Representations of Finite Classical Groups: A Hopf Algebra Approach

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

869 Andrey V. Zelevinsky

Representations of Finite Classical Groups A Hopf Algebra Approach

Springer-Verlag Berlin Heidelberg New York 1981

Author

Andrey V. Zelevinsky Institute of Physics of the Earth Department of Applied Mathematics B. Grouzinskaya 10, 123810 Moscow, USSR

AMS Subject Classifications (1980): 16 A 24, 20 C 30, 20 G 05, 20 G 40

ISBN 3-540-10824-6 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-10824-6 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter I. Structural theory of PSH-algebras ...........

12

1. Definitions and first results ..............

12

§ 2. The decomposition theorem ..................

21

§ 3. Universal PSH-algebra:

the uniqueness theorem

and the Hopf algebra structure ............. § @. Universal PSH-algebra:

27

irreducible elements..

49

.........................

71

§ 5. Symmetric polynomials ...... . ................

71

§ 6. Representations of symmetric groups .........

86

§ 7.

93

Chapter II. First applications

Representations of wreath products .........

Chapter III. Representations of general linear and affine groups over finite fields ................... 107 § 8. Functors

iU,e

and

ru,@

................. 107

9. The classification of irreducible representations of

GL(n,Fq)

.......... , ............... 110

~I0. The P.Hall algebra .......................... 115 911. The

character values of

GL(n,Fq)

at unipotent

elements ........ . . . . . . . . . . . . . . . . . . . . . . . . . . .

128

912. Degenerate Gelfand-Graev modules ............ 138

IV

§ 13. R e p r e s e n t a t i o n s

of general

and the branching

rule

.................

A p p e n d i x 1.

Elements

A p p e n d i x 2.

A combinatorial

proposition

Appendix

The composition

of functors

3.

References

of the Hopf algebra

theory

143

....

149

............

155

r

and

i .

...... . ...................................

Index of N o t a t i o n Index

affine groups

167 177

...................................

180

...............................................

182

Introduction In this work we develop a new unified approach to the representation theory of symmetric groups and general linear groups over finite fields. It gives an explanation of the well known non-formal statement that the symmetric group is "the general linear group over the (non-existent) one element field". This approach is based on the structural theory of a certain class of Hopf algebras. The original plan of this work was to apply the technique developed by J.N.Bernstein and the author for the investigation of representations of general linear groups over elds ( ~ 1 3 , L 2 J )

p-adic fi-

to the representation theory of the groups

GL(n,Fq). The main tool of f ~

L2j is the systematic use of

the functors

and the general theorem on

iv, e

and

rv, e

their composition ( LIJ , §§ 1,5). These functors generalize those of induction and restriction while the composition theores generalizes the Mackey theorem. The main results of ~I3 and E2] were obtained by the formal manipulations with these functors. This technique may be applied to the groups

GL(n,Fq)

without any difficulties. Moreover, the simplifications caused by the full reducibility of representations, allow one to obtain for finite groups the more complete results than for p-adic ones. For example, in E3 ~ there was described the restriction of representations of

GL(n,Fq) to the subgroup

Pn

consisting

of matrices with the last row

(O,O,...,O,1);

in

a description is still unknown. Unfortunately, in E3~ are not self-contained: ween the representations of ric groups

Sn

p-adic case such

the reasonings

they use the relationships bet-

GL(n, Fq)

and those of the symmet-

as well as the representation theory of the sym-

metric groups. In fact, the technique of LIJ , L2J can be applied directly to the representation theory of

Sn. This approach allows

one to obtain from the unified point of view the main classical results. It turns out that the most convenient language in our approach to the representation theory of

GL(n, Fq)

and

Sn

is

that of the Hopf algebra. Consider the sequence of the groups Gn

(n~/ 0), where

Sn. Let

R(Gn)

Gn

is either

GL(n, Fq)

(q

be the Grothendieck group of the category of

finite-dimensional

complex representations of

Gn, and

~ R(Gn). Functors of the form iv, e endow n ~0 structure of an algebra over 2 , while functors

R =

R

is fixed) or

R

with the

rv, e

make

into a coalgebra (in the case of symmetric groups one has the

usual functors of induction and restriction,

see 1.1 below). The

composition theorem transforms into the statement that

R

is

a Hopf algebra i.e. that the comultiplication

m :R--~R~R

a ring homomorphism.

in the case of

The ring structure on

R

is

symmetric g~oups was considered in classical works by A.Young while for

GL(n, Fq)

it was introduced by J.Green E ~

the Hopf algebra structure on tence for for

Sn

GL(n, Fq)

R

• Although

is also very natural, its exis-

was mentiomed only quite recently (see ~ S J ) it seems to be new.

while

The Hopf algebra

R

satisfies two extra axioms of positi-

vity and self-adjointness. The positivity means that multiplication and comultiplication

are induced by the operations on

ordinary representations, i.e. they take representations to representations (not only to virtual ones). The self-adJointness means that the functor

rv, e

is adjoint to

iv, 8

(a generali-

zation of Frobenius reciprocity). The formal definitions will be given in 1.4. The crucial observation for this all results on representations of the groups

work is that

Gn, obtained by

our method are in fact based only on these properties of the Hopf algebra

R. So the problem arises to develop the structural

theory of Hopf algebras satisfying the positivity and selfadjointness axioms (we call them PSH-algebras). This problem is completely solved in Chapter I of the present paper. Chapters II and III contain some applications of this theory. Before a detailed description of the contents of this paper let us make three remarks. I) If we put

Gn = GL(n, F), where

then the Hopf algebra structure on

R

F

is a p-adic field,

and its positivity re-

main valid while the self-adjointness fails. It would be very interesting to find a weakened form of the PSH-algebra axioms such that the corresponding structural theory includes the representation theory of the groups results of the present of

GL(n, Fq), have

work

GL(n, F). Note that many of

related to

the representations

p-adic analogues ( E2~, ~6J).

2) This work is closely connected with that of D.K.Faddeev

7~.

In ~7J , the representation theory of the groups

GL(n, Fq)

is treated

by an elementary method based on the

theorem on the intertwining

number of two induced representa-

tions. This theorem in the case of finite groups is equivalent to our composition

theorem,

so the method of D.K.Faddeev

is

similar to our one. Some of the arguments in Chapter I are similar to those of ~ 7 ] and this Chapter can be considered as an axiomatization

of [ 7 J .

3) This work is entirely elementary and contained.

practically

In Chapter I besides usual linear algebra, we use

only some general properties found i n ~ 8 ] ~ f o r

of Hopf algebras, which may be

convenience,

these properties are given in

Appendix I. In Chapters II and III only some basic facts the representations half of the

self-

of finite groups are needed;

book~9~is

on

the first

more than sufficient.

Now we describe the contents of this paper. As indicated above, Chapter I (§§ I-@) is devoted to the theory of PSH-algebras. In § I the axioms of PSH-algebras preliminary

are given and some

results are proved. First of all we consider the

model example of representations

of symmetric groups which

motivates the subsequent formal definitions. theory of PSH-algebras

The structural

is developed in §§ 2-4. In § 2

prove that every PSH-algebra naturally decomposes tensor product of "elementary"

we

into the

ones, namely of PSH-algebras

with only one irreducible primitive in § 3 we prove that an "elementary"

element

(Theorem 2.2).

PSH-algebra

R

is unique

up to an isomorphism and change of grading. Furthermore the Hopf algebra structure on

R

is described explicitly

(Theo-

rem 3.1). This universal PSH-algebra

R

is studied in detail

in §§ 3-4. It has some natural bases. Each of them is parametrized by partitions. We compute explicitly all transition matrices between these bases as well as the action of multiplication and comultiplication on them. Note that

R

can be realized

as the representation algebra of the symmetric groups so all results on

R

can be reformulated in terms of representations

of symmetric groups. It is useful to keep this in mind in the course of reading §§ 3,~, although

the applications to symmet-

ric groups are discussed separately in § 6. Chapters II and III contain various applications of the results oK Chapter I. Although the notion of a PSH-algebra is motivated by representations of the groups it has some another applications

Sn

and

GL(n, Fq),

(an usual advantage of the

axiomatic method! ). In this work two more applications are given, to symmetric polynomials and to representations of wreath products. I am sure that another applications are possible e.g. to topology and representations o~ real and complex groups, and hope to consider them in the future. In § 5 the with the algebra

algebra A

R

from §§ 3,~ is naturally identified

of symmetric polynomials in the countable

set of indeterminates over ~ interesting structures on results of §§ 3,4

. This realization induces some

A. As the

immediate corollaries of

we obtain new proofs of some classical re-

sults on symmetric polynomials e.g. the Aitken theorem, the Littlewood-Richardson rule and the Littlewood-Roe theorem. Section 5 is closely

connected with the paper of L.Geissin-

6 ger f 5 ] where the Hopf algebra structure on

A

and the property

of self-adJointness are discussed. Note that in L 5 J some interesting applications are given. It would be interesting to compare the present approach to the ring based on the theory of ~ -

A

with that of D.Knutson,

rings (see [10J ).

In § 6 the representations of symmetric groups are considered. In 6.3. we give a dictionary translating notions of Chapter I

to the language of the symmetric groups. This dictionary

allows one to derive from the results of §§ 3,~

all main clas-

sical results on the representations of the symmetric groups including the Branching rule and the Murnaghan-Nakayama and Frobenius

Character formulas. We conclude § 6 with the "Hook

formula" for dimensions of irreducible representations of

Sn

In § 7 we extend the results of § 6 from symmetric groups to wreath products. If duct

Sn[G]

with

G n, on which

G

is a finite group then the wreath pro-

is defined to be the semidirect product of Sn

acts via permutations of coordinates

(see 7.1 below), For example, if then of

Sn[G~

G

consists of 2 elements

is the hyperoctahedral group, i.e. the Weyl group

type

the group

Sn

Cn

(or

Bn). As in the case of symmetric groups,

R(S[G] ) = ~ j R(Sn[G] ) is naturally endowed with

the structure of a PSH-algebra. Its irreducible primitive elements correspond to irreducible representations of

G, so for

its investigation one must apply the decomposition theorem from § 2. We give the classification of irreducible represenrations of the groups

SnCG ]

and compute the character table

? of

Sn

in terms of that of

G. Note that the classification

of irreducible representations of

Sn~G J

and even more general

wreath products is due to W.Specht; for detailed exposition see E12~ . I hope the present approach is in some aspects more transparent. In Chapter III we apply the theory of PSH-algebras to the representation theory of general linear and affine groups over a finite field. In § 8 we define the functors

iv, @

and

rv, e

and obtain their main properties. In § 9 the classification of irreducible representations of the groups

GL(n, Fq)

of cuspidal ones is given. Fix a finite field G n = GL(n, Fq) (n = O,J,2,...). Set functors of the form

iv, ~

and

Fq

in terms

and set

R(q) = ~ R(Gn) ; the n~O

rv, e

endow

R(q)

with the

structure of a PBH-algebra (9.1). The results of § 2 imply that

R(q) decomposes into the tensor product of subalgebras

R(~), where

f

groups

(9.2, 9.3). Note that the important notion of

Gn

ranges over cuspidal representations of the

a cuspidal representation appears very naturally in our approach. According to the results of § 3, each of algebras is isomorphic (up to change of grading) to the algebra

R~) R(S)

of representations of the symmetric groups. This result was proved by D.K.Faddeev ~7~ by a similar method. There are two ways to identify

R(S)

with

R(f)

as PSH-algebras. To

choose

one of them we use the results by I.M.Gelfand and ~ I . G r a ev EI3J and S.I.Gelfand ~ J tent subgroup of

Gn

and

. Let ~

Un

be a maximal unipo-

a nondegenerate character of

Un. For each representation S~

of

~

denote by

the

dimension of the subspace of vectors

v

~ ( u ) v = ~ (u).v

is called nondegenerate

if ~ ( ~ )

~ 0

for all

U~Un;

~

in S~

~(SU)

such that

and degenerate otherwise. I.M.Gelfand and

M.I.Graev proved that ~ & O ) sentation ~

of

Gn

~

I

for each irreducible repre-

while S°I.Gelfand proved that all cus-

pidal representations are nondegenerate. We determine the isomorphism of

R(S)

and

R(~)

uniquely by the requirement that

it takes the identity representation of representation in

Sn

to a degenerate

R(~), Note that in § 11 we give independent

proofs of theorems of I.M.Gelfand and M.I.Graev and S.I.Gelfand. They make our approach to the representations of

Gn

entirely

self-contained. In § 10 the P.Hall algebra ~

is considered. By defini-

tion, ~ =

~ ~ where ~ is the space of complexn~O ~ ' valued class functions on Gn supported on the unipotent elements. One has the projection each representation of

Gn

p:R(q) "---~ ~

the restriction of its character

to unipotent elements. We endow a Hopf algebra over

C

R(C) C

tations of the groups

~

such that

morphism. We prove that the subalgebra

R(q)

assigning to

with the structure of p

becomes a Hopf algebra

restriction of

p

to the Hopf

generated by the identity represen-

Gn, gives the isomorphism ~ ( L I ~ ~ ,

(Theorem 10.3). By this isomorphism we identify ~ R~C

where

R

is the universal PSH-algebra from

This allows one to obtain very simply

with §§ 3,$.

a lot of results o n e ;

in particular, the Green polynomials arise naturally, and we obtain their main properties.

At the beginning of 9 11 the theorems of I.M.Gelfand and M.I.Graev and SoI.Gelfand are proved by means of the technique of 9 10. Then we obtain the J.Green formula for the values of irreducible characters of

Gn

at unipotent elements. In our

terms, the problem is to compute explicitly the morphism p:R(q)----~

; it is done in Theorem 11.7. As a corollary,

we compute very simply dimensions of irreducible representations of

Gn

ments.

and

their character values at regular unipotent ele-

(Proposition 11.10)~we give also a very simple proof

of the Macdonald conjecture for

GL(n) (Prop.t1.11). It would

be very interesting to obtain the complete J.Green Character Formu;a by methods of this work. All results of 99 1Ot11 are well-known

(see E 4 ] ,~5~),

but I hope that the present approach makes them considerably more transparent.

The conclusive §§ 12 and 13 contain more

fresh results. In 9 12 we consider the representations of from various one-dimensional representations unipotent subgroup

Un

Gn

induced

of the maximal

(we call them degenerate Gelfand-

Graev modules). I.M.Gelfand and M.I.Graev in b 3 J every irreducible representation of

Gn

proved that

can be embedded in

one of these modulas. We obtain the more precise result

com-

puting decomposition of these modules into irreducible components (Theorem 12.1). As a corollary, we construct for each irreducible representation

~

fand-Graev module containing ~

of

the degenerate Gel-

with multiplicity I (Proposi-

tion 12.5); this realization of ~O nerate Whittaker model for

Gn

is an analogue of a dege-

p-adic groups, obtained in F21

10 As an application of this realization, we prove that the Schur index of each irreducihle representation of position 12.6); for char Fq ~ 2

Gn

equals 1 (Pro-

this was proved by Z.OhmoriL16 J

in a considerably more complicated way. In § 13 the relationships between the representations of the groups

Gn

and those of the general affine groups

above) are considered. The group direct product of

Gn_ I

Pn

Pn

(see

decomposes into the semi-

with the abelian normal subgroup

The classification of irreducible representations of

Pn

V n. is

easily derived from the general representation theory of such products ( [9 3 , 9.2). Irreducible representations of

Pn

hap-

pen to be in a natural one-to-one correspondence with irreducible representations of all groups

Gn_l, Gn_2,... , G o

(see

13.1, 13.2; another proof is due to D.K.Faddeev [17~ ). In terms of this classification, we compute explicitly the restriction of irreducible representations of of

Pn

to

Gn

to

Pn

and

Gn_ 1 (Theorem 13.5). It is interesting that these

restrictions always are multiplicity-free. As a corollary, we describe explicitly the restriction of irreducible representations of

Gn

to

Gn_ 1

(Corollary 13.8); this

restriction

was computed by E.Thoma L18J by a quite different method and in quite different terms. The results of § 12 seem to be new;

a

half of Theorem

13.5 was announced in ~3~ • Some technical results are collected in

3 Appendices. In

Appendix I we prove all general statements on Hopf algebras

11 needed in this work. In Appendix 2 a combinatorial proposition is proved, on which our proof of the Littlewood-Richardson Rule is based. I believe it is of independent interest. We use the beautiful reformulation of the Littlewood-Richardson in F19 J . Using it, the author

Rule given

recently has obtained the gene-

ralization of this rule ~20~ (see Remark A2,6 in the end of Appendix 2). In Appendix 3 the general theorem on the composition of functors

~, ~

and

rv,~ is stated and all its appli-

cations used in this work are collected together. It is a pleasure to express my deep gratitude to J.N.Bernstein who played

very important role in this work. He has

read carefully a number of original versions of this work and suggested a lot of valuable improvements

(the main ones are

referred in the body of the paper). It was J.N.Bernstein who suggested to me to apply the theory of PSH-algebras to general wreath products (in the original version only hyperoctahedral groups were considered),

degenerate Gelfand-Graev modules,

Schur index of representations

of

Gn

the

and to the new proof

of the Gelfand-Graev theorem. At the various stages of this work it was presented at the seminars of A.M.Vershik

(Leningrad State University)

and of

D.B.Fuchs (Moscow State University). I am grateful to A.M.Vershik, D.B.Fuchs and the participants of their seminars for their interest in this work.

Chapter I.

Structural

theory of PSH-algebras

§ I.

Definitions and first results

1,1.

First we discuss the basic model example concerning rep-

resentations

of symmetric groups (in more detail it will be dis-

cussed in § 6). After classical works of A.Young it becomes clear that one must study complex representations together,

of all these groups

taking into account their interaction.

tion is carried out by operations

This interac-

of induction and restriction. I

Let

Sn

be the permutation group of the set ~ l , n j

f

~

C

~

=

x

bedded in

Sn

as the stabilizer of the subset

[1,kJC

It allows us to construct from representations of

Si

the representation

product representation Conversely, the subgroup Sk x S1

of

~

Sn

3~

b,n J .

of

Sk

and

induced from the tensor

of the subgroup

one may restrict a representation S k x S1; since any irreducible

6

S k x Slt" S n. of

Sn

to

representation

is a tensor product of representations

of

Sk

of

and

S1,

one obtains a sum of such 9ensor products. Let

R(Sn)

be the Grothendieck group of the category of

finite dimensional

complex representations

of

free abelian group generated by the equivalence ducible representations morphism

of

Sn ;

it is a

classes of irre-

S n. The tensor product gives an iso-

R(S k x S1) = R ( S k ) ~ R ( S 1 )

and the operations of

induction and restriction described above give rise to the I linear maps ik,l: R(Sk)@R(SI)---mR(Sn)

and

rk, l: R(Sn)---~R(~) ~

R(S I)

13 It is convenient to consider all these maps together. Consider the graded group R(S o) = 2

R(S) = ~ R ( S n)

(here

S O = ~e} , so

)- Define graded group morphisms

m : R(S)@ R(S)

m R(S)

and

m

: R(S)--~ R ( S ) ~ R(S)

mI R(Sk)~R(Sl)

= ik, l'

We consider

as a multiplication; it makes

by

m

a graded algebra over making

R(S)

2

m*IR(Sn) = k trk, 1 k+l=n

R(S)

into

. Similarly, m " is a comultiplication

into a coalgebra. These structures happen to be

compatible and they make

R~into

a Hopf algebra over ~

(for

definition of a Hopf algebra see [ 8 J ; we shall recall it later). The fact that Hopf algebra axioms are valid is non-trivial; it expresses in a condensed form the essential properties of representations. The most essential is the statement that are compatible, i.e. that

m

: R(S)--~R(S)~R(S)

m

and

m

is a ring ho-

momorphism. Really, to prove this one has to compute the composition

m ~o m, i.e. the composition

rk ~ o i k , 1

for all

k,l,kt,11.

