in studying their role in particle physics that Yamanouchi

introduced what we now call sords (to be correct, Yama~

nouchi used the reversals ofprtesent—day Y~wordsz of. [H],

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Section 7-7). In particular, the LittlewoodsRichardson rule has a nice interpretation in terms of interactions between

systems of particles. We therefore append to this section a short discussion of elementary quantum mechanics.

The Schrodinger equation for a one—dimensional problem

is

u"+(E-V(x))u = O 3 Where E is the eigenwalue, u the

eigenfunction, and V(x) the potential..As the dimension is one, to each eigenvalue 6 there must correspond just one

solution u(x) (i. e., the solutions are "non—degenerate"). Suppose that V is an even function; then replacing x by -x, one sees that if u(x) is a solution belonging to E , then

'u(+x) is too. The non-degeneracy then.implies u(—x)=cu(x) for some constant c, so that

u(x) = cu(-x) = 02u(x)

,

c = i1

.

In other words, the eigenfunctions u(x) must be either even or odd. Or else: a symmetry property of the linear Operator

leads to a classification of the solutions according to the same symmetry property.

In general,

in order to study an atomic system, it is

necessary to find the symmetry group of the Hamiltonian,

that is, the set of transformations which leave the Hamiltonian invariant. If u is an eigenfunction belonging

to the energye , and d is an element of the symmetry group G, then on is ‘degenerate'with 11. Unless ou = cu for all 0‘, the level is degenerate. The eigenfunctions belonging to a

given energy 8 form the basis for a G-representation, usu—

ally an irreducible one. Only in rare cases, for special

choices of parameters, one has 'accidental' degeneracy, so that sets of functions belonging to different irreducible

representations coincide in energy. However, no symmetry

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i".A C C

considerations force these sets to be degenerate with one

another, and we may restrict our attention to the irreduci—

ble representations. Let us now choose a particular type of symmetry group. Assume that we have a system of "equivalent" particles

,

that is, the Hamiltonian of the problem is invariant under the interchange of all coordinates of such particles. Taken any eigenfunction u(l,2,...,n) belonging to a fixed eigen—

value, any element of G=Sn will produce an eigenfunction belonging to the same energy, by permuting the coordinate

vectors of the n particles (denoted by l, 2,

..., n , resp).

Thus if we permute the first two particles in u, we can ob— tain the linear combinations

u(l,2,...,n)+u(2,l,...,n) = (e+(12)) u(l,2,...,n) u(l,2,...,n)+u(2,l,...,n) = (e—(12)) u(l,2,...,n)

,

which are also eigenfunctions, and are respectively symme—

tric and antisymmetric in particles 1 and 2. The permuta—

tion operator e+(12) is called the "symmetrizer", and e—G2) the "antisymmetrizer" of the particles 1 and 2. Similarly, we can attempt to construct, from the initial eigenfunction

u, equivalent functions that are symmetric (antisymmetric) in larger sets of particles. This process can meet some ob-

stacles. For instance, let

b = 2:.0' 66$“

and suppose that bu

turns out to be identically zero; then u is not completely

symmetrizable, i. e., u is antisymmetric in any one pair of

particles. So the best we can hope for is that (b')u£b , where

b' = T 0’ (we mean Sn—lssn acting on particles 1,

Gasm-

i

2,-oo’n—l)o

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59 As an example,

consider the case of three particles and

an eigenfunction u(l,2,3). If 3.

buao ,

If

buéo , u has "symmetry type"

then we may succeed in symmetrizing in the

particles 1 and 2 (of. the linear combinations given aboveL

or in the particles 1 and 3. The symmetry type is then(2,1). If also

(b')u50

for both pairs, then u is antisymmetric

in the three particles, and its symmetry type is denoted by

(1,1,1).

In general, if Al is the maximum number of particles in which we can symmetrize u (and we renumber the particles

so that they are precisely the first A1 particles), A2 is

the maximum number of the remaining particles Kl+1,...,n in which we can symmetrize u (and we renumber the particles

so that they are precisely the particles Al+l,...,KI¥A2Lzmfi so on, we eventually arrive at an equivalent eigenfunction

of symmetry type a partition ?\=(%1,X2,...,%S)

of weight n.

Such function is called a "normal symmetric form" ("normal

S—form"). Clearly, one could also consider normal antisymf metric forms of type a partitionik: it turns out that a

-

function of symmetric type A can be converted by permuta—

tions and linear COmbination to a function of antisymmetric

typeyi=§(. And what the whole process shows is that the normal S-forms of type % are the basis functions for the

irreducible Sn—representation associated to the shape7\.

We now come to the Littlewood-Richardson rule. For simplicity we just consider an example (the one given in Section 4). Assume that we have two separate systems, the first

consisting of particles 1,2,3, and the second con—

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1,. O

taining particles 4,5,6. Also assume that all particles are equivalent. First we suppose that the systems are not inte_

acting with each other. Hence the states of the first sys— tem are classified by the irreducible representations ofS 37 and the same happens for the second system. Let us say that

both systems are in a two—fold degenerate level (2,1). When the two systems interact with each other, we have

to classify the states of the amalgamated system, according to 86'

(We say that we are dealing with the "outer product"

(2,1)n(2,l).) Notice that.if we just permute 1,2,3 among themselves and 4,5,6 among themselves, the outer product is

an irreducible representation of this particular subgroup of 86, which is S6-reducible. Thus the total number of

basis functions of (2,1)n(2,l) can be found as follows: there are

(§)=20 ways of splitting the six particles into

two systems of three particles each;

two basis functions

every system carries

(the two standard tableaux of shape

(2,1) with values in the particle indices {i,j,k}, where i