The computation is based on the Mackey Theorem on restriction of induced representations (for detail see Appendix 3). The Hopf algebra structure on information on representations of

R(S)

accumulates essential

Sn. But this structure doesn't

differ ordinary representations from virtual ones. Consider the r

2-

%

basis J L

in

tions of all groups

R(S)

consisting of irreducible representa-

Sn, and the basis ~ X

Ordinary representations of the groups to positive elements in

R(S)

~

in

Sn (Sk x S I)

(R(S)~R(S)),

R(S)@R(S). correspond

i.e. to non-negative

14

linear combinations of elements of ~

(AX~).

The impor -

rant property that induction and restriction take representations to representations, means that multiplication and comultiplication are positive i.e. take positive elements to positive ones. By Frobenius reciprocity the functors of induction and restriction are adjoint to each other. To express this fundamental fact in terms of the group bilinear form

< ) >

on

R(S) R(S)

we consider the I

-valued

such that all subgroups

are mutually orthogonal and for representations .]~) ~ same

Sn

R(S n) of the

we have



= dim H O m s n ( ~ ) ~ ).

The Schur Lemma implies that J ~

is an orthonormal basis of

R(S). Clearly, the inner product < ~ > has J L ~ J L

on

R(S)~R(S)

which

as an orthonormal basis, has the same meaning in

terms of representations. Therefore the Frobenius reciprocity means that operators i.e. that

~ x,m(y)>=

m

and

m

are adjoint to each other

~m*(x),y>

for

x ~ R ( S ) , y ~ R(S)~

@ R(s). 1.2. Now

we give formal definitions motivated by the pre-

ceding example. A trivialized group (briefly T-group____ ) is a free ~-module consider ~

R

with a distinguished ~ - b a s i s as a T-group w i t h _ ~ ( ~ )

of each family of T-groups and the

=~I}

A

= --~(R). We

. The direct sum

tensor product of a finite

family of T-groups become themselves T-groups via

15

n

In particular, any sum

T-group

of T-groups ~ , ~

T-group

R

,

R ~

n

decomposes into the direct e ~(R).

A T-subgroup of a

is any subgroup of the form ~

is a subset of

_~(R).

~'~

, whereA'

~Oe~'

Put R+ : ~ ~ D ~ elements of

R+

I m~o >~ O ~

;

are called positive. We write

x ~ y

if

x - y ~ R +. A homomorphism between two T-groups is called positiv___~e(or T-group morphism) if it takes positive elements to positive ones. For each T-group ~>

on

R

R

define the ~ - v a l u e d

bilinear form

by

= ~O

for

The form ~ ~ >

~e

is symmetric, nondegenerate and positively de-

fined; we call it an inner product on

R. We will freely use

geometrical terminology; e.g. elements of A

may be charac-

terized as positive elements of length I. Elements of A T-group such that

(R)

R. If ~ m~o >

- - ~ D ~ 0



>0

E R+

then the e l e m e n t s ~ < ) E A

are called irreducible

clearly the condition or as

are called irreducible elements of a

m.o> 0

constituents o f ~ ,

can be written down as ~ . ~

16 1.3. Now we recall the terminology on Hopf algebras. Let

K

is a graded

be a commutative ring with unit. A Hopf algebra over K K-module

m : R @ R--~R on),

R

=n~oRn~. with

(multiplication), m

e : K----~R (unit)

and

the following six axioms

~)RI)

(R~ R

: R--~R~R

(comultiplicati-

: R---~K (counit) satisfying

(G), (A), (U), (A'), (U*) and (H). m, m , e

(G) (Grading). Each of graded modules

e

K-module morphisms

and

K

and

e

are graded by

and K = Ko). (A) (Associativity). The multiplication (U) (Unit). The element

e(1)~ Ro

The axioms (A*) and (U*)

is a morphism of (R@R) n = ~ ( R k @ k+l=n m

is associative.

is unit of the ring

R.

are the associativity of comultip-

lication (= coassociativity) and the property of counit. In general, if (X) tain diagram

is a property expressing the commutativity of a cerD

constructed by means of morphisms

we write (X*) for the

e

by

and

e,

property of commutativity of the diagram

obtained by reversing all arrows of D and replacing and

m

m

by

m

e . For example, the axiom (U*) m~ans that the diag-

ram R@R idle*/!j

* ~ ~ d

R

R@ K~ f-~< commutes. (H) (Hopf Axiom). The operator homomorphism (the multiplication in

by (X@y)' (x~y °)

= xx e @

yyJ).

m

: R-----~R~R R~R

is a ring

is defined as usual

17 Removing axioms

(A) and (A*), one obtains the definition of

a quasi - Hopf algebra.

A (quasi) - Hopf algebra

R

is called

connected if S

(Con) Operators

e : ~ R

and

o

e

: Ro

> K

are mutually

inverse isomorphisms. A Hopf algebra

R

is called commutative

(Com) The multiplication cocommutative

m

is commutative;

(or with commutative

axiom (Com*) h o l d s , i . e .

if it is called

comultiplication)

if the

i f t~he diagram

,I,

R - ~m

( d (x~y)

R~)Od

= y@x)

is commutative. I.~.

Now we define the main subject of this work~a positive

self-adjoint Hopf algebra (briefly PSH-algebra). A (quasi)- Hopf algebra

R

over ~

is called positive

if it satisfies axioms (T) and (P). (T) Each

Rn

and hence the whole

1.2); in other words, R

is a free ~ -

R

is a T-group

(see

module with a distin-

f ~

guished basis - ~ L

consisting of homogeneous

(P) (Positivity). positive

All morphisms

elements.

m, m , e, and

e

are

(see 1.2).

A positive

(quasi)-Hopf

algebra is called ~

if S

(S) (Self-adjointness). and

Operators

m

and

m

(resp,

e

e*) are adjoint to each other with respect to inner pro-

ducts ~ , structure

~

on

R, R ~

(see 1.2).

R, and

~

, induced by a T-grou~

18 A PSH-algebra is a connected positive self-adjoint Hopf algebra over ~

.

1.5. Let us introduce some notation. We denote the unit of R

i.e. the element

e(1)~R o

simply by 1. By axioms (Con)

and (P), I is an irreducible element of

R, and

write

n. Axioms (G), (Con),

xy

for

m(x~y)

and set

I =~R

Ro = Z '

1. We

vt ~ U

and (U*) imply that for

m'(x) where

=

X@I

m (x)~ I @ I .

x ~ I:

+ I@X

+

(x),

m ~ +

An element

xE I

is called primitive if

+

m * (x)

:

x~1

+ 1@x,

i.e.

of primitive elements in

m ~

(x) = O. Denote by

R. + Set

subgroup generated by products

12 = m(I~I), xy, X~Rk,

P

the subgroup 12

i.e.

Y~R1,

is the

k,l~O.

1.6. Proposition. Any positive self-adjoint quasi-Hopf algebra over Z

is a PSH-algebra, i.e. the associativity of mul-

tiplication and comultiplication follows from other axioms of PSH-algebra. Moreover, any PSH-algebra is commutative and cocommutative. Proof. Evidently,

each of properties (A *) and (Com') follows

from (S) and the corresponding property'~ithout asterisk", i.e. (A) and (Com). Now apply the following. 1.7. Lemma. Axioms (G), (U), (U*), (Con), (T), and (S) imply that

P

is the orthogonal complement of

respect to the inner product ~ ~

12

in

R~R

with

,

Proof pf the lemma. According to (T), all subgroups in

I

are mutually orthogonal. Hence,

by (S),

Rk~R 1

19

~x,

m(y)>

= ~m*(x),y>

= ~ m (x),y ~

for

x~l,

y~l@l.

+

It

follows

form

that

x N~. t

m(y), Y E I ~ I ,

is

iff

orthogonal

to all

m*(x) = O, i.e.

elements

of the

xEP.

+

Q.E.D.

Proposition 1.6. follows at once from this Lemma and the general theory of Hopf algebras (see Appendix I, Lemma AI.3). 1.8o Remark. Proposition 1.6 has some applications. For example, the commutativity of the algebra

R(S) (see 1.1) is an

essential (although rather simple) property of represemtations of the symmetric groups. Another less trivial a~plication concerning representations of the groups

GL(n, Fq)

will be given

later in § 9 (see 9.1). 1.9. Let x

: R---~ R

R

be a PSH-algebra. For any

x~R

denote by

the operator adjoint to that of multiplication by

s

X

i.e.

x

is defined by

x'(y),z~

= ~y,xz >

,

y,z~R

.

(according to Part (b) of the next proposition, defined). The operators

x

(a) Let

in particular, x*(Rn) = 0 with

x~ for

their main properties.

R k. Then

x (Rn)CRn_ k

is an inner product with

(b) The operator

x

for

n~O~

n ( k. Identifying naturally

(see 1.3 (Con)) we see that the linear form

Rk--~R o = ~

is well-

will be the main tool in our study

of PSH-algebras. Let us summarize Proposition.

x

: R----~R

x

Ro

:

x (denote it by ~ x ~

equals the composition

).

20

(c) For any

x,y~ R (xy)

= y o

R

In particular, since form

x

x

is commutative, all operators of the

commute with each other. S

(d) If

x~

R÷

then the operator

(e) If

x , y , z ~ R, and

m (x) =

x

is positive (see 1.2).

ai@b

i

then

i

~a*

x * (yz) =

i (f) If

~

~

R

i

(y) b* (z) i

is primitive (see 1.5) then ~ :

is a derivation of the ring ff (yz) :

(g) If ~ x (~)

~

R, i.e.

y):z + y,

Rn

R ~ ~R

(z)

is primitive,

0 ~ k ~ n, and

x ~

Rk

then

: o.

Proof. Parts (a)-(d) follow at once from definitions while (f) is a particuSar definition of

x

case of (e). The statement (g) follows from and Lemma 1.7. It remains to prove (e).

Clearly, the operator tiplication by

a~b,

(S) and (H), we have

<x

R~

is

R-----~R~R e

a ~

adjoint to the mul-

b • Using this fact and axioms

(u~R):

(YZ),U>==<m(y~z),xu>=~jy~)z,m'(xu)

=~y@z, =

m'(x),m'(u)>=~ (y) @ bi(z) , m ' ( u ) > =

=

bi(z) , u > . &

y•z, ~m

=

(ai@bi).m'(u) ~ i(Y)~b~(z)

,

o>=

21 Since

u~ R

can be chosen arbitrarily, Part (e) follows. Q.E.D.

§ 2. The decomposition theorem 2.1. In this chapter we shall develop a complete structural theory of PSH-algebras. We shall see that

this theory is quite similar

to the well-kno,~n theory of Hopf algebras with commutative multip lication and comultiplication over a field Recall that any such Hopf algebra

~

K

of characteristic~

is naturally isomorphic

to the symmetric algebra of the space of its primitive elements

(see L8

, G 5

or Appendix ~ below). In other words:

(I) The Hopf algebra

~

decomposes into the tensor pro-

duct

where each Hopf algebra

~

(2) The Hopf algebra ~ is isomorphic to the algebra indeterminate m*(x) = x @ 1

x, where + 1•x.

has only one p r i m i t i v e element. with only one primitive element K~x]

deg x z k

Thus, ~

of polynomials in and

x

one

is primitive, i.e.

is essentially unique.

In this section we shall prove the analogue of (I) for PSHalgebras. The role of "elementary" algebras

~

will be played

by PSH-algebras with only one irreducible primitive element. 2.2. Let (Re( I

~ ~ A) be a family of PSH-algebras. Define

the tensor product

o(EA

to be an inductive limit of the finite ~ensor products

~

R~

22 ranges over finite subsets of A). Clearly,

(S

R

is a PSH-algebra

with the set of irreducible elements

s each

R~

dab

is naturally embedded into

R. Our first main result

on PSH-algebras is the following. Decomposition Theorem. Any PSH-algebra

R

decomposes into

the tensor product of PSH-algebras with only one irreducible primitive element. ~ore precisely, let ~O

irreducihleprimitive A(~)

Then

= ~O~j~/

R(~)

~ s

a

elements in

=J~ ~

P

be the set of

R. For any ~ 6

< ~ , ~ 0

for some

PSH-subalgebra in

R

n>~0/

we

set

and

with the set of irreducib-

le elements ~

(~), ~ is the unique irreducible primitive element

of

R

R(~), and

as a PSH-algebra is a tensor p r o d u e t ^ ~ R(~).

This is proved in 2.3-2.7, Proposition. Let

2.3.

. ,

<X unless

r

=

s

~4)::,, ~ ) ~ 4 ' .,~ ,~&' ~ ~

)

,

,

. Then

,3Z'> = 0

and sequences

(¢,

~* "

-)

are equal up to a permutation. Proof. Apply 1.9 (f):

~=4

,:

+4 ' ' "

and

= 0

, equals~

, and exactly

#

k

of ~

equal F4

then

cJ

The proof is completed by induction on

r. Q.e.D.

2.~. Remark. Our proof of Proposition 2.3. doesn't use the irreducibility of the elemen~

~L

and ~jl and is based only gJ

on the fact that distinct elements among them are primitive and mutually orthogonal. This will be used later. 2.5. Denote hy ~ , ~ ~

S(~; ~ t

the semigroup of functions

with finite support. For any ~ S

; Z'F~

set

The axiom (P) implies that all ~ Denote b y ~ i Z ( ~ )

are in R+ (see 1.2, 1.~). ) the set of irreducible constituents of ~

and put

Proposition. (a) ~he set J ~ =J~(R)

(b) The group i.e.

R =@ R ( ~ ) ,

R

is a disjoint o~ion

is graded by the semigroup ~ S(~

~

)

. This grading i ~ compatible

with the Hopf algebra structure, i.e.

24

t">' ~ Proof.

I

tl

-

Ca)The disjointness of ~ J L ( ~ ) f o l l o w s

at

once

from 2.3. We must verify that for any ~ K J L

there e x i s t s ~ i o ~ J (see 1.2). The statement is ~rivial if either ~ ~ or ~ = q.

So we suppose that ~

~

I (see 1.5) and

~

is not primitive.

Lemma 1.7 implies that there exist two elements

~

of positive degree such that Using induction on d e g ~

~(~I,~A.~II

C~3I) ~C)II~'~

•

, one may assume that there exist

t

j~l~/CO and ,~:,, ~ d~,L.O'~Hence

Q.E.O. (b) The equality

R = ~

R(~)

follows at once from (a)

while the inclusion m(R(~l)~ R(~O~) C R ( ~ t + ~ u) follows from the proof of (a). The corresponding property of comultiplication follows from the self-adjointness of

R.

Q.E.D. 2.6. Proposition. Let

(32)~/ ~_.~F_.~(~ ~E ''t'-)

have dis-

joint supports. Then multiplication

establishes an isomorphism of T-groups. In other words, the element s ~

~a

where

DI^

~)~.~-J/-[L{J~ ~ ,I

,0//,,,I) (.L) I~.JL~f,t/j

are irre-

25 ducible, mutually distinct, and any irreducible element in

Proof. ~et ~ , ~ 2

~

must prove that ~ , ~ : unless ~ 4 : C ~ 2 ) ~ : ~ .

.k_c

~,' ~ '

~

_S7_~'7>')

. We

is irreducible and t h a t ~ 4 ' ~ ~ 2 # It suffices to verify the equality

Apply axioms (S) and (H):

According to 2.5 (b), m*(~) nents belonging to

R(~)~R(~4)

~ +~--~ R(~+~)

~ R(~+~45

(mi(t~)~)) is a sum of compo(R(~t)~R(~4~) , where % +

The product of such components lies in (see 2.5 (b)). Therefore, it is orthogonal

Since supp~ ~ supp ~#= J~ , the equations

~'+ ~'_- $~', 7

,

have the unique solution

The corresponding components of respectively

~4 @ I

and

m * (~4) and

I @ ~41.Hence

m * (~) equal

26 as desired. It remains to prove that any element of A 1

the form ~ O ~

where

(~+~

iS of

(,~--A(~), (~ 6.../ZL/~ J, Let t

be the decompositions of

j'~

and I ~ !

"

l

J

sums of irredu-

into

cible elements. Then

is the appropriate decomposition of

JJ~99+~l , so the elements

exhaust all irreducible constituents of

K T ~.~!

,

Q.E.D. 2.7. Proof of Theorem 2.2. Under the notation of 2.5, the subspace

where ~

R(~)c R

o

(see 2.2) equals

S(~) I)

+

is the characteristic function of the sub-

set ~ p ~ d ~ - -~ ~ . By 2.5 (b)

R(~),LL is a PSH-subalgebra of

Clearly, any ~ ~ S(~') ~ )

R .

has the form

p g' Applying several times Proposition 2.6, we see that the multiplication establishes an isomorphism of T-groups m :

Since that

@ R(nf ~R(~O).

R =~R(~), m

~

S(~ ;

(see 2.5 (b)), it follows

establishes an isomorphism of PSH-algebras

27 The last statement to be proved is that ~ irreducible primitive element in

R~).

is the unique

This follows at once

from Proposition 2.3.

Q.E.D. § 3.

Universal PSH-algebra:

the uniqueness theorem and

the Hopf algebra structure 3.1. In this section we prove for PSH-algebras an analogue of the statement (2) from 2.1. We show that a PSH-algebra with only one irreducible primitive element is essentially unique, and study in detail its Hopf algebra structure. Fix to the end of Chapter 1 a PSH-algebra

R

with the

unique irreducible primitive element ~

. According to Theorem

2.2, any irreducible element

is an irreducible cons-

tituent of then

n

~n

for some

~ R

n ~ - - ~"r. Z In particular,

is divisible by d e g ~

grading on

R

without loss of generality that d e g f

Theorem.

(b) For any and

Yn

in

x2 n~O Rn

x2(Y n) = O,

and

, and assume

= 1. Thus,

are summarized in the following.

(a) The element ~ 2

ducible elements

xn

R

Rn ~ 0

. Therefore, we can change the

, dividing all degrees by d e g ~

Our main results on

if

is a sum of two distinct irre-

Y2"

there exist unique irreducible elements such that Y2(Xn) = 0

(see 1.9).

28 (c) If

O~ k ~ n

then

X k (Xn) = Xn_ k , If ~ O ~ A

Yk (Yn) = Yn-k

is distinct from

the similar holds for

xo,xl,...,x n

ad (x n) = o;

Yn" In particular, for

Yk (Xn) = xk (Yn) = 0 (d) For

then

n~ I

k ~ 2.

we have n

m

(x n) = ~

x k ~ Xn_ k

k=O n

m

(Yn) = ~ k=O

(e) The ring

Yk ~ y n - k

R is a ring

~I,x2,...~

of polyno-

mials in indeterminates

(Xn/ n ~ 1 ) ; similarly, R = l ~ 1 , y 2 . . . J ,

The elements

( y n ) satisfy the relations

( X n ) and

n

(n~/1)

k_~ (-1)k xk Yn-k = 0 (f) The algebra automorphism

t(Xn) = Yn

R

has a unique non-trivial PSH-algebra

t. One has

and

t(y n) = x n

(g) Any PSH-algebra

for

n~1.

R I with the unique irreducible pri-

mitive e l e m e n t ~ i of degree I, is isomorphic to

R

as a PSH-

algebra; according to (f), there exist exactly two PSH-algebra isomorphism between

R

and

RS .

Theorem is proved in 3.2-3.1%.

29 3.2. Proof of 3.1 (a). Apply 1.9 (a), (f):

= = ~xn, (A.)6 > = (similarly for

yn ). Therefore our assertion follows from 3.1(c).

32 3.6. Before proving the remaining assertions of Theorem 3.1j we derive some consequences from the already proven ones. It is convenient to define

Xk = Yk = 0 (we recall that

for

xk

and

Yk

for all

k~by

k~O

Xo = Yo = I, x I = Yl m r '

and

Xk ~ Yk

for

k ~ 2 ) . According to 3.1 (d) and 1.9 (e),one has

' - *X n

(ab)

=

xk (a) x I (b), k+l=n

y

(ab) =

Yk (a) Yl (b) k+l=n

(a,b~R). Define the operators •

'

Y

and

=

k

Y

on

R by

map R n

into

~ *

,

Xk

X

X

Yk

;

k

clearly, they are well-defined and

~

Rk

Ogk~n

cording to (~), X

and

Y

Ac"

are ring homomorphisms. They will

play a crucial role in the sequel. g

~

We define the linear forms ~ setting for

and ~

from

R

to Z ,

a~R n .

(a) = xn (a), C C Clearly, ~ and O~

:

(a).

are positive; by (~)they are multiplicati-

re, i.e. are ring homomorphisms. Furthermore, fy the "normalization condition"

~

and ~ d

satis-

33 The following proposition which will be useful in applications, shows that ~D~ and g~ may be characterized by these properties. Proposition. Let ~

: R~ ~

and normalized form (i.e. ~ ¢ )

Proof. Since ~

be a positive, multiplicative = I). Then ~ equals either gz~

is multiplicative and normalized, one has

~(~2)= I. Therefore positivity of S (x2) = 0 ~A~?~n

or

implies that either

S (Y2) = O. Suppose that

then

Y2 (~)) ~ 0

S (Y2) = O. If

hence ~ < ~ Z " ~ - 2 ,

Therefore,

v t - Z th

_-o.

Since ~ ( ~ " ~ stituents of ~ ~ ~

= I, and ~ )

= 0

for all irreducible con-

except Xn, one has

. Similarly, if

) = I. We see that

~ (x2) = 0 then

----"~ ° Q.E.D.

3.7. Now we introduce the notation and

terminology rela-

tive to partitions and Young diagrams. Denote by ~ of families

(1I,. .. ,lr) where

ii ~ ~

the set

~ ; two families, dif-

fering by an order or the number of zeros, are identified, i.e. determine the same element of ~

. Elements of ~

are called

partitions. They will be denoted by Greek letters ~ , ~

Y

etc.

while their parts by appropriate Latin letters e.g. ~ ~_(ld,.. lr). For

k~1

denote by

which are equal to

(Iri, 2r2,...)

rk = rk(~) the number of parts of A

k; we shall sometimes write ~

as

. Put

=

Z

i.e. r(~) is the number of non-zero parts of ~

. For each

34

X=

set

(1I, ..., I r)

Any ~

can be written as

~ l r ; put also

Is = 0

for

(ll,...,1 r)

(it,12,...,1 N)

will be called a canonical form of ~ :

11~12~.~ "

s ~ r . The sequence (11,12,...)

as well as each its initial segment N~r(~)

where

with

(the notation is

(11,12,..)).

A Young diagram is a finite subset of

~x

~

with each point (i.j) all points (i¥,j I) such that (recall that ~

containing iI~ i , jI~ j

as usual stands for the set of positive integers);

Assign to a partition ~

~

with

c.f.(~) = (11,12,...) the

Young diagram

Clearly we obtain the bijection between (J-Pand rr~ the set of all Young diagrams. We identify this set with ~JJvia this bijection, and use the same notation for a partition and the corresponding Young diagram. For example, we write (o)E

~

for the partition

D.

When we show diagrams graphically, we assume the i-axis to go downwards while the j-axis to the right (as if

(i, j) were a

matrix index). We shall use the corresponding geometrical terminology. The transposition = (j,i). Evidently, acts on ~ .

t

t : ~X~'---~-~x~

acts by (i, j)t =

takes Young diagram to Young diagrams i.e

Furthermore,

t2 = id

i.e. ~ A ~ ) ~ = A

for all A ~

35 If

A

= (1'''''ir) ~ J

we define the partition ~

and

ii ~ 0

for

i = 1,...,r

then

(JDby

X~- = (1I-I, 12-I,..., lr-1); put a l s o ~ ~_ from A

~

. In terms of Young diagrams, ~ v- is obtained

by removing the first column (and shifting by I to the

left). Set also

A','

t

this means that

)

is obtained from

~

by removing the first

row and shifting by 1 upwards. The following two assertions follow at once from definitions.

~ ) c.f.

~A~) : ( r ~ ) , r ~ h

rC,~" )

(2) I f c.f. ~ ) = ( 1 1 , 1 2 , 1 3 , . . . )

then c.f.(~ ~) = ( 1 2 , 1 3 . . . ) .

Example. ~ = (I,~,2,2) = (1,22,~); c.f.(~) = (4,2,2,1,0,0...)

~--

xx xx x

3.8. Let us return to the proof of Theorem 3.1. For any = (1I,..., I r) x~

=

=

set

Xll Xl2... Xlr

in particular, x ~ that for any

~

= y~

n>~o

,

y~

=

Yll ... Yl r

;

= I. The first assertion~ in 3,1(e) means

tho monomials (x~ I A ~ ~ )

form a basis of

R n. We apply the following general Lemma. Let

e1,...,ep

be elements of an arbitrary T-group R.

36 Then

el,...,e p

is a basis of a certain T-subgroup

in

R

(see

1.2) if and only if the Gram determinant d e t ( < ei, ej >)i, j=1 , ... ,p equals 1.

Rl ~

Proof. Suppose that

el,...,e p

form a basis of a ~-subgroup

R. This means that

el,...,e p

are linearly independent

where~4,~,~

)t~are

distinct irreducible

that the transition matrix el,...,ep} Therefore

is invertible,

Conversely,

between the bases f & £ ) 4 ) , t , ~ a n d

and

A

and

A -I

are both integral.

) = det(A.A t) : (detA) 2 = 1.

let

det(<ei,e j>)

linearly independent. el,...,e p




For any

A

elements of

and

, and

x 6

R

Let

R~

= 1. Clearly,

el,...,e p

are

R ! be the subgroup generated by

the orthogonal

complement

of

R p in

R.

consider the system of linear equations

a~ei,ej~ = ~ , e j >

, j = 1,2,...,p,

~:4 with unknowns

al,...,~.

det (< e i , e j > )

The determinant = 1

so there exists the unique solution ers. This means that

R

of this system equals

(ai) , and all

ai

are integ-

decomposes into the direct sum

R = R I ~ R ~. Let now

~

~

R

be an irreducible

element. One has o O = ~ 2 ~

37 where ~ l

R,I

~

~ E : R J'. Therefore

:

If follows that one o f ~ in

R # or in

! and~--isl

R -~ We see that

zero, i . e . ~ O

R j and

R~are

lies either

T-subgroups in

R.

Q.E.D. 3.9. Fix ~6

~0.

We apply Lemma 3.8 to ~ e l , . . . , e ~

cally higher than c . f ( ~ ) . e1,...,ep

form

Put

a Z-basis

in

/

~,, ~ p 1 ~ ebe all elements of ~ r,

. ~ore precisely, l e t ~

ordered in such a way that for

= [g~

i< j

c.f(~

ei = x ~

)

is lexicographi-

. We must prove that

R n. Let us derive this from the

equality (~)

det ( ~ e i , e j ~ )

: I

Indeed,according to Lemma 3.8, (~) implies that ~ e l , . . . , e p ~ is a basis of a certain T-subgroup

R I ~ R n . To prove that

it suffices to verify that no irreducible element ~ be orthogonal to all

~

R r- R n

Rn

can

e i. But this is clear since

It is easy to calculate the inner products ~ ei,e j ~

explicitly

(see 3.17 (c) below), but a direct computation of det ( ~ e i , e j ~ ) seems to be rather tedious. So, in order to prove (~) we apply the following trick due to J.N.Bernstein. Put f ei = y~,

(i = 1,...,p)

We shall derive

(~)

(a) The matrix

(~ei,

diagonal; in particular, (b) All (ei).

l

ej

.

from the following two statements. el j

~)

is triangular with 1's down the

its determinant equals I.

are integral linear combinations of the elements

38 Indeed, by (b) P l ej --~-

aij ei

, where

aij~Z

i=1

Clearly, the matrix fix

(i ei,ej>)

is a product of the Gram mat-

( ~ e i 'e j/ > )

and the matrix

(aij). According to (a), one

has: 1 = det (<ei, e~>) = det (<ei, ej>).det

(aij) .

Since the factors on the right-hand side are both integers, each of them equals Z1. Since the Gram determinant is always non-negative, we see that

det ( <ei,ej> ) = 1, as desired.

Summarizing, we see that the equality R =

~[Xl,X2,... 3

is derived from the assertions (a) and (b). The assertion (a) is proved in 3.10 while (b) in 3.11-3.12. 3.10. The assertion 3.9 (a) follows at once from the next. Proposition. If ~ ) , ., ~ ,~~ ~

and c.f. (~) is lexicograp-

hically higher than c . f . [ ~ ~]

then

y~ (xA ) = o . Furthermore,

Y~e

Proof. Let

(xA ) = 1

for all /k &

r(~) = r. The formulas 3.6 (~) and }.1 {c)

imply that Yr (x~) = x ~ -

and

@

Ym (x~) = O

for

m ~ r

(see 3-7).

39 Now apply 3.7 (I). Q.E.D. 3.11. The assertion 3.9 (b) follows easily from the relations 3,1 (e) between

(Xn)

and

(yn). Indeed,

these formulas

can be rewritten as Yn =

(-I) k-lXkYn-k

(n>jl)

k>~1 Using induction on

n

y~ ) are polynomials

one obtaims that all in

xl,x2,..,

Yn

(and hence all

with integral coefficients,

as desired. To prove the relations 3.1 (e) we shall use the important notion of the conjugation of a Hopf algebra

(see L 8 ~ ,

§ 8, or

Appendix I, Proposition AI.6 below). In the theory of PSH-algebras it is convenient the morphism tiple

to replace the conjugation

t : R--~R

differing from T on R n

T : R--~R

by

by the mul-

(-I) n. Proposition.

(a) The map

t

automorphism of the PSH-algebra (b) For all

n ~Z

t(Xn) : yn, Proof. automorphism

(a) Clearly,

is a (positive)

invQlutive

R.

we hav e t(y n) = x n t

• T

as well as

is an involutive

of the Hopf algebra

R

(see Appendix I, Prop.

AI.6). It remains to verify that

t

takes irreducible

to irreducible an isometry of

ones. Let us prove that R. We recall that

by the property of commutativity

T

T

(and hence

elements t)

is

is uniquely determined

of the diagram:

40 R

m ~ R ~ R - ~ d ~ T~ R ~ R ~ R \

(D)

~

/

Consider the diagram (D*) obtained by passing to adjoint operators in (D). The axiom (S) implies that (D=) is obtained from (D) by replacing since each

Rn

T

by the adjoint operator

( T

exists

has finite rank). It follows that T

Since

T

=T

.

T 2 = id, we see that T * =T=

hence

T

is an isometry, as desired.

Clearly, an e l e m e n t ~ ~b~C~)~= I

and

T(~) = - ~ fore, if ~

T-1

is irreducible if and only if

~00~ ~ > > 0 .

hence ~

~

Rn

By definition of

t ( ~ ) = ~ , so

t(~

= ~D

for

T

we have n ~ 0 . There-

is irreducible then

and

4t(~,f hence

> =zt(w),t(f3 > :4~,.n>P

>0

t(~O) is irreducible, as desired. (b) By definition of

T, we have

m O ( i d @ T ) o m" (x2) = O. Apply 3.1 (d) : 0

: m~ (id@T) (1@x 2 + ~

+ x2~I) : T(x2) - ~ +

Therefore, 2

t(x2) = T(x2) = ~

- x2 = Y2"

x2.

41 From the fact that

t

is an isometry, we easily derive

that (~)

(a ~ R).

(t(a))* = to a*o t-d

Hence

(x ot) (xn) : (toy ) (xn) :o By 3.q (b) this implies that

t(Xn) = Yn

for

for all

n ~ O. n ~ O. Simi-

larly, t(y n) = x n. Q.E.D. 3.12. The end of the proof of 3.1 (e). By definition of m°(id@T)

o m" (xn) = O

for

T~

n~l.

Applying 3.1 (d) and 3.11 (b), one obtains

the formulas 3.1 (e).

We have already derived the assertion R = Z~Xl,X2,-..

~

from these formulas (see 3.11). The proof of the equality R = ~EYd,Y2,..- J is quite similar; another way is to apply the automorphism

t.

Q.E.D. 3.13. Proof of 3.1 (f). The existence of

t

3.11; it remains to prove the uniqueness. Let ~ an automorphism of the PSH-algebra suffices to prove that either (xn) = Yn

for all

is proved in : R---~R

be

R. According to 3.1 (e), it

~ ( x n) = x n

for all

n , or

n. But this assertion follows at once

from Proposition 3.6 applied to the form

~=~

o~. Q.E.D.

3.14. Proof of 3.1 (g). Write

R

I

as a polynomial algebra

,x2,... according to 3.1 (e). Consider the ring homomorp4 l hism ~ : R--~R' which sends x n to x n . By 3.1 (d), j is a Hopf algebra morphism. We must prove that j takes irreducible elements

~o irreducible

42 ones. It is easy to see that the formulas 3.1 (c) and 3.6 (~)

allow one to c o m p u t e

the inner products

explicitly(we will do this later in 3.17). This implies that

j

is an isometry. Now apply the arguments similar to those at the end of the proof of 3.11 (a). Q.E.D.

Theorem 3.1 is completely done. 3.15. Let us find all primitive elements of Proposition. (a) For any mitive element

Zn~ R n

n)1

= I

The subgroup of primitive elements in

RK = R ~ K

there exists the unique pri-

such that Zn,Xn>

(b) For each field

R.

K

•

Rn

equals

of characteristic

~,z n .

0 the Hopf algebra

equals K gz I,z2,... J .

Proof. (a). According to 1.7 and 3.1(e), an element is primitive if and only if it is orthogonal to all

xA

z~

Rn

except

xn. Clearly, any such element is proportional to the one with the least positive value of < Z,Xn>. det ( ~ x ~ ,

Since the Gram determinant

x ~ ) ~ j ~

equals I (see 3.9), this latter value is 1. Our assertion follows. (b). This follows immediately from (a) and the Theorem quoted in 2.1 (see also Appendix I, A1.1 and AI.2). Q.E.D.

43

3.16.

= (11,...,i r) ~

For any ~ :

Zll Zl 2

.- Zlr

where

put

;

put also

z~

= 1. By 3.15 (b), the elements

basis in

R~

= R~

~

ii ~

z~

form a ~

-

. Let us describe the relations bet-

ween this basis and each of the

bases

(x~

)

and

(y~). We

shall use the method of generating functions. Proposition. Let

R [[~]]

series in one indeterminate

X(}),

Y~),

over

ZO'~) of R~E~3

and

n~0 Then in

~

be the ring of formal power R. Define the elements

by

n~O

R~[~]J

n~O

one has and

Z(~) = d~

in X(~) = (d~

X~))/ X(~) .

(clearly these formulas allow one to write down the elements and

zA

in the basis (x~)).

Proof. The first equality is equivalent to the formulas 3.1 (e). Let us prove the second one. Denote temporarily by ~

Z

=

Zn+ I n>~O

the power series din

X (~) =

Since the constant term of

d

X(~)

/ X~)

X(~) equals

.

x o = I, the series X(~)

44 is invertible in Zn6R n

for

R[[~]]

hence

~(~)~ RC~J

1

. Clearly,

n~1. It remains to prove that all elements

are primitive and satisfy

< z%,Xn> = I

zn

(see 3.15 (a)).

We shall apply the following assertion which can be verified directly. Y

~ :~ ~ ponding morphism ~ [ ~ J ~ - ~ ~ J J o b t a i n e d

~'~ the corresby applying

(~) For any ring homomorphism

coefficientwise (it will be denoted also by ? ) commutes with the operator of logarithmic derivative. First we apply (~) to the homomorphism S

m

For any

in

U =ZUn

(R~R)LL~JJ

: R

~

~

......

~R~)R.

R[[~]]

denote by

UI (U2)the series

. Under this notation one can rewrite the asser-

tion 3.1 (d) as m

X ~ ) = XI({),X2(~)

.

Therefore, m

=

Z(~)

d

dl

= m

in X(

ln~X1(~),X2(r)~L_&.~_3

d = ...........in . m

X(

) =

d lnx I(~) +~lnx2(~) dE

z1(~) + z2(~) This means that all coefficients

Z'~n are primitive.

Now apply the assertion (~) to the homomorphism

45

~ -~I~--'--~Z

(see 3.6).

By definition, e

n~O

Therefore,

I~:~.~" d-

X(~)J d~ in[ ~X(~)~=

= d~ In (I-"~-~ = ¢~_~ )-I = ~0 This means that

(zn) = I

for

n~1,

~

.

i.e. ~Xn, Zn~:1,

as desired.

Q.E.D.

3.17. We conclude this section by computing the inner products between all elements

(x~),

(y~), and

pret these results consider the dual bases (z~)

(z~). To inter-

(x~),

(y~) and

defined by

x~

~

•

~cco ~o~ ~o ~ ~ e c~ o~ ~e basis of

R~

o~s ~

~ ~ f/~n~M/

. Clearly, the inner products of a vector with

all elements of certain basis are its coordinates with respect to the dual basis. Thus, we shall compute the transition matrices between each of the bases

(x~),

(y~)

and (z~) and each

of the dual bases. Proposition.

t(xA)

(a) For any ~ ~ ~

= YA ' t ( y A ) = xA

(see 3.7, 3.11).

one has

, t(zA) = (-~)~t-r(~}

zl

46 (b) For

n>.l one has

X zn = z n + 1, The operator

Y z n = z n + (-1) n-1

z n : R--~R

the generators

xk

is a derivation of

and

Yk

(see 3.6) R

and acts on

by

S

Zn (Xk) = X~-n (c) Let

~

= (ll,,..,1 r) and

titions with all the set of

' Zn (Yk) = (-1)n-1 Yk-n

Ii

r × s-

and

mj

~

Z +

their entries over the rows equal over columns are

= (ml,...,m s)

being non-zero.

matrices over

" be two par-

Denote by

M~

such that the sums of

ll,...,1 r

while the sums

m 1,...,m s . Then

while

~

x A, y~ >

M~'~

with entries

is equal to the number of matrises from 0

and

the number of matrices from

1. Furthermore~ < x ~ , z ~ N~'~,

>

equals

which have only one non-ze-

ro entry in each row, while

Finally, < h


=

)

YA ( x ~ )

and < z A , x ~ >

=

e

= zl (X~)). X1 ( X / ~ )

Apply (b), 3.1 (c) and 5.6 (~): =

Yl (x~) =

~x(ml_kl,m2_k2,...,ms_ks ) (kiE l ; ~ k l + . . . + k s = l ) (ml_kl, m2_k 2' ... ,ms_ks)

(ki=O,1;kl+...+ks=l)

48

z~Cx~ > :.~: xCm1,~,...,mi_l,...,ms> . Iterating,

we obtain the desired values for ~

YA ' x ~

< y~, ,

. Now use Ca) and 3.11 Ca):

y#>: =<xa, ~ p ~

,z~> = = (_~)¢~i-rCA),< ~p,~i> ' ~he fact that all 4-

Proposition

z~

are mutually orthogonal,

follows from

2.3 with the account of Remark 2.4. Furthermore,

computations

the

in 2.3 show that

< zA' zA~ =~k>.1rk : ~i:~:i ' It remains to verify that zn, zn >

= n .

It follows from 3.15 that


xn

zn

in the decomposition

of

equals the coefficient

in the basis

(x~).

By 3.16:

r This implies that our coefficient

equals

n , as desired. Q.E.D.

3.18. Corollary. For any

k~ Then

z~

=

I

m

cA

Z~

e

= (I

rl

,

2r2

,...)~

put

of

49 § 4. Universal PSH-algebra: irreducible elements. 4.1. We retain the notation of § 3. Thus,

R

is a PSH-

algebra with the unique irreducible primitive element f have two Z - b a s e s (z~) in

R ~

(xA)

and (y~) in

, parametrized by

R

and the ~ -

. We basis

partitions ~ ~ °

In this section we shall study the basis ~

in R, consisting

of irreducible elements. We shall compute explicitly the transition matrices between ~

and all

bases from § 3. First we

show that irreducible elements of

R

are also naturally

para-

metrized by partitions. Proposition-definition. For any ~ ~ ~ unique irreducible element and

in R ~

there exists the

such that

x~ ( ~

0

~ 0. We have @

In other w o r d s , ~ } can be characterized as a unique common irreducible constituent of

The map ~---~ ~

x~

and

y~, and it satisfies

is a bijection between ~

Proof. The condition

x~ ( ~ ) ~

andS°

0, clearly, means that ~ @

is an irreducible conslituent of

x ~ . We know thgt

y~

is

positive and t

y~

(xA) = I

(see 1.9.(d) aria 3.10 ).

This implies the existence and uniqueness of ~

and the equa-

lities

Now let ~ ) ~

6 ~and

~@~

; we can assume that c.f.(~

50 is lexicographically higher than c . f . ~ ~) (see 3.7). Applying again the positivity of •

~e

and Proposition 3.10, we obtain

hence ~'A~ z~z~ .

Finally. by 3.1 (e) rkzRn = I[~ I, this implies that the elements ~ I ments of

exhaust all irreducible ele-

R. Q.E.D.

Corollary. For ali A

Proof. Apply 3.11 and 3.17 (a) Q.E.D. ~-3- Now we compute the action of the operators

X

and Y

(see 3.6) on irreducible elements. This result will play the crucial role in the sequel (see 4.~). Let us give two combinatorial definitions. Let ~

if ~

C ~

~ ~ ~

C ~

be two Young diagrams (see 3.7). We write

. This means that ~

at most one point from each row of ~

instead of ~

-~ ~

ent to,,zk _ ~ -

A

is obtained by removing . We write

; olearly~the relation , ~

is equiva-

and means geometrically that ~

tained by removing at most one point from each column of ~ Theorem. For all ~ & ~

we have

is ob.

51 This is proved in 4.5-4.10. ~.$. The next

proposition shows the significance of Theo-

rem 4.3. Proposition. For each

n ~ 1 the operators

X* -1 = Z 4

and

Y" - 1 = Z y k

k~l are injective on

k~l

R n-

Proof. It suffices to consider only adjoint operators, we see

(X" - I). Passing to

that our assertion is equivalent to

the following: for

n~1

which acts on

the operator from Rn_ k

~

Rn_ k

l~k.4n

by multiplying by

to

Xk, is an

Rn, epimorphism.

But this follows at once from 3.1 (e) Q.E.D. This proposition allows us to reduce the verification of various identities in the operator

X -1

Rn

(or

to that in

Rk

for

k0

then

r(~gr(il~r(~)

then

r ( ~ ) l r (~). On the other hand, if

r)r(~) + I

then

(see 3.1 (c) and 3.6 (~)) hence

r(~)~ r ~ ) + d

(c) Let

r = r(~). By 4.5 (~), Yr (

xi (

- xi °Yr (

According to (b), if xi (

)

ai(~,~

ai(~,~)~0 ~

~)=r

) =

ai(~)

then

that

ai ( ~ ) ~ ) (d) Let now

)"

"

is uniquely determi-

and the partition

= a i ( ~ v~) ~ )~,

) Yr (

r(~)~ r, so we have

It is easy to see that a partition ~ ned by the number ~ I

, so

. It follows

as desired.

r = r(~) + I. Using 3.1 (c), 3.6 and 4.5 (~)

we obtain that Yr (xi "

= xi-1 '

.

56 Hence

xi- I

~i

~

A

According to (b), if

ai(~.~

)>0

then

The same argument as in (c) shows that

r(~)~r,

SO we have

ai(~)~ ) = ai_1(~ ~

).

Q.E.D. Theorem 4.3 follows at once from this Lemma by means of induction on I~l • 4.11.

JL

Before investigating the relations between the basis

and the bases of § 3, we will extend J L

of positive elements in

to a larger class

R, parametrized by so called skew diag-

rams.

this notation is justified by the fact that the element ~ \ ~ } is non-zero only when I D ~

(as Young

diagrams) and depends

only on the set-theoretic difference between the diagrams and ~

(this will be proved in 4.1}).

By 1.9 (d), al! elements ~ \ ~

and for

I~I---- I~l

we have

Finally, we derive from 1.9 (b) that (~)

m

are positive. By 1.9 (a):

57 4.12. Let us introduce some combinatorial terminology. Define the partial order pl ~

on

u

(i,j)~il,jt)~--------~

/~/×/~

by

i~ i~ j ~ j l •

A skew diagram is a finite subset ~ ~ x ~ /

which is con-

vex with respect to this order, i.e. such that

,

a,b ~

a~pc~

b

-~

c

~.~.~.

Clearly, a (non-empty) Young diagram can be defined as a skew diagram, containing the point (1,1). It is easy to see that the difference ~ \ ~

, where ~

are two Young diagrams, is

a skew diagram; conversely, any skew diagram has such a form.

I~':~'I

As in 3.7, we write ~

for the number of points o f ~

, and

for the transposed diagram. Let ~

be a skew diagram. A s u b s e t ~ l ~

is reg__ular

if it contains with each point a all points of ~

, which are

greater than a (i.e.

P Clearly, i f ~ l ~ i s

regular t h e n ~

diagrams. We shall w r i t e ~ l - - ~ tained from ~

I and~\~

(~/

~ ~

/ are skew

) if ~ I

is ob-

by removing a regular subset which has at most

one point in each row (column). Obviously these definitions are compatible with these given in 4.3 for Young diagrams. Moreover, i f ~ = A \ ~ ted as a difference between two Young diagrams ~ diagrams ~ l _ _ ~

(the similar

are just those of the form ~ / \ ~

holds for the relation

,-.Z. I!

).

is representhen skew where

58 4.13. Proposition. Let A The element ~ \

ff~

and ~b

be two Young diagrams.

is non-zero i f l a n donly ~ if ~ ~ L ,

depends only on the skew d i a g r a m ~ : / ~ \ ~ b pends only on the shape o f ~

. Moreover, it de-

, i.e. does not change if ~

shifted by a vector of the lattice L × ~ to each skew diagram ~

and

is

. Thus, we assQciate

the positive element ~ ~

~Z~P

We

have:

X'(~)

= Z

~

and Y" ( ~ )

~'_L~e

=Z

~Z~.

~/q~

Proof. By Theorem ~.3: -

sum

q'A

over

such that

) --

)

@

(similarly, for

Y-I).

Using induction on I~I , we can assume that our proposition is already known for all

~/k

,

where / ~ _ L ~

) k4/~

.

Therefor e, (X'-l)(~Xff~)

= / rams a ~ / such that ~ ~ 2 The fact that ~ \ ~

~/~

(the sum is over skew diag• ~ / = ~ see ~.12). depends only on the shape of ~

follows at once from Proposition ~.4. It remains to prove the equality

"Jg2(~'~-~) ~- ~ 2 / ~

This can be rewritten as

hence it follows at once from ~.2 and 3.11 (~). Q.~.o.

,

59 4.14. Let

~

~e a Young diagram and ~

a skew diagram.

We compute the inner products

~,D~,) ~ . ~

~'~y)~'~j~ ,

and

Once the answer is known ~one can compute the decomposition of

~

(and so of any irreducible element in

each of the bases CJ~3y3")

and C c ~ J

if~=/tk~

and ~

, where X

definition of ~

and

yy

(see 3.17). Furthermore, are Young diagram, then by

one has

In particular, letting ~ = xy

R) with respect to

~

we obtain the decomposition of

with respect t o ~

By a numbering of ~

.

we mean any mapping ~:~----->~/

which is a morphism of partially ordered sets where ~ is or~t dered by lt ~ p and ~ as usual (i.e. ~ is such that

In terms of @.12j ~ : ~ - - - ~ i s any

k ~ ~

a numbering if and only if for

the set

is a skew diagram, and

~

-4

e~<J) its regular subset ~:the eqoi-

valence of these definitions is obvious). We say that ~

type

(n1,n2,...) if

nk = I ~-4(k) I

Geometrically, a numbering

for

has

kE~.

~; ~----~/

will be shown

as an array of integers obtained by replacing each point

x

of

60 by the number

~

(x). Such an array represents a numbering

if and only if its numbers are non-decreasing along the rows and down the columns o f ~ number

k~

occurs

; it has type nk

(nl,n2,...) if any

times. We say that a numbering

is row-strict (column-strict) if the numbers increase along the rows (down the columns) o f ~ . Proposition. The inner product

~2~y~c~> (~)~:~)))

equals the number of column-strict (row-strict) numberings o f ~ of type c.f. (~0 (see 3.7). Proof. Let c.f. ( ~

= (nl,...,nr) ; we can assume that

I~l = I~I = hi+...+ n r. We have

}

o...o4

By ~.13,

(the sum is over skew diagrams ~ I~i~=I;~l-~).

Applying several

tain that ~C~}/~ ~ e q u a l s

such that ~ I ~

times this equality, we ob-

the number of chains

¢ = ~ o ~a~1 ca¢ z c , . . c ~¢~ = ~ , where all ~ ; for

and

are skew diagrams, ~ - 4

-Lc~'k '

/~

and~kt-

k = 1,...,r . We associate with each such chain of diag-

rams the numbering ~ - - - ~ The answer for

~

which equals

~)~>follows;

k

on~

~}/)~iS

in a similar way or by means of the automorphism

\~_4~

computed t.

Q.E.D. ~.15. Remark. Proposition lary: for any skew d i a g r a m ~

~.14

has a combinatorial corol-

and the sequence ~

= (nl,n2...)

of non-negative integers the number of column-strict numberingS

61 of ~

of type J

does not depend on the order of numbers

nl,n 2 .... (i.e. it depends only on ~ = (nl,n2...) ~

and the partition

). Indeed, by 1.9 (c) this number equals

Dcy 4.16. Now we decompose the elements ~ irreducible elements of

R) with respect to each of the bases

(x~) and (YA) i.e. write ~ and in

and

(in particular~all

as polynomials in

xl,x2,...

YI'Y2' . . . .

Proposition. (a)Let ~

,~

c.f.(~) = (ml,...,mr)

(see 3.7). Define the numbers

LI>L2>...

>1.r>/O

and

~

~,

c.f.(~) = (11,...,i r)

MI>M2>...>Mr>IO

L i = li+ r - i ,

by

M i = m i + r - i.

Then k

= det ( ~Li_M j

,

,

,

In particular, an irreducible element ~ det (Xli_i+j)i,j=1,...,r (b) If #

c.f.(

I

Li = i i + s - i

= and

/

Mi = mi + s - i

in particular, ~} (recall that

= det (Yl~-i+j)i,j=1,...,s Xk = Yk = 0

for

Proof. Put (temporarily) ~\~=

"

' ...,Is), c.f.( 1

det (XLi_Mj),

k~O).

equals

then

) = (m ,...,msl),

62 It is easy to see that E ~ \ ~ Li~i

for all

E~ \ ~ ]

: I : i~"

can be non-zero only if

i = 1,...,r , i.e. only i f A D ~ t!

; furthermore~

Using ~.4, ~.13 and induction on I A k f f l ,

we see that the desired equality ~ A \ ~

= E~\~.7 follows from

the formula

wzk To prove (~) of l e n g t h

we need some n o t a t i o n .

r, we w r i t e

4 ~• 2 ~ "

A ( 4

IfJ4;-

" ' )• ' ~ )

'

nant of the m a t r i x w i t h these rows, For any ~(L)

the row

(XL_MI,XL_M2,...,~L_Mr).

We apply the operator is a ring homomorphism,

and

X*

are rows

f o r the d e t e r m i L E Z

denote by

Thus~by definition,

to this determinant, Using that X"

X~(Xl ) =

obtain:

- -)~

(see 3.6, 3.1 (c)). We

xn n~l

,I~sl

,2..L 2 ~ ~2>... > ~ r ~ r In other words,

such that

•

63

where the sum is over all partitions Y

, whose canonical form

c.f.(~ = (nl,n2,...,n r ) satisfies (~)

ll~n l~12~n 2~-..~lr~nr~0

•

It remains to verify that ( ~ ) holds if and only if ~ l ~ But this is clear from the geometrical description of the relation I~~ l ~ l

(see ~.3). Indeed, the condition that ~

is ob-

tained by removing at most one point from each column of ~ means that all points removed from each row of ~

,

lie to the

right of the end of the next row. This is just ( ~ )

!

(b) This follows from (a) by means of the automorphism

t.

Q.E.D. ~.17. Now we shall compute the inner p r o d u c t s ~ y ~

~

(we use the notation of $.1~). Taking into account 3.18, we see that this gives a solution of the following problems:

(a)

Decomposition of the elements ~ 3

all irreducible elements ~ bases (my)

and

~

(in particular, of

) with respect to each of the

(z~):

(b) Decomposition of the products lar, of the elements

zy

z¥°~p~

(in particu-

themselves) with respect to the basis

Jl,

b

=Pe2P

By a skew-hook we shall mean any non-empty finite subset of ~ × ~

of the following form:

64 XXXD~

X X XX X

xX~D~x more precisely, a skew-hook is a set (in,in)}, where for either

(ik+1 , jk )

k =1,...,n-I or

where

i

the point

(ik+1,Jk+l)

is

(ik,Jk-1). Clearly, each skew-hook is a

diagram. For any skew-hook ~

s( 9

(il,Jl),(i2,J2),...,

put

(-I) i-I

is the number of rows, intersecting with ~

le, the skew-hook displayed above meets 6 rows, so

° For examp-

s(ag :-4.

Proposition. (a). We have

_- Z where the sum is over all skew-hooks ~ ! ~l

is a regular subset of ~ (b) The inner product

such that

#,~q~l~=

n

and

(see 4.12).

~ zW ~ @ ~ _ ~ >

equals

Z~(~)'

where

.J

the sum is over all numberings ~ that all subsets

- ~ "(k)

for

m ~/ k ~ ~

of type c.f.(~) such are skew-hooks, and

Proof. Part (b) is derived from (a) as in the proof of Proposition 4.14. So we have only to prove (a). Write ~ =~\ff

as

difference between two Young diagrams: ~ - -

(see 4.12). Let

LI>L2>...>Lr>~0 be coordinates of ~

and and ~i

I

M I > M 2 > . . . >Mr>~0 , defined in 4.16. By 4.16 (a):

65 ~

= det (XL

i,j=1,...,r i

We apply the operator a derivation of

R

zn

to this determinant using that

zn

is

8

and

zn (xk) = Xk_ n

(see 5.17 (b)). We ob-

rain:

(~)

where

A

zn

(

=

io=1~io

io = det (XLi_no~i,io_Mj). We fix

1~io~r

and inves-

tigateA~o • If

Lio-n = Lil

for some

i1> io

then

ALO

has two iden-

tical rows hence it equals O. So we assume that Lil> Li ° - n>Lil+l for some

il>ji o. Interchanging the

sively with the

(io+1)-th

io-th

one, (io+2)-th

A row of AA~O

succes-

one,..., i1-th one,

we see ~hat

A . 1,,o = where the "L-coordinates" of the partition

X # are

(L1 ,L2, ... ,Lio_ 1 'Lio+l,..., Li I ,Lio-n, Lil+ 1 , •.., L r) • Passing from these coordinates to usual ones, we see that if c.f.(~) = (ll,..,lr) then c'f'(~/) = (ll'''''lio-l'lio+l-1, l i o + 2 - 1, • . . ,

li 1 - 1 ,

lio-n+il-i o, li1+1 ,.'., Ir ). Geometrically, this means that X I is obtained from ~

by

moving the skew-hook a~ ! with 18~II = n

meets

such that ~ l

re-

66

il-io ). the

i-th

Clearly,

row if and only if the condition

subset of ~

, grid, as

all such skew-hooks.

io~ i ~ i I

(so

s(~

=(-I)

- n ~ means that ~ ! is a regu~az Ll o Li I i o varies from I to r, 3g' runs over

Summarizing,

we see that

and (~) gives our assertion. Q.~I.D. 4.18. Now we decompose the b a s i s ~ L

where

V

the elements

with respect to

~

, i.e. compute the inner products

is a Young diagram and ~

a skew diagram.

Once the

answer is known, we obtain the action of multiplication ltiplication

in the basis J Z

and comu-

, namely

#,yeP (see 4.11). Proposition.

The inner product ~ ) ) ~ ) ~ e q u a l s

ber of column-strict fying the following (J) I f ~ 4 ; ~ ' "

numberings

~

of ~

the num-

of type c.f.(~)

satis-

condition: ~n

is the order of the numbers

ading from right to left along the first row of ~

~Dc)

, next right

to left along the second row, etc., then for any

m = 1,...,n

and

~~

k ~

the number of

less than the number of

k's

k+1's

among~4~)--" among

re-

0~4~2_)--')c~y~ •

is not

67 Example. y

= (3,2,1), ~

= (3,3,1)\ (1). There are exactly

two column-strict numberings of ~ 1

~

=

1

122 ,3

and

f2

1

=

1 2

1 .3

2

The s e q u e n c e ~ 4 ) - - - ) ~ f-- ~

of type c.f.(y), namely

it is

for f = 94

is

1,1,2,2,1,3, while for

1,1,3,2,1,2. Since in the latter sequence the

number ,3 precedes to all 2's ~

does not satisfy (J). Thus,

(by %.1~ we also obtain

= 2).

Proof. Denote the number of column-strict numberings of of type c.f.(~) satisfying (J), by unless I~I=I~I

. we must prove that

This

for

is

evident

I~1

= O, w h i l e

g~,)/ . Clearly

for

I~(~[ >

0

g~)/

= 0

this follows

from the equality

(see %.~). By 4.13 and induction on I~l , the left-hand side of

(~)

where ~I__~

is

~I

runs

skew diagrams such that ~ / ~

and

. Similarly, the right-hand side of (~) can be rewrit-

Sen as

V

68 where y

runs over all Young diagrams such that y - ~ ~

Comparing coefficients of each ~ y }

and Y ~ ,

, we see that (~) is equivalent

to the following combinatorial statement: for any skew diagram ~

and

Young diagram ~2 with

IY~ ~ ~ I

This will be proved in Appendix 2. Q.E.D. %.19. We conclude this Chapter with a parametrization of irreducible elements in any PSH-algebra. By a graded set we shall mean a set

X

with a function deg: X

We denote by

S(X; ? )

~Z + , the set of functions

3.7) such t~at consider

~

: X

~ (x)r_ ~ but a finite number of 2 S(X;~) as a graded set by deg ~

= ~ x~X

)~

(see

x ~ X . We

deg x • ~(2L)I~

and put Sn

(X;~)

Now let

R

= be a PBH-algebra and

primitive elements (we treat ~

as a graded set, the function

deg being induced by the grading of

and each ~ (~)

~

R

). By 2.2 and 5.1,

as a PSH-algebra is isomorphic up to grading to

our universal PSH-algebra ~

its set of irreducible

R. According to 5o! (f), for any

there are exactly two PSH-algebra isomorphisms

R£~(~

69 let us choose one of them and denote it by ~ n , Proposition. For any ~ f 6 ~

~

~

(~f(~)~) ~- ll~

~

The correspondence ~ - - ~ f ~ J

put

(see 4oi)o

is a bijection of_ S ( ~ j ~

the (graded) set of irreducible elements of modulo the choise of isomorphisms % Rn

) onto

R. In other words,

, irreducible elements of

are naturally parametrized by the set

Sn(~3).

This follows at once from 2.2 and 4.1. 4.20. Remarks and complements.

(a) The present proof of Theo-

rem @.3 is due to J.N.Bernstein. In the original author's proof the formula (~)

7~}

= det (Xli-i+ j )

(see 4.16) was proved first; the computations in 4.16 show that (~) readily implies formulas 4.3. But formulas 4.3 look simple and natural, while the appeazance of (~) in our approach seems rather mysterious, so we prefer the present proof. Trying to understand (~), J.N.Bernstein has obtained the following beatiful formula. For any Sl:R

~ R

If ~ ~

define the operator

by " o Yi

S1 = (here Xl+ i

l~0

stands for the operator of multiplication by , c.f.(~ ) =

(,,,,)

=

(ll,..,1 r) 1,11,12,.--,lr

and

l~l I -

This implies that ~

= Sll°S12 ~ .-. O Slr(1 ~,

then

Xl+ i ).

70 which is essentially equivalent to (~). The formula ( 5 )

can be

easily derived from (~) by expanding the determinant expressing ~l,ll,...,lr}

up to first row. Since ( ~ ) will not be used

in the sequel, we don't discuss it in more detail. (b) In the original author's proof of Theorem ~.3 the important role was played by the following partial order relation on

~,

By definition, ff ~ A

if ~

can be obtained from

by a chain of operations, each of which is a replacing of a pair

~ l,k} of parts ~l+1,k-1~,

of a partition, where

l~k~O,

by the pair

This relation is well-known (see e.g. [10J,~21J

The next statement shows its significanae. For ~ ) ~

~ ~

). the

following three conditions are equivalent:

This follows readily from the results of this section; we leave the proof to the reader

(see[21J).

Chapter II. First applications

5. Symmetric polynomials In this section we realize the universal PSH-algebra

R, as

an algebra of symmetric polynomials in a countable set of indeterminates. 5.1. Let ~ b e algebra

~

a commutative ring with unit. eonsider the

= ~~4~2~.~-JJ

of formal power series in a

countable set of indeterminates ~ 4 ~ 2 ~ %~

~ over ~ .

Recall that

consists of expressions of the form

where ~

runs over all multiindices

(al,a2,a3,...)

in

and only finite number of them are non-zero),

and ~

~ ~

number I~l

. The expressions = ~

ak

~

and on the algebra ~

by

Consider~ the subalgebra

A~o c~

~

, invariant under the

~4,~ ~

The group

,

action of

S~,

and such that

F, are uniformly bounded.

are called symmetric polynomials

over

• The most interesting for us will be the case when

S~

consisting of power series

degrees of all monomials occured in Elements of J ~

are

acts on multiindices by

(al,a2,.--) = (a~-~1), ~-~2),-.-)

F ~ ~

ak

are called monomials, the

a degree of the monomial ~ .

of permutations of ~

(all

, so we shall write

for

;

in

72 The algebra ~

(in contrast to ~ ~ s

graded by degrees

of menomials; the correspondence ~

is a functor from

the category of commutative rings to the category of commutative graded rings. Note that even in the case when ~

itself is

graded, we forget this grading when considering the grading o n ~ . For any partition ~ u~ ~ ~

~

we define the s2mmetrized monomial

to be uA

=

where the sum is over all multi-indices O~ under the action of

Soo

conjugate to c.f.(~) is a free

(see 3.7)o Evidently, J ~

~ - m o d u l e with the basis ~ A I ~ e 2 ~ . as a projective limit

It is also useful to realize J ~

of rings of symmetric polynomials in a finite number of indeterminates. More precisely, denote by

the graded algebra of symmetric polynomials over ~ determinates ~ f ) ~ l ) ~ . . ( for

N~N

in N

in-

?~A/. There are natural morphisms

)n

r (

n

(~i ~

>0

for

and

(

n

>

i>N); denote any such morphism

by ~ N . It is easy to see that these morphisms allow one to identify

(~n

with the projective limit lim ( ~ n

0

Evidently the symmetrized monomials r(~)~ (~n

~A/:(~n

>("~)n

and

are isomorphisms. To avoid a confusion we shall sometimes writ@ "

indicating explicitly

)

the set of indeterminates.

If JLr~4,G~'~

is another countable set then any bijection ~ : J L ~ 4 ) ~ ; ' ' ~ induces an isomorphism

clearly, this isomorphism does not depend on ~ . We shall denote 2 all such isomorphisms by C and identify canonically all J ~ < ~ , 5.2. Now we construct an isomorphism of graded algebras R----~J~=%Z,

where

R

is the PSH-algebra from

9§ 3,@. The

following construction is due to J.N.Bernstein. (I) First, we construct the morphism of graded algebras

P SR For any

N~I

>J~R

define the mapping

p(N) : to be

R

the composition PI

~R~R~..

N times

.... , g v 3

~R

P2 ..

74 where

P1,P2, and

P3

The operator in

are defined as follows.

P1

is obtained by iterating comultiplication

R, i.e. it equals the composition~

m R----~R@

m~id R ............... " R ~ R @ R

The operator o~erator

~

words, if

P2

m~id ...

is the tensor product of

;~ 3 vi ~ R a i

m @ id >

, se~ding any

v ~ Rn

>R~R~...~R. N

copies of the

to

w ~ n. In othe,

then

P2(v1®... ~ v N) = v 1 ~ al ~ v2~ aa 2 ~ ... ~ v N ~ ~gA/ , Finally, identifying

we define

P3

to be the natural projection. In other words, P3

acts by

2

e3(v1~ ~ v 2

~ 2 ~

...~VN~

Clearly, PI,P2,P3, and so p(N) rings (for

PI

2~

) =vlv2

~

are homomorphisms of graded

this follows from the Hopf axiom (H) from 1.3).

Commutativity and associativity of comultiplication in

R

imply

that p(N) ( R ) ( z ~ N R i.e. that

p(N)

is a morphism of graded rings

R R

Evidently, ~oP tending

N

(N~) = p(N)

for

N'>N

;

to infinity, we obtain the desired homomorphism

PER--

~A~

75 (2) Consider the ring homomorphism

~ac : R

3.6). Since the correspondence ~ - - - ~ ~

~

is a functor,

(see ~z5

induces the graded ring morphism

which will be denoted also by hism

p:R

~ ~

Sx. We define the graded ring morp

to be the composition

R

~ 2 o R ....

.

5.3. Propositi °n. (a) The morphism ments of

R

p

acts on various ele-

in accordance to the following table

i

v~R

p(v) 6

!

I ~ uk JAl=n

xn (see 3.1)

(the sum of all monomials of degree

Yn

(see 3.1)

U(In )

n)

(elementary symmetric polynomial)

zn l x~

(see 3.15)

U(n )

(see 3.17)

(power sum) u~ ~,C~K_ ( ~ r u n s over all

(see ~.12,~.13)

column-strict numberings of the skew diagram ~

(b) The morphism

p:R

t%

; see ~.I~)

is an isomorphism. We

76 shall identify

R

and ~

via this isomorphism and use the

same notation for an element Proof. Let fact that ~

of

R

and its image i n ~ .

v ~ R n. Using the definition of

~

and the

is multiplicative (see 3.6) we can readily com-

pure (see 5.1,5.2) We obtain that (~)

p(v) =

~

x~(v) u~

=

All formulas of the part (a) follow at once from (~) and the results of

§§ 3,@ (see 3.17, @.1@, and 4.15). Part (b) fol-

lows at once from the equality

p(x~ ) = u~

(see 3.17). Q.E.D.

5.4. The isomorphism

p:R

~

algebra. All results of §§ 3,@

makes ~

into a PSH-

are transformed to the asser-

tions on symmetric polynomials. All these assertions are classical. We will not formulate them since this can

be done

automatically; let us only mention some names. The symmetric polynomials Schur functions

or

~

where / ~ J ~

S-functions , while the ~ ,

, are called where ~

is

a skew diagram, are called skew Schur functions. The formulas 3.16 are Newton formulas , the expressions 4.16 (a) and (b) of S- functions are the Jacobi-Trudi and the Naegelsbach-Kostka formulas, while the identity det (XLi_Mj) = d e t is the Aitken theorem.

(y~

MI )

i- j

Proposition @.18 gives the Littlewood-

77 Richardson rule

for multiplying S- functions, and the Little-

wood-R0e theorem, which says that the coefficient of ~ } ~\~

in

(when written as a linear combination of Schur functi-

ons)LIJis equal to the coefficient of

~} in ~/t~},~y} ,

The classical proofs of these statements can be found in [22],

Ch. VI (two another approaches are developed in ~ 0 ]

and [23] ). Our proofs, which are based on the systematic use of the "lowering" operators

a , sometimes seem to be more simp-

le and natural (especially, those of the Littlewood-Richar~son rule and the Littlewood-Roe theorem). The lowering operators in~

were used by a number of authors e.g. by MacMahon, Van

der Corput and

Foulkes; the references and the discussion of

them from the same point of view of Hopf algebra can be found

5.5. In the remainder of this section we shall express the g l

various structures o n ~ and

arising from the identification o f ~

R, in terms of the "coordinates" ~ t

bedding o f ~

i.e. using the em-

into the power series ring

First we want to write down explicitly the comultiplication in ~

. We introduce two countable sets of indeterminates

we realize ~

@ ~

as the subalgebra in ~L[~,--- ~ J ]

~

=

consis-

ting of those series which are symmetric polynomials separately with respect to

~

and ~

.

78 Proposition.

If ~

multiplication

where

C

is realized as above, the co-

m':A

> ~@~

has the form

is the canonical isomorphism (see 5.1).

Proof. It suffices to verify the equality when

x

m~(x)--C(x)

varies over some set of generators of the ring ~

.

We choose as generators the power sums zn =

(n = q,2,...)

(they are generators of By 5.3 and 3.15 definition of

only, but this does not matter).

m (zn) = Z n @ 1 + 1 @ z n. On the other hand, by C

C(z n) =

we have

~ +

= zn ~

I + I @

Zn-

t

g

Q.E.D. 5.6. Now we discuss the inner product < ~ >

. We re/% call that it is induced by the T-group structure on J ~ (see 1.2); by definition, thenS-functions ~ }

~

in ~

~)

form

an orthonormal basis in jug • We want to describe ~ ~ >

more

explicitly in a "coordinate" form. It is known from

standard linear algebra that an inner

product on a finite dimensional vector space rized by a nom-degenerate symmetric tensor if ~ > basis of

is an inner product on

V,

V

can be characte

g g V ~ V. Namely,

~e1'''''eP~-- is any

V, and { e_L 1,...,ep.A~ is the dual basis of

respect to < }> g = A

then ei ® e

EV~V

V

with

79 I~ is easy to see that ~

>

is determined by

g

depends only on < ~ > g; in fact,

g

. Conversel~,

determines naturally

I

the inner product on the dual space tify

V

and

V , this allows one to iden-

V', and hence to obtain the inner product on

We will describe explicitly the tensor responding to the inner product ~ ~ > Propositi0n. I f ~ @ ~

gn~n~J~n n on O~ n •

is realized as in 5.5 then

equal to the sum of all monomials of degree

n

V. cor-

gn

is

in the set of

indeterminates

Proof. By 5.} it suffices to verify the identity

One can do it by a straightforward compmtation (see ~10J, pp. 57-38), but we prefer another approach based on the homomorphism

from 5.2 (I). We identify ~

so ing

P

with

becomes a homomorphism from R

with ~

R

to

R ~

by means of the isomorphism

homomorphism

By definition of

~ ~ ~

P

and 3.1 (d),

via

. Identifyp, we obtain the

80

P (Xn) = ~ X% @ UA IAI=n Therefore the identity 5.7. Lemma. If ~ hism

P :A

A

=IC(n)l/Isnl so

Zn = ISnl / IC(n)I ~ %(n)

"

Now we apply the formula for the action of induction on class functions (see E 9 ~ , 7.2 or 8.2 below): if

H

te group

H, and

G, C is a conjugacy class in

gacy class in (~)

Let

G

containing

ind G (IHI/ICI, % C H

~

= (ii,...,i r ) ~

H = Sll x ... x S l r ,

is a subgroup of a finiD

is the

conju-

C, then ) = IGI/~DI,%D

. Applying

"

(1)

to the case when

C = C(ll) x ... x C(lr)

, and

G=S n,

D = C A , we

obtain: z;

=

/Snl/ic;l'%~

It follows that

= ~: z~ ,z~) = ( I S n l / I c ~ l ) 2 . ~ % ~ , ~ , ~ > _ _

cA

t s nl/Ic,~t •

This is the assertion I0. The formula 9 follows now from 3.18. Q.E.D. 6.4. Using the dictionary from 6.3, we may transform all results of §§ 3,4 to the assertions

on the representations

of symmet-

ric groups. Let us mention three important classical results. (I) The Littlewood-Richardson

rule (Prop. 4.I8). It describes

the restriction of irreducible representations groups

Sk x S 1

as well as the induction from

of

Sn

to the sub-

Sk x S 1

to

Sn .

(2) The branching rule (Theorem 4.3). By the formula 6 from the table 6.3, this means that for any irreducible representation of

Sn

the representation

of

Sn_ k

in the subspace of

S k-

91 invariant vectors of ~

is multiplicity-free and equals

2~r/~ .oo

c~ ~

l p/:

~

o

rained by removing one point from the Young diagram ~

.

(3) The Murnaghan-Nakayama character formula (Prop. 4.I7). By the formula 9 from the table 6.3 the inner product

~ zy, f ~

computed in 4.I7 (b) is the character value of the irreducible representation ~

at the class

Cy .

Another expression for character values is given by the Frobenius formula: if ~

,~

are the coordinates of value of

~

~

~

, N ~ n , and

LI>L2>,,,>LN~0

described in 4.I6 then the character

at the class

CV

equals the coefficient of the

monomial

4

"~2

in the alternating polynomial

~V(~'I'"''~A/)' Ac=~4'. . . . 6.5. Now we compute tations of the groups

(see 5.3,

5.I2).

dimensions of irreducible represen-

Sn .

Proposition. (a) Let tee of ~

, A/)

~ 'J~

and

L I > ... >I~r

be coordina-

described in 4.I6. Then

n: ~

i<j

(Zi-L j)

Li: Z2: ... Lr'. (b) Pot any point

a = (i,j) belonging to the Young diagram

the cardinality of the set

92 or is called

hook length of a and denoted by dim ~ I =

i, >~i, j , =J } h(a). Then

n.' ~h(a)

(this is the hook formula obtained in [~T~). Proof. (a) It follows from 6.3 (x) that the mapping, sending each ~

&

~

(Sn)

to

dimJU/ISnl = dim/U/n: , extends to the

ring homomorphism d

Applying

d

: R(s)

-----*Q

.

to the equality ~

=det

(see 4.16),

(XLi.r+j)

we see that dim ~ 3 - - n :

det(

I (~i-r+J) '

(we agree the convention that tiplying the

I/(/i-r+J),, = 0

i-th row of this determinant by

if

Li-r+j< 0). Mul-

L i : , we obtain

that det (Pr-j (Zi))' LI ' •

where degree

e e e

•

(L-k+l). Since

Pk(L) = L (L-I) ... k

Lr '

with the leading

Pk

coefficient I, the

is a polynomial of j-th column

(Pr.j(~1),-.., Pr.j(~r)) equals (z{-J , L~'J, ..., ~ - J

) + (linear combination of previous columns).

93 It follows that

det (Pr.j(Li)) = det (Zr'j) = [~I (Zi " Lj) i i~J (the well-known Vandermonde determinant),

as desired.

(b) Under the notation of (a) it is easy to see that for any i = 1,2,...,r

the number

j>i is a product of hook lengths of all points of the

i-th row of ~ . Q.E.D.

§ 7. Representations

of wreath products

7.1. In this section we extend the results of § 6 to the representation theory of wreath products. Fix a finite group the wreath product of

G

by

the semidirect product of 6:

S n (the notation

Sn

by

(gi,...,gn) --------~ (gg-~(1)

In other words, elements of gn), where ~

6

Sn [ G ]

Gn

G. By

S n [ G ~ ) we mean

with respect to the action

,..., gg-~(n)

(6aSn

' gi ~G)-

are the expressloms 6 (gI"'',

Sn, gi ~ G; the multiplication

in

Sn[G ]

is

defined by the commutation relation (x)

(gI'''''gn)'6

We consider mal subgroup a r)

of

n

Sn

of

=

~" (g6 (I) ''''' g 6 (n)

as a subgroup of

S n [ G ~ , and

S n [ G ] . For any ordered partition

we set =

Sa2

-..

Sat

.

Gn

as a nor= (ai,...~

94 and identify

S~ EG~

with the subgroup

S~' Gn

in

Sn ~G~

(see

consists of two elements, then

Sn [ G ~

is the

6.3). Example.

If

G

Weyl group of type

Cn

(or

Bn) ; these groups are also called hyper

octahedral. It is useful to keep this example in mind while reading this section. 7.2. Let us consider the graded group R (SEG~)

=~ R (Sn[G ~ ) n~o

(see 6.I).

It is a T-group with the set of irreducible elements ~(S n [G~)

(see 1.2, 6.I)

n~0 We define the multiplication and comultiplication in

R(S[GJ)

in a quite similar way as for symmetric groups. Namely, for k+l=n

we identify

R(Sk[ G] ) ~

R(S 1 [G~ ) with

R(S(k,1 ) EG~ )

(see 6.I, 7.I) and define the multiplication m

:

R(Sk[~]) ~ R(SI[G3)

to be the induction from

S(k,l ) E G ]

)R(SnLG~) to

Sn CG~ , and the comul-

tiplication

k+l=n as Res k+l=n Proposition.

Sn[G3

S(k,IFG]

This setting makes

R(S [G~ ) into a PSH-algeb-

ra. Its set of irreducible primitive elements is _ ~ ( S I CG~ ) = = 3 ~ (G). The proof is quite similar to that of Prop. 6.2 (the axiom

95 (H) will be verified in Appendix 3, A3.3). Q.E .D. 7.3. By Theorem 2.2,

R(S[G~ )

as a PSH-algebra decomposes

into the tensor product of sub~lgebras each

R(~)

R(f), ~ C ~ ( G ). By 3.Z (g)

as a PSH-algebra is isomorphic to the algebra R(S)

from

§§ 3,4 (we will in this section represent

R

identification from 6.3). Let us describe

R(p) and define its iden-

tification with

as

R

via the

R(S) more explicitly.

By definition (see 2.2), R(~) n

for every

n~O

is spanned by

irreducible constituents of P ~ . We have

)

Ind

(see 7.2),

Gn so by the Frobenius reciprocity representations of contains ~ @ ~ @ ~ ,

R(~)n

is spanned by irreducible

Sn [ G o , whose restriction to the subgroup ~,

As in 6.3, an isomorphism of PSH-algebras uniquely determined by the image of tion of

~

(x2) of

R(S)~R

(F)

is

x2, the identity representa-

S 2 . We choose the isomorphism ~

representation

~n

sending

x2

to the

S2~G ] = S2'G2~such that

• Now we describe ~

and the inverse isomorphism more explicitly

from a functorial point of view. Let us define the additive fUnctore (Sn [G~ )

and

~

:~(Sn~G

~)

- - ~ ~2C (Sn) o (see 6.1). Let

V

be the space of F

and

~

V

be that of the

repre-

96

~9@~®., • ~

sentation~= q~

of

Sn

in the space

Gn. Define the representation

of

~)a V by

"~ (6) (vi@v2@... @ Vn) = v6-~(i) @ %-~(2)@ ... ® ~_4(n ) (I) Let ~ _

A

representation E

(Sn)

= ~C~)

and

W

be the space of J~ . Define the

~ -~(S n [~]) in the space W~(~"V)

by

~(gl,...,g~) (~

~ id~

(~(~)~ ...~

~ ( ~ n ))

(gi~).

is well-defined according to 7.I (m)). (2) Let ~

~

=~i~)~(bs)~.T~

w~ (Sn[G]). Define the representation 3~ = in the space

HomGn ( ~ " ~

~

) by

~(6)A = ~ (6) OA" "C(6-4) (the inclusion

~

Clearly, ~ e

(6) A e HomGn ( ~ , and ~

~ ) follows from 7.I (~)),

are additive functors. According to 6.I,

we have the corresponding morphisms of graded T-groups : R(s)

~R(s[a])

and

: R(S[~])

pro;osi%ion. (a) The operators ~

and ~

............~....~(S). . are adjoin% to

each other with respect to the inner products < ~ > .

c~> ~ao~ o~ ~

~

~

~ ~ ~-~o~

tire Hopf algebra morphism. (c) The composition

~

o~-

R(S) -------~ R(S)

is the identity operator, while

mo~m

~.e. ~ ~o~-

97

is the orthogonal projection of R(S~G]) onto R(~). In particular, ~and

~

induce the mutually inverse isomorphisms of PSH-algeb-

ras R(S) and R(~). Proof. (a) We will prove the more precise statement that the functor ~px is adjoint of ~ ~ & ~ ( S n E G ~)

(x)

Let

there exists a natural isomorphism

HOmsn

(.T6, T~ /::~))"~'~

W be the space of ~

finition, A

i.e. that for every~ ~ o~ (Sn),

, and A ~ Homd(J'l:::,~

~:~) ). By de-

is an operator from W to HomGn (~n~ > :~C ). Define

the operator ~ : W@(@nv) ~ A (w@vi®

HOmSnFG ] (G~(j-g),j~).

...

@ v n) = A(w)

One can easily verify that correspondence A ~ (b) Clearly, ~

A

by (v I ~

... @ V n ) .

~_HOmSnE.G~(C~pC,1Z)~::~ ) and the

establishes (~).

and ~

are positive; definitions imply at

once that they are coalgebra morphisms. It is rather tedious to verify directly that follows at once from

and

are ring homomorphisms, but this

and the axiom (S)from 1.4: since ~

is a coalgebra morphism, its a d j o i n t ~

~)

C~)

is a ring homomorp-

hism. (c) By (a), for every ~ ~

~

(Sn)

we have the natural morp-

hism A:~-._.___~ ~ (apply (z) for ~ : ~

°~

~#g) and choose A

CJ-C~ corresponding to the

identity morphism at the right-hand side of (~)). Similarly, for every ~ ~ ~(SnEG~)

we have the natural morphism

98

-

It suffices to prove that

A

is an isomorphism while

A

maps

~ ) ° ~ i (~) isomorphica!ly onto the sum of all constituents o f ~ , belonging to

R(~). The straightforward proof is left to the reader. QjE.D.

7.4. Combining 7.3 and 4.19, we obtain the classification of irreducible representations of the groups

Sn[G ] in terms of that

of G. Namely, irreducible representations of zed by the set

Sn(~(G) ; ~ )

7.5. By the isomorphisms ---,R(S)

(see 3.ii) to all

Sn~G~are parametri-

(see 4.I9) via the formula

~

we transfer the operator

R(~)

t:R(S)~

and then extend it to the invo-

lut lye automorphism

t,R(S[G])C lear ly,

t

> R(sCG]).

acts on irreducible represemtations by t(~)

=

~

, where

~'~(~) = ~ ( ~ ]

(see 7.4, 4.2). Propositio n .

If ~

G4 (Sn~G])

(inner) tensor product of J~ which equals

~

on

Sn

and

then

t(Eg) = % ~

and the character ~

on

~

of

~

, the SnEG ~

Gn .

This follows at once from the description of

t

in

R(S) (the

formula 5 from the table 6.3), the explicit construction of in 7.3, and the obvious fact that the operator J~F---~ ~5@ ~ tends to an algebra morphism

R(SEG ])

~ R(S[G]). Q.E.D.

ex-

99 7.6. Proposition 4.18 allows us to compute explicitly the action of multiplication and comultiplication in

R(S[GS) on irredu-

cible representations. In other words, we know the induction from S~[G3to

Sn~GS

and the restriction from

Sn[GS

to

S d L G S . F,r

example, we obtain the following. Proposition. (Cf. 6.4 (2)). The restriction of an irreducible representation$~

of

(see 7.3)to the subgroup

Sn [G~

S(n.l,i ) [G ]= Sn_I[G ] x G

is multiplicity-free and equals

where the inner sum is over %(~')= ~(~') for f ' ~ f , and

~

~ Sn_ I ( ~ ( G ) ; ~ ) ~

such that

(~) is obtained by removing one

point from the Yottng diagram ~(~), 7.7. Now we consider the characters of the groups

SnC 3

goal is to construct the character table of that of

G

and of the groups Sk

conJugacy classes of As in § 6

Sn[G ~. Our

in terms of

(see 6.4 (3)). First we describe

Sn[G 3.

(see 6,I) we identify

R(S[G3 ) with the lattice

in

C(S[G~)= ~ C(Sn[G~) , and consider C(S[G~) as a Hopf algebn~O ra over C . To each conjugacy class C of Sn[G ] we assign the function

where

% C is the characteristic function of

primitive if ~ C

S(k,1)[G ~

C

is

is a primitive element of the Hopf algebra

C(S[GJ); clearly, this means that group

C. We say that

for

conjugacy classes of

C

intersects trivially any sub-

k,l> O, k+l=n. Denote by G

(it is graded by

K(G) the set of

d e g ~ I; see 4.19).

100

Proposition. in

(a) For each

n >~

Sn~G Z are parametrized ~y the

C(n;K)

in

primitive conJugacy classes

set

K(G): the

Sn[G 3 , corresponding to a class

elements of the form

K& K(G), contains all

6~'(gi,g2,...,gn) , where

= (n---->n-I ---->n-2---->... ---~I is an

primitive class

• n) ~ S n

n-cycle, and g1'g2"

We shall write

~n,K

"-- " gn 6K

for

~C(n;K)

(b) ConJugacy classes in Sn(K(G) ~ ~ )

ction

K),K

~ (K) = ( l i ( K ) , 12(K), . , ° ) .

tains the

C (~)

corresponding to a fun-

, is defined by

K~K(G)

where

"

Sn[G J are parametrized by the set

(see 4.I9). The class

: K(G

•

'

In other words,

C(~ )

con-

conjugacy class

K k of the subgroup

c

Sn[ 3

(see 6.3 ( 1 ) ) . Proof. (a) It is easy to see that every primitive conjugacy class in

Sn~G]

where

is an

6

consists of elements of the form n-cycle in

conjugate to one of the form by an element of of

ZS~n),G n

6.(gI,...,gn) ,

Sn. Evidently, every such element is 6~'(gl,''',gn)

under transformation

Sn. It remains to find the orbits of the action by inner automorphisms on the set

6n.G n

(here

101

ZS (6 n) is the centralizer of ~ in Sn; it is easy to see that n it is the cyclic subgroup ~ 6 ~ generated by 6~ ). Define the mapping

N

~

( ~(gl,g2,...,gn

: ~G n

~ G

by

~) = glg2 "'" gn

To prove our statement it suffices to verify that two elements and

h

of

6n ~ Gn

if and only if

g

are conjugate under the action of ,G n

~(g)

and

~(h)

are conjugate in

G.

By 7.I (~) we have

(~)

~"

(~-~

( 6~(gI,--',gn))'6-1 n

=

~ ,(g2,g3,...,gn,gi) ;

(hI,-.-,h n) ' (6n,(gi,...gn)) "(hI,...,hn )'i = -I -I -I -I ). = 6~' (hngih I , hig2h 2 , h2g3h 3 ,--. ,hn.lgnhn

It follows that for

g = 6n,(gl,...,gn)~

6~'G n

we have

(~ g~ -i ) ~ g~i. n ( g ) gl , n

~(

(hi,...,h n) g (hi,...,hn) -I ) = h n' ~ ( g ) ' h n I

This proves the part Gnly if" of the above statement. The part "if" also follows readily from (~). (b) The shortest way is to apply the structural theory of Hopf algebras over fields of characteristic zero (see 2.I). By (a), the elements ~ l , K

(l~I, K~K(G)

primitive elements in

)

C(S[G] ) . According to 2.1 (I), (2),

is a polynomial ring ~ [ ~ ~ 3 _ This means that the elements ~ C ~ )

basis in

form a basis of the space of C(S[G~J

(see also Appendix I, AI.I-AI.2). (~, 6 S ( K ( G ) ; ~ )

)

form a

C(S~]).

It follows that the conjugacy classes

C(~) are pairwise dis-

102 tinct and exhaust conJugacy classes of all groups

SnEGJ , as desi-

red. Let us give another proof. The elements

~l,K

and mutually erthogonal, so by 2.4 the elements tually orthogonal. Hence all classes 7.4 the number

of such classes in

reducible representations of

are primitive ~C(~)

are mu-

C(~ ) are distinct. But by Sn~G ] equals the number of ir-

SnEG ] . Q.EoD.

7.8. By definition, ~he character value of an irreducible representation ~ }

of

SnEG] at a conJugacy class

C(~)

(see 7.4,

7.7) equals the inner product

For any ~2 e

z

S(..i~.(G);~ ''~ ) set yT,

(see 3.16, 7.3). According to 7.3 we have

the right-hand side is known character table of each

from 4.I7. We see that to find the

Sn~G ]__

it suffices to compute the transi-

tion matrix between the bases ~ C ( ~ ) } Fix

and ~ Z ~ }

in

C(S~GJ).

n ~ l . It is easy to see that each of the sets ~(Zn)

I ~Z(G)~

and

~ ~n,KIK~K(G)}

is a basis of the space of primitive elements in

C(Sn~G]) (see

3.I5, 7.3, and 7.7). We shall compute the transition matrix between these bases (evidently, this enables us to compute the transition

103

and ~

matrix between the bases ~ Z ~ that this matrix does not depend on racter table of

and equals simply the cha-

For every

~

_~

(G)

and

K & K(G)

the character value of the representation ~

jugacy class

K. Then for

~n,K =

out

G.

Proposition. (K)

n

(~)l " It turns

~

n ~I

~ '

denote by at the con-

we have ~(Zn);

o) (here " ------ " Proof.

stands for complex conjugation).

We shall show that for e v e r y ~ 5 ~

It suffices to consider the case when 3~ tation of

C(Sn) ,

is an ordinary represen-

Sn . In this case the left-hand side of (~) is the cha-

racter value of

~

(3~)

at

C(n;K), i.e. the trace of the

)

operator X = where

k~K

on the space

~(3~)

(~'(k,e,e,...,e)),

(see 7.7). Under the notation of 7.3 (I), W~(~nv)

X

acts

by

X (w~vi4~...@Vn) = ~ ( 6 n ) w @ v 2 @ v 3 ~ . .

. @Vn~f(k

) vI

,

The trace of this transformation can be computed directly, and we obtain: trX =

trJU(E~) • try(k) = < 3 ~

(see 6.3), as desired.

,Zn>O ~ ( K )

104

Evidently, the elements ~

(zn)

for

p ~ J £ (G)

ly orthogonal ( s i n c e they belong to the d i s t i n c t

Substituting ~5

= zn

R(~)

into (~) and using that ~

are mutual-

). Therefore

is an isometry,

we obtain the first formula to be proved. The second formula follows from the first one and the orthogonality relations on

G-

Q.E.D. 7.9. The theory is considerably simplified in the case when G

is abelian. The construction of the functors ~

and %

)

is

)

simplified by the fact that all irreducible representations of

G

are one-dimenslonal (see 7.3). The classification of irreducible representations of

Sn~G]

(see 7.4) in this case can be derived

directly from the general representation theory of groups with an abelian normal subgroup (see ~9], 9.2). Moreover, when

G

is abe-

lian, some of results become more beautiful. For example, we have the following Branchin~ Rule

(cf. 6.4(2)). If

G

is abelian then the res-

triction of any irreducible representation ~ } the subgroup

(the

~ Proof.

Sn.iLGj

of

Sn[G]

is multiplicity-free and equals

are as in Prop. 7.6). By 7.6, this restriction always equals

to

105 (even if

G

imp = 1

is not supposed to be abelian). If

for a n

G

is abelian then

Dl Q.E.D.

7.10. Now we compute the restriction of representations of SnLG ~ to

Sn

and the induction from

Sn

to

Sn~G~. We define the

graded group morphisms

I:RCs) - - - - ~ ( s ~ ] )

~

~: ~(sEQJ) ---*R(s)

to be I -- Ind Sn~G] Sn

on each

R(Sn)

R = Res Sn~G~ Sn

on each

R(S n[GJ).

Proposition ~. (a) The operators

and

I and

R

are adjoint to each

other and are both PSH-algebra morphisms (i.e. positive Hopf algebra morphisms). (b) Suppose

G

Re

=

e I = id : R(S)

In other words, if we identify each 7.3 and so identify

~

to be abelian. Then for any

R(S~GJ) = ~

R(~)

> R(S)

~(~)

with

with

~

_ ~ ( G ) we have

(see 7.3).

R(s) as in R(S)

then the

operator I : R(S) -

)

~

R(S)

is induced by comultiplication in the operator

PI

R : @

R(S)

(more precisely, it is

from 5.2) while

R(s)~.~(s)

f is induced by multiplication in

R(S). It is amusing that multip-

106

lication is now realized as restriction while comultiplication as induction. Proof. (a) The adjointness of

I

the Frobenius reciprocity. Evidently E

and Y

R

follows at once from

and

R are positive and

R

is an algebra morp-

is a coalgebra morphism. The fact that

hism, can be derived readily from the Mackey theorem on the restriction of induced representations ( ~9J, 7.4; see also Appendix 3, A3,4 below). Since axiom (S) from 1.4

l

and

R

implies that

are adJoint to each other, the i

as well as

R

is a Hopf al-

gebra morphism. (b) Since ~

is

one-dimensional, the equality

follows at once--from the construction o~ ~ p

)

ty

oI = id

RO~

= id

~ee 7.3). )The equali-

is obtained by passing to adjoint operators (see

7.3 (a)). Q.E.D. 7.II. In conclusion, we compute dimensions of irreducible representations of

Sn~G~

Pr__~osition. ~

of

Sn~G 3

The dimension of an irreducible representation equals

~

(note that when

@

~ (a)

G

~(~)

(see

6.5 ( b ) )

is abelian, the second factor is ~).

This follows at once from 7.3, 7.4, and 6.5 (b). Q.E.D.

Chapter III. Representations of general linear and affine groups over finite fields

§ 8. Functors IU, e

and zN, e

8.I. In this section we introduce the functors generalizing induction and restriction of representations, and outline their main properties, Let groups such that e :u -

C*

G

M

be a finite group, and

normalizes

U

and

be a character of

U

M ~ U

M

and

U

its sub-

=~ e~ . Let

normalized by

M, i.e. such

that e (mum -~ ) = e(u)

for

m~M,

u~U.

In this setting we define the fumctors iu,0

: ~(M)-

-J~(G)

~e -induction")

> ~

("e-restrictlon")

and

ru, e : .~ (o) -

(M)

(see 6.1). (~) Let

V

be the space of

the representation

(u) = e(u).~v

for

~

of

u~u.

~ ~

P = MU

~

to

(M). We extend f

in the same space

V, setting

~t

iu,e ( ~ ) = Ind G (~) . P In other words,

iu,e(?)

is the representation of

translations in the space of functions f(mug) = e(u). ~(m)f(g)

for

f:G - - - ~ V mEM, u~U,

G

by right

such that g~G

.

108

(2) Let

E

be the space of

J~

6 A(G).

We set

for all Clearly,

the subspace

ru,e(YO

is the representation

When

EU,e~ E

is of

in

i~, e

U = S e ~ , the functors

induction and restriction.

M-invariant.

M

9

u&U

By definition,

E U'e .

ru, e

and

become usual

Their main properties remain valid in

the general situation.. P ropositi0n ~ (b)

ru, e

~6j~(G)

is adjoint of

iu, e

and

ru, e

iu,e, i.e. for any

are additive.

Q ]

& ~/~ (M),

we have a natural isomorphism

~om (c) Let ter of

(a) The functors

(r~,e(Jg),~) =

N

and

¥

~om ( ~ ,

be subgroups

V

such that the functors

iv,e,,

J~(N) ---~ ~ (M)

make sense. Define the character e°(uv)

= e(u).e'(v),

of

~d

iu, e

(?)

M

and

rv,e'"

8 ° of u ~ u,

.

8 t be a charac-

~-~ (M) ---~

U ° = UV v ~ v

~ (~)

by

.

~hen

iu,8~iv,01= iuo8o , (d) If iu, 8

and

H

rv,e , o ru,e

is a finite group, and "i-~

ru, e

commutes with the functors

= rUo8 o A i, j

(H)

then each of

;z ~

and

~; ~ ® ~ . prpof.

Part (b) follows readily from the Frobenius recipro-

city while other assertions are immediate consequences

of the

definitions. Q.E.D.

109 8.2. Since the functor

iu, e : ~ ( M )

'"~J~6(G)

is additive,

it induces the positive morphism

%,e and the

: R(M) --.----,R(G)

~ - linear operator

iu, e : C(M) -----~C(G) (the similar holds for

ru, e ). All assertions of Prop. 8.I

can be

obviously reformulated in terms of these operators. Let us give explicit formulas for the action of

iu, e

and

rU, e

on class functions. proposition. g~

Under the notation of 8.I we have for % 6

G

:~,e(;;z :) (~) = (the

for

C(M),

s=

~

is

E

over the set

,t

IMl.Iul

~

% (m)~e(u)

(gi,m,u)~G x M x U / g = g { mug~I~) and

~

C(G), m £ M rU,e ( ~ )

I (m) = - ~

e -I (u), ~ (mu)

The proof is left to the reader. 8.3. Let a character of

M,U,N, ans U

and

V

be subgroups of a group be a character of

V

G, @

be

such that the

functors

make sense. Under some extra assumptions one can compute the composition

rv, ~ ~ iu, e

110 Since the formulation is rather cumbersome, it will be given in Appendix 3 (when

U = V =~e~

one obtains the Mackey theorem; see

[9J, 7.4). The computation of such a composition in various situations will be one of our main tools. In the body of the text the results of this kind will be only formulated while the proofs will be given in Appendix 3. § 9. The classification of irreducible representations of

GL(n, %

)

9.I. Fix a finite field ~/

)

and

graded

R (q)

T-group

= R (Go) = ~ ) .

~/

. We shall write

Gn

for

GL(n,

for

~ R (Gn) (see 6.I); thus, R (q) is a n~O (we agree the convention that G O = ~ e ~ so R(q)o=

We shall make

As in I.I, we must for all

R(q)

into a Hopf algebra.

k,l,n

with

k+l=n

define multip-

lication

m : R(%x

Gl)

;R(G n)

and the component of comultiplication

w mk, I : R(G n) ------*R(Gk x G I ) . Let us embed

M -- Gk x G I

into

Gn

as

the subgroup of cellular

diagonal matrices and consider the subgroup

U = U k , l C Gn

ting of unipotent upper triangular matrices

u = (uij)

uij

can be non-zero only

malizes

U

and

if

i = j

or

i.~2. Evidently,

Ln_ k ,

115

]3y z.9 (a), (b):

-4

(Ln)

Y2 ( ~ hence by 3.I (b),

9.7. Remark.

for all

) : o

n,

L.~ = ~c (~)"

Q.E.D.

The isomorphism between

R(L)

and

R(S) is due

to R.Steinberg C253 ; the non-degenerate irreducible constituent of the r e p r e s e n t a t i o n ~n=

Ind

Gn

~ ,

Bn i.e.the element

~(yn

)

i s t h e well-known S t e i n b e r g r e p r e s e n -

tation, § I0.

The P.Hall algebra

I0.1. Denote by

the subspace in

C(G n)

(see 6.I) con-

sisting of functions with support in the set of unipotent elements of

Gn. Put n~O

so ~

~

is a graded subspace of ~) C(G n) = R (q) •

~

(see 6.I, 9.I) .

Let p :R(q)@c

....

;~

be the operator of orthogonal projection onto ~ ; if ? ~

C(Gn)

then

unipotent elements of

p(f)

is the restriction of

in other words, f

to the set

Gn °

By the theory of the Jordan normal form, unipotent conjugacy classes of the groups

Gn

are parametrized by partitions: to a

of

116

partition

= (iI,'',ir)~ 9

~

corresponds

there

the class

consisting of elements with Jordan cells of orders Denote by % ~ ~ ~ ~

I~

~

the

li,12,...,1 r

characteristic function of

is a basis of ~ .

[Z(K~ )I

For ~

K~

K~ , so

we put

=)Gn)/IK~] ,

~n the next section we shall compute explicitly the map so the character values of vestigate ~

Gn

p, and

at unlpotent elements. Now we in-

in detail.

Proposition. p : R (q)~

(a) ~

~

~

is a Hopf subalgebra in is a Hopf

R (q) ~

C , and

algebra morphism. We call

the Hall algebra. (b) Let

o~

and

g

be structural constants of~

, gi-

I

yen by

Then

while u~K~

g~v F ; then

can be computed as follows. Let

I~I

= n

and

k

g~v

equals the number of subspaces

V~

~

inva-

r

riant under the class

u K~,

and such that the action of while the action of

u

u

on

on /g

V

belongs to

belongs to

KV.

Proofo (a) The definition of multiplication and comultiplication on

R(q) ~

~

at once that ~ i s ment

in terms of characters (see 9.~, 8.2) implies a Hopf subalgebra, while its orthogonal eomple=

is am ideal in

R(q) ~ C

such that

117

It follows that

p

is a Hopf algebra morphism.

(b) The expression for

g~y

can be derived directly from

definitions (see 9~, 8.2). The coefficients

g ~ Y also can be

computed directly, but the easier way is to use the self-adJoint hess of

R(q). Consider the inner product < ~ > ~

restriction of < ) > ~IZ(K~)I'~

I~

from ~

R(q) ~

in ~

on ~

, the

. Clearly, the basis

is dual to ~

with respect

to < ~ > ~ . So, by the axiom (S) from 1.4 the matrix of

m~in

Q.E.D, 10.2. Remark. The coefficients nomials (they are polynomials in the basic field). The constants

g ~ y are called Hall poly-

q, the number of elements of I Z(K~ )I

are computed in~53.

Let us evaluate I Z(K(n))I , which will be needed later.Choose an element

u gK(n ) i.e. a Jordan cell of order

n

with the

eigenvalue ~. Clearly, I Z(K(n)) # equals I ZGn(U) # , the number of elements of the centralizer of see that

ZGn(U ) consists

where

is a polynomial over

F

u

in

Gn . It is easy to

of elements of the form ~

F(u-~),

of degree at most

n-I

witk

non-zero constant term. It follows that I Z(~h~ I

= qn-~ (q-l).

iO.3. Now we show that the Hall algebra ~ as a Hopf algebra to

R~ = R ~ ~

, where

R

is isomorphic

is the universal

PSH-algebra (see §§ 3,4). Define the Hopf algebra morphism

PL

1t8

to be the composition

Rc ~

P

~(q)~

Theorem. (a) The morphism

(see 9.6)

p~ : RC ----->~

is an isomorp-

hism of Hopf algebras. (b) For

n>/l

we have -

P~(~) = ~

A

)

(I'q)(~'q2)'"(I-qr(~)'I)'%A

(see 3.7)

Theorem is proved in I0.4-I0.6. I0.4. Pirst we show that I0.3 (a) follows from the fact that all

p~(Zn)

are non-zero for

n ~ I ; in particular, it fel-

lows from I0.3 (b). Indeed, since ~

is an epimorphic image of

multiplication and comultlplication in ~ Theorem quoted in 2.I, ~

R(q)~ , the

are commutative. By

is the symmetric algebra of the space

of its primitive elements (see Appendix I, AI.I and AI.2 (a)). Since

R~

= ~ Ezi,z2,...7

(see 3.15 (b))

is primitive, the assumption that all plies that

p~

it follows that

p&(z n)

and each

are non-zero, im-

is an embedding. Since

p~

is an isomorphism, as desired.

I0.5. The identity

PC (zn)

119 follows at once from the fact that representation of

Gn

~

(xn)

is the identity

(see 9.6). To prove two other identities

from I0.3 (b) we recall some yell-known combinatorial results. For

n I>0

put (n) = (q-II(q2-1l...(qn-I).

~or

Og k % n

denote by

q the number of space over

k-dimensional subspaces in a n-dimensional vector

~q (we put also

(n) k_q = 0

if the condition

Ogkgn

does not hold). It is easy to see that the number of (ordered) k-frames in

~n q

equals

(qn l](qn q)... (qn qk-l) = q ~ i ~

~(n)/~(n-k)

.

In particular, we have

IGnl

n--~i~" n ~(n) =q

From the latter identity

(or,

directly, by definition of

,~,/n)q)

one can derive the relations

These relations in turn imply (by induction on

n) the "q-bino-

mial formula" : n

(a+b)(a+qb)(a+q2b)'...o(a+qn'I b ) = ~

J-~

(~)q.an-kb k

120 N

I0.6. Denote temporarily by sides of the identities

N

Xn' Yn' and zn

the right-hand

I0.3 (b), and consider the formal power

series X(~) =

~m

,

Y(~)=

J~.0

~ ~~n+z' ~"

z(~) =

Yn ~

and

m>~0

m$O ~

~n ~ L L ~ J ] .

~ . ~ , to ~ove th, ~ont~t~e° PL(Yn ) = Yn

and

p~(z n) : zn

it suffices to verify that

x(~) ~~(-~) = ~

d ~ ))' ~Y(-~) = ~ ). C%

:d

Expanding the products, we see that we must prove the identities m

(x)

( - I ) k Yk X-n-k : 0

and

('I)k(n'k)

k:0

- k : Zn

k:0

(n~I). One can easily compute the products

yk.~n_k

by means of

I0.1 (b). We obtain: _

yk,Zn.~

:

q

'

~L=m

(we use that for

u ~ K~

dim Ker(u-l) = r(~) ). Substituting

these expressions into (x), we see that it suffices to verify the identities

('l)k

q ~

k q

: 0

(r>~z)

;

F.>~O

(lq(lq2> L>~O

(lqrl

(r~z),

121

The first identity is a special case of the q-binomial formula for

a=I, b=-l, and

n=r; to prove the second one it suffices

to differentiate the q-binomial formula with respect to then to substitute again these values of

a,b, and

a

and

n.

This completes the proof of Theorem 10.3. Q.E.D. 10.7. We shall identify ~ Po

with

~

via the

and denote identically an element of . All structures on

red to ~

R

R$

isomorphism

and its image ~n

and results of §§3,4 can be transfer-

; for example~

On the other hand, we have another structures on ~

, namely

the inner product < ~ >9/ induced by the embedding ~ and the basis ~ ,

orthogonal with respect to < ~ > ~ ( t h e

inner product on ~

simply
).

The interaction of these structures gives rise to many interesting relations. Let us discuss some of them. 10.8o Since ~

is identified with

(see 3.6). induces the

C-linear form

which will be denoted also by ~ Proposition.

~c

.

acts on ~ n

,~oo~. ~e~n. t~e ~o= ~: ~ ~(~>

R~ , the form ~3c

- 0

S (%C,~) : 1

for I for

as follows:

~ ~ ~ ~, ~I~,~2~,... n~ O.

122

¢ The expression for

g~Ay

from I0.I (b) implies that

F

multiplicative. Since the value of

~

at any

O

is

xn = ~ ~

Q.E.D. This proposition can be reformulated as follows: for any irreducible representation

60 ~

R(~) n

identity one, the character value of ~ O. In II.IO

at the class

K(n )

is

we shall generalize this assertion.

As to the form

on ~

(see 9.6) except the

~

, by definition (see 8.2, 9.5) it acts

0

by

It would be interesting to calculate the right-hand side more explicitly. 10.9. NOW we compute the inner product ~ ~ >q/ elements of various bases of ~

. Note that ~

between

~satisfies the

axiom (S) from 1.4 with respect to each of inner products ~ ~ and ~

>O# " It follows that all arguments of Chapter I, which

are not based on the axiom (P), ma~e sense if one r e p l a c e s ~ > > by ~ ~Q,. In particmlar, all assertions in 1.9 except 1.9 (d) remain valid for operators ~ - - - * ~ ,

adJoint to the opBrators

of multiplication with respect to the inner product ~ ~ >Q/ , Proposition. extending

t : R-

(a). The operator ~R

t : ~---*~

obtained ~ by

by ~ - linearity (see 3.If, I0.8)

preserves the inner product

~

>q/ ,

(b) We have

~ X n , X n > q =~ Yn,Yn> q =

_q ~(n)

123

I

< Xn,Yn> q

I

< Xn,Zn> q Yn, Zn> q

=

(-I)

n-I

.

I

qn-I

< Zn, Zn> q

(o) For any

n

qn-I with

I : (ii,...,i r) ~

ii ~ 0

put

~q(~) = ~ (qli . I ) i=I Then for any

x 6~=

B~

we have

<x'zA> q = I__/___. ~q(~) < x,zi >

Combining this equality with 3.I7 (c) and 4.17 (b), we obtain the values of ~ x ~ , z l > q , < ~y,zk> q , < z ~ , z ~ >

q

and

kij (kij"I) (kij)~Mi'~ ~ j

~Dq (kij)

)

< xA, Y~> q =

~ q (the set

M~'~

J

is defined in 3.17 (c)).

Proo ~. (a). The proof is quite similar to that of 3.11 (a).

124

(b) The expressions for ~zn,Yn> q

follow

< Xn,Yn> q , ~ y n , Y n > q , and

at once from definition of ~ > > ~

(see 8.2)

With the account of I0.3 (b) and I0.5. By (a), < X n , X n > q =4 yn,Yn > q

~ Zn,Xn> q = (-l)n'I~zn,Yn > q

and

(see 3.II (b), 3.17 (a)). Finally, the equality Zn,Zn>

n

q =

qn-I is a special case of (c). (c). By means of the axiom (S), we reduce our equality to the case when 1.7, for ~

~

= (n) and

(n)

we have

x



runs over some basis i n ~ n. By

= O;

it follows from (b) and 3.I5 (a) that

q =

I qn-I

~Zn,Xm>,

as desired. (d) The proof is quite similar to that of 3.17 (c). Q.E.D. IO.IO. Corollary. equals

The number of unipotent elements in

Gn

qn(n-I) .

Proof. By definition of ~j and I0.3 (b) the number of unipotent elements in

~n

equals q

Now apply I0.9 (b) and I0.5. Q.E .D. This is a special case of the well-known theorem by R.Steinberg.

128 I0.II. NOW consider the elements ~ Since we know the decomposition of ~ (y~), and ( z ~ )

(see 10.7).

~

in ~each of the bases ~ ) ~

(see 4.16, 4.17), Proposition 10.9 in princip-

le enables us to compute the inner products ~ A ~ ~ ~ > ~

a n d ~ , ~ ~

~

. For a general ~

~K~~ , the

author does not know simple formulas for these expressions (except ( ~ , ~ _ ~ > ~ /

; see 10.9 (c)). Let us compute ~ A ~ ) y ~ > ~ /

(in II.IO this will be applied for the computation of dimensions of all irreducible representations of ~

linear form

Gn ). Denote by

~

the

A

~-~C~

acting on each

dq (x) = ~ X,Yn>

proposition. (a) (b) For any A E ~

The form

~by

q

dq : ~ - - - ~ C

is m~Itipllcative,

with c.f. (~) = (11,12,...) (see 3.7)we have

(qh(a) 1) (here h(a) is hook length of a; see 6.5 (b)). Proof. (a) The proof is quite similar to that in 3.6. (b) Under the notation of 4.I6 (a) we have

dot (x i r+j)i,j=l, Applying the mu!tlplicative form dq

r to this determinant, we

obtain that dq(~i\~ ) - det(~q(Zi.r+j) (see I0.9 (b)). The computation of the determinant is quite si-

126

milar to that in 6.5

(the details are left to the reader). Q.E.D.

I0.I2. One can use the information on ~

to obtain a lot

of interesting identities° Let us consider one example. By 3.I6 we have

Let us apply to this identity the multiplicative form ---~

dq : R --

coefficientwise (see 3.16 (1)) and use I0.9 (b). We ob-

tain the identity in ~ [ [ ~ 3 exp( ~ k~I

which can be rewritten as

~ k(qk-'I'') )= n> ~

0

~ (nq--~) ~q

~Yl

10.I3. Now we consider the relationships between the basis ~%~} in ~ -

and various bases in

R (see §§3,4). Let (Q(~>~))

be the transition matrix between the bases C ~ )

~he+ coefficients

and (z~),i.e.

Q(J ,~ ) are called Green p o ~ o m i a l s

happen to be polynomials in

(they

q) ; they play the important role

in the character theory of the groups

Gn

(see ~ ] and II.8

below). By I0.3 (b), for ~ ~ 0 n )

we have

= (Z-q) (I-q 2) ... (z-q

obviously, all values

Q(~ ~ ~ ) can be expzessed in terms of these

and the structural constants

Each of the bases ( ~ A ) pect to < ~ > ~

);

, and we have

g~w

" In particular,

and (z~) is orthogonal with res-

127

.l-J+l,,(1)

]Z(K)I

(see I0.I, I0,9 (c), and 3.18). So we have the inversion of (1):

(:ll~.),

~1~, a =~

. IZ.(.K~i~lii

,Q(~

) r'r

Combining (~) and ( ~ ) with the results of §§ 3,4, we obtain the transition matrices between ( % ~

)

and all bases from §§ 3,4

in terms of Green polynomials.

The main p r o p e r t i e s of

Q(]~)~ )

are given in

tISJ

,Prop.

I

5.16; all of them become very transparent via our approach to the Hall algebra ~ .

For example~comparing (x) and (~-)#we ob-

tain the orthogonalit 2 relations:

We shall give two more identities on compose

In

Q(~ j ~

with respect to the basis ( ~ )

). Let us de-

by two different

ways. By I0.3 (b),

On the other ha~d, by 3.17 (b), 3.18, and (x) we have

Comparing these expressions, we obtain the identity

128

# Similarly, using

Yn

instead of

Xn, we obtain the identity

§ II. The character values of

G•(n, Fq)

at unlpotent ele-

ments II.I. In this section we compute the character values of all irreducible representations of the groups

Gn

at unipotent ele,

ments. Clearly, this problem is equivalent to that of explicit computation of the morphism p : R(q) We recall that

with

R

> ~4~

R(q) = f ~R(~),_~ . ~ and each

via the isomorphism ~

hand, ~

is identified with

10.3). Since

(see I0.I)

p

R(~)

is identified

(see 9.3-9.5); on the other

~)via

the isomorphism

PU

(see

is a ring homomorphism (see IO.I (a)), to com-

pute it explicitly one has omly to compute for each f £ ~

the

composition

Denote this composition by

p~: R k

~D " Clearly, p~

is

a Hopf algebra morphism, but it does not preserve the grading : if

d e g ~ = k i.e. ~ E

_ ~ (Gk)

11.2. Our definition of (~) = I proof

~

then

pf

maps

R n to (~)kn "

is based on the identity

from 9.4. Now we prove this identity. The present

is due to J.N.Bernstein; it is based on the technique

129

o(~.,']>lJ

developed in § I0. Another proof of the inequality

(i.e. of the S.I.Gelfand theorem) will be given in 13.4. The proof finishes in II.6. Since ~

is identified with

R C , the f o r m ~x

(~)from

3.6 induces the C -linear mnltiplicative form ~ - - - * ~ will be denoted also (~bY

~ (~-'~)

Lena. The form g) on

R(q)

, which

(see 9.4) is equal to the

composition /%

Proof. Remembering the definition of

~t

(see 9.6), we

see that Lemma is equivalent to the fact that S the orthogonal complement ~ ± of ~

in

vanishes on

R(q)c , i.e. on the

space of functions, taking value 0 at unipotent elements. This fact follows at once from definition of ~

and 8.2. Q.E.D.

the composition

11.3. Denote by Sw: R(q) -----~

By I0.8, for every ~ 6 character value of ~

A

(Gn) the value ~' (3T) is the

at the class

K(n ) .

Lemma. (a) There exists an involution

60~-- ~ 004 of the set

_~

(Gn) such that ~4z__ ~

and ~(bO)-- _+ ~'(bO~) (b) For

n~l

for

every

for every cuspidal ~

,

60 6 ~ ( G n ) .

we have

2. qn-l(q -1). Proof. (a) For each ~ e ~ aut omorphism

consider the unique non-trivial

of the PSH-algebra

R(~9 )

(see 3.I (f)).

130

We extend these automorphisms to the automorphism ~)~ > ~ of the ~SH-algebra

R(q) ~ ~ g~ R(}p ) ; clearly, p ~ = ~

~g~, By definition of

t

(see 3.II), for any d ~

for any ~

(Gn) we

have

OO £ where

=

±

T(~),

T : R(q) .... ~ R(q)

is the conjugation of

other hand, by definition of ~ac and ~

R(q). On the

~ for any

J(2~,~

we have

where

T~:,~---~

is the conjugation of ~ .

By definition

(see Appendix I, AI.6), T commutes with any Hopf algebra morphism; in particular,

~ o p : p~T : R(q} ----*~6 (see IO.I (a)). S ~ r i z i n g

for all ~

& ~

and applying Lemma II.2swe see that

(Gn) , as desired.

(b) By (a), it suffices to verify that

Set ~I

= I g(K(n))I'~(~1) ~ ~

(see IO.I)

By definition,

Therefore, the left-hand side of (x) equals

131 as desired (see I0.2). Q.E.D. II.4. Now we are able to prove the S.I.Gelfand theorem,i.e. that

~(~) ~ 0

for any f ~

~

prove that ~ I ~ ) ~ O, i.e. that Since ment

p : R(q) -----, ~ p(~) ~ ~

nal to

zk . But

d e g ~ = k.

is a Hopf algebra morphism, the ele-

~$5 (Zk) = I

p(~)

(see 3.I5), so

p(~) = O. The equality

the character of ~

is proportio-

~aC (P(~)) can

p(~) = O

means that

takes value O at all unipotent elements

Gk; in particular, dim ~

that

~ c (F(~)) ~ O. Let

is primitive. Therefore,

be zero only if

in

. By II.3 (a) it suffices to

= O. This contradiction shows

(~) ~ O, as desired. II.5. To prove the Gelfand -Graev theorem, i.e. that

~(~3)% I

for any

6&~ ~ _ ~

(Gn) , we need the following well-

k~lown. Proposition. For each sentations of P~q[TJ

Gn

n~ I

the number of cuspidal repre-

equals the number of irreducible polynomials

of degree

n

with leading coefficient I and non-

zero constant term. Proof. First we compute the number of conJugacy classes in Gn . It is well-known that these classes are in a natural oneto-one correspondence with isomorphism classes of dules

V

such that

dim~q V = n

and the operator

Fq ~T] - moT : V --*V

is invertible. The structural theory of modules over principal ideal rings implies that each such module

V

decomposes into

the direct sum of primary cyclic ones, and the isomorphism class of

V

is determined by this decomposition. Any primary

cyclic module over

~q IT]

has the form

/ (p1) ,

132 where

P

is an irreducible polynomial

its leading coefficient to be I) and

over l~O

~q

(one can assume

.

Summarizing, we obtain the following description of conjugacy classes in polynomials

Gn. Denote by P~q[T]

~f

the graded set of irreducible

with leading coefficient I and non-zero

constant term (see 4.I9). The conjugacy classes in rametrized by the set ~ : P ~

Sn (~l,) ~

~

~ (Ii(P) , 12(P),...)

~q~T~ /

(plk(P)

For example, the unipotent class the function ~ for

are pa-

): to a function

there corresponds the isomorphism class of the ~)

Gn

such that

T

• q KTJ - module

).

K~

from IO.I corresponds to

(T-I) = ~

, and

~

(P) =

P ~ T - I. Since the number of irreducible representations of

Gn

equ-

al8 its number of conJugacy classes, it follows that

1Sn(~, ;ff))l =

I Sn

((~'; if))/

for

n>/I

(see 9.4; mote that we don't use the explicit form of the leomorphisms

but only their existence). From this we easily

derive by induction on and

n

that for every

n>~I

the sets

have the same number of elements of degree

n.

Q.E.D. II.6. Proof of the Gel~andrGraev theorem. proof is to give a lower bound for the

The idea of t~e

sum

g

we shall show that the assumption that

~ (~J)> I

for some

133 60 6 A ( G n ) ,

implies that S >qn-I (q-I)~

This contradicts II.3 (b). We shall use the notation of 2.5,By 2.5 (a),

where Sn(~: Z + )

= ~6

S(~'~Z +) # d e g ~

= ~(f)'de~

= n}

Let us prove that ; = qn-I

I SnC~

(q-I)

Indeed, by II.5 we may replace the set ~

by

. Associating

T ~ s(~", Z +) the polynomial

to ~

pe~'

we obtain the bijection of nomials

F~q

[TJ

Sn(~li,~.-n~/+) with the set of poly-

of degree

n with leading coefficient I

and non-zero constant term. The number of such polynomials ale

equ-

qn-I (q-l), as desired. ~'or any ~ ~ S

(~ ~+)

W

put

A

By definition, an element of .-/~-(~) is an irreducible censtituent of

We have already proven[th~that ~Cf) > 0 II.4). Since

for

i~ =~tiplicative, we have

(s.°

~Cscg)>O.Since

134 is a positive form on

R(q)

that for any ~ there exists ~ E

with integer values, it follows ~(~)

such that ~ ' ~ ) ~

I.

Therefore,

s cT)>.l

for all TE 5( ;Z +)

,

It follows that

S = ~--

S(?)~

~ I

deg~ =n moreover, if

= qn-I (q-I) ;

deg~=n ~'(~)>I

for some ~ O E ~ ( G n)

then this ine-

quality becomes strict. It remains to apply II.3 (b). Q.E.D. II.7. Now we compute the morphism

p : R•

~

Theorem. L e t ~ P~ (Zn) =

Proof.

Since

zn

(see II.I).

-~ and

deg~ = k. For

n>iI

we have

(I) n(k-l) Zkn • -

is primitive and

p~

is a Hopf algebra

J

morphism, it follows that

p~ (zn)

is a primitive element of

J

(R~)kn , hence it is propertional to

Zkn

(see 3.I5 (a)). To

find the coefficient of proportionality we use Lemma 11.2. We have ~(Pf

(Zn))=

~(~(z

n) ) = ~ ( Z n ) =

(-I)n'l ,

while ~ (Zkn) = (_Ifkn'I (see 3.I7 (b)). Hence the coefficient equals (-I)kn'n = (-I~ Q,E.D, 11.8.

Theorem 11.7 implies at once the J.Green formula

for character values of irreducible representation of

Gn

at

135 For any f f = (mi,..,m r) ~ ~

unipotent elements. denote by

the p a r t i t i o n

( ml,

(~,~L i.,-Z~A) of

any finite family

and k E ~ ; for

partitions denote by

.~_L

the union of these partitions (for example,

(4,2,I) _II (32,2, I) = (4,32,22,12 )

).

The J.Green formula says that the character value of the irreducible representation i ' ~ see 9.4-9.5) at the class

Here the

Q(~ ~

)

K~

of

Gn

(here ~ ~ S n ( ~ - ~

);

equals

are J.Green polynomials (see IO.I3)

!

the inner products ~]]°~ ~ / ~ are computed in 4.I7 while the ! constants c~c in 3.I7 (c) (note that the values ~ ~V~> ~ / ~ are the character values of irreducible representations of symmetric groups; see 6.3). To prove the J.Green formula it suffices to decompose the element

withtlnrespect to the basis ~ ' ~ _

and then pass to the basis ..-F~J~ ,

of " ~ ; the details are left to the reader. Note that the very simple formula of Theorem II.7 is essentially equivalent to the J.Grsen formula. II.9. Corollary. Let ~

and ~ )

be such that

deg~--I~l = n. Then the character value of ~ K~

equals

at the class

136 in particular, this value does not depend on ~ . This follows at

once from II.7 and I0.3 (b).

II.IO. Now we apply II.7 to compute the character values of irreducible representations of K(n )

Gn

(the character value of ~

dimension of ~

at the classes at

K(in)

and

K(in) Ss, evidently, the

).

Proposition.

Let

~ ~

Sn(~

) and ~

ponding irreducible representation of

be the corres-

Gn . (see

(b) The character value of

~~

at

K(n )

is

IO.II)

O unless any

has only one row; in this case this value equals (.i)n'~i~(~)I Proof. (a) Define the form

,~

d : R(q)

to be the

composition

R(q) ........ D ~

dq r ~

(see I0.II)

Clearly,

d(~ =

dlmJ~

for any ~

By IO.I (a) and IO.II (a), d

~

(%)

is multipllcative. Therefore, our

assertion follows from the identity

By I0.9 (b), dq on

can be defined as a multiplicative form

R ~, whose value at

zn

is

(-I)n'I/qn. I

(n~I). So

(z) follows at once from II.7. (b) The proof is quite similar to that of (a). One has only

137

to replace

d

S

by the form

t (see II.3), and the identity (x)

by the assertion that

on

Rn

.

Q.E.D. II.II. We conclude this section with

a very simple proof

of the Macdonald conjecture (see C24~, 6.II). proposition.

For any 6 ~ ) ~ _ ~ ( G n)

the sum of character

values of 6~ over all unipotent elements of 'dime0

for some

Proof. equals

Gn

equals

±qm,

m62.

Under the notation of § IO, the sum in question

IGnl, < p ( ~ ) , X n > q q

(see IO.I. I0,3 and I0.7). We have

= IGnl, q

= ±IGnl, ~ p(~)t),yn>

q

(see 10.9 (a), Proof of II.3 (a) and I0.3 (b)). It remains to prove that (?e

dim~t/dimO0

Sn(~,- ; ~

for p ~

) ) then

is a power of LO t = < ? t ~

q. Clearly, if C O = [ ~

, where

~t(~)=

(~(~))£

. Our statement follows at once from II.I0 (a) and

IO.II (one has only to observe that the set of hook lengths of any Young diagram coincides with that of the transposed diagram). Q.E.D. II.12. Remark. In [26~ C.W.Curtis defined for any finite thevalley group

G

the dualit~

operation

R(G) -----~R(G) ~)~">&Ow)

The definition easily implies that ~0~ = (-I)n, T(~) ) where

T

is the conjugation of

for any

&0 6 R(Gn) ,

R(q). In g26~,~27] the main

138

properties

of

T, namely that

T

automorphism and an isometry,

is an involutive Hopf algebra

are generalized

Chevalley groups. Using these properties

for all finite

and one result of

D.Alvis in [27] proved the weakened form of the

T.A.Springer,

Macdonald conjecture

for all finite Chevalley groups

(the argu-

ments in [27] are similar to those of II.II). § I2. Degenerate

Gelfand-Graev

modules

I2.I. By degenerate Gelfand-Graev modules we mean representations of

Gn

induced by various one-dimensional

tions of the subgroup matrices.

U = Un

of unipotent upper triangular

I.M.Gelfand and M.I.Graev

reducible representation of these modules.

of

representa-

Gn

in ~I33 proved that any ir-

can be embedded into some

We obtain the more precise result, computing

explicitly the spectrum of any of these modules. For each ordered partition the character

(ki,...,kr)

~ kl,...,k r

((uij)) the sum is over all

i

except

- kr. I , For example, ~

= ~

of

=

U

group in

~ ki~...,k r

(

ui,i+1 ),

n-kl, n-kl-k2,...,n-k I -... (see 9.4) while U

~4,4,,,,~4

: I.

is conjugate to

under the action of the diagDnal sub-

Gn. It follows that any degenerate

le is isomorphic

n , we define

by

It is easy to see that every character of one of the

of

Gelfand-Graev modu-

to one of the modules

• ..

=

ki,...,k r Un

Theorem.

Let

~

~

Sn( ~

~ "~)

and f ~

be the cortes-

139 ponding irreducible representation of

~T}~4,,,~,~

multiplicity

Gn (see 9.4, 9.5). The

of ~

in the module~),~,j~

is

where the sum is over all functions ~----~(ll(~),12(~),...,Ir(~)

) from

~

to (Z+) r

such that ii(~)'deg~ (the inner products < ~ ) ;

= kI ~A>

for

i = I,...,r

, appearing in the answer,

I

are computed in 4.I4). Theorem is proved in I2.2-I2.3. I2.2. By definition, ~kl,...,k r where

I

= iu, ~ki,...,kr

(I),

stands for the identity representation of the identity

group (see 8.I). According to ~.I (b), for any 6 0 ~

~

(Gn) we

have

We want to express structure on

-- rU'Tkl,. .. ,kr (~) '

rU,~kl,...,kr,

in terms of the PSH-algebra

R(q). Define the operator

D : R(q)

> R(q)

to

be the composition

id@~ ~ R ( q ) ~ 7 / - - ~

R(q) - - ~ R ( q ) @ R ( q ) (see 9.1, 9.4); denote by

Dk

R(~)

the homogeneous component of

D,

140

acting from Len~.

R(G n) to

R(Gn_k).

The operator

ru, ~

: R(G n)

> R(Go)= ~

ki,...,k r coincides with the composition Dkr o Dkr_IO ... o Dki Proof, Uk =

Put u = (uij) ~ U n

I uij =

Remembering the definitions of

m

and

ij

for

I~i~n-k~

(see 9.I,9.4)

and

using 8.I (d), we see that Dk : R(G n) -----~R(Gn_ k) equals the composition ruf ~ k,~

o

rUn-k,k,I

To compute the composition several times Prop.

Dkr o Dkr_i o • -. o Dki we apply 8.I (c). The easy computation shows that

this composition equals

rU, ~ki,..,kr Q.E.D.

12.3. According to 9.5 and I2.2,

So Theorem I2.I follows at once from the next. Lemma. (a) Dk ( ~

vi) = ~DmI(vI),Dm2(V2),

where the sum is over all

,,,,Dmp(Vp),

(mi,...,mp) 6 £~+)/~with ~-~mi= k.

141

(b) Each subspace hence under all Dm

is

0

on

im

R~)

D m. If

m

R(q)

is invariant under

is not divisible by

R(~); but if

m = 1. deg~

deg~

D

and

then

, then the

, l ~ +

op erat or

coincides with

~o

~

,

pro0f. Part (a) means that

D

is a ring homomorphism. This

follows from the Hopf axiom (H) of

R(q)

and the fact that ~"

is multiplicative (see 9.I, 9.4). Part (b) is an immediate consequence of definitions of

D

and

~

(see 9.5).

/

Q.E.D. I2.4. Corollary. (k~,... ,~) 3~5ki,..,kr

If two sequences

(kI,...,k r)

and

are equal up to a permutation then the modules and ~ k ~ , . . . , k rt

are isomorphic.

12.5. By means of the Frobenius reciprocity we can reformula te the Gelfamd-Graev theorem (see 9.4) as follows: any non-degenerate irreducible representation of

Gn

occurs in ~ n

with

multiplicity I. We extend this result to all irreducible representations of Proposition.

Gn . For any CO E _ ~ (Gn) there exists a degene-

rate Gelfand-Graev module ~

, containing CO

ty I. More precisely, let CO ~-

, where ~ ~

with multipliciS(~ ) ~

1, and

c.f.(

choose ~

=

~ k i , . . . , k r , where

ki ~

mi(~),deg ~

(i = 1,2,...)

This follows at once from Theorem 12.I and the fact that

142 0 , ~X >

if c.f.(~) is lexicographically higher

than

(

Iif

e

) )

(,ee 4.i). Q.E.O. 12.6. By a Je~en~rate Gelfand-,raev model of ~O ~ D l we mean a realization of ~ kl,..,kr

as a submodule of the module

from 12.5. This is an analogue of the degenerate

Whittaker model for representations of the groups a

(%)

~ - a d i c field, obtained i n ~ 2 ~ .

GT,(n) over

Let us give an application

of this model. Proposition. The Schur index of any irreducible representation of

Gn

Proof.

equals I. According to the theorem by R.Gow

~28J

, any degene-

rate Gelfand-Graev module is rational i.e. is defined over ~. We recall that the Schur index of ~ where

KI

is defined as ~ K I : Ko~.

is the field of definition of ~ 3 ~ K

2

is the mini-

mal field containing all character values of ¢~). It follows that if 6~ J~

ocuurs with multiplicity I in a rational representation

then the Schur index of

of J~

onto

~

CO

is I (indeed, the projection

is just the character of 64) , considered as an

element of the group algebra; see e.g. ~ 9 J

). Therefore, our

assertion follows at once from I2.5. Q.E.D. When char

~q ~ 2, the

proposition is proved by Z.Ohmori

LI6J. The proof in ~I6~ is also based on the R.Gow theorem but it is rather roundabout.

I43

§ I~.

Representations of general affine groups and the branching rule

I~.Io Denote by

Pn

the subgroup in

rices with the last row

Gn

consisting of mat-

(O,0,...,O,I); evidently, Pn

can be

realized as the group of all affine transformations of an (n-I)dimensional affine space over irreducible representations of

~q. In this section we classify Pn

of irreducible representations of to

and describe the restriction Gn

to Pn ' and that of

Gn. I . Clearly,

Pn

has the abelian normal subgroup

Vn=Un_i, I

(see 9.1), and decomposes into the semidirect product =

Pn

Pn =

Gn_I'Vn. It is easy to verify that the set of characters of

Vn

has exactly two orbits under the action of

Gn. I , namely

the orbit of the identity character I and that of the character (see 9.4). Clearly,

stabGn.i

I = Gn_ I ,

stabGn_ I ~

= Pn-I

The next proposition follows at once from the well-known description of irreducible representations of a semidirect product

G°V, where

V

is an abelian normal subgroup in G.V

(see ~9~ , 9.2). Proposition. The operators r V n , I ~ rv n,~ : R(P n) ------~R(Gn_I) ~ R ( P n _ I and IV n " ,I e i V n , ~ : R(Gn_I) e R(Pn_I) ,,

> R(P n)

)

144

(see 8.I) are mutually inverse isomorphisms between T-groups R(Pn) and

R(Gn_I)@R(Pn.I).

13.2. Applying Prop. 13.q successively to we obtain that

R(P n)

Pn,Pn_l,...Pl ,

is isomorphic as a T-group to

R(Qn.I)@R(~n.2)@

... @ R ( % ) .

More precisely, let us define the operators R = R n : R(Pn) -----~=~I R(Gn,k)

and

n I = In : ~ R(Gn. k) ------~R(Pn) k=I by induction on Rn

n

as follows:

is the composition

rVn,I @

R(Pn)

id • Rn. I R(Pn_I) ............ )

r V n ~ R(Gn_I ) @

n

R(Qn. I) ~ ~9 (the definition of Proposition.

In R

between T-groups

and

R(Pn)

R(Gn_I_ ~ ) =

k-4

is quite similar). I

are mutually inverse isomorphisms

and

L~4

R(Gn_ k) .

This follows at once from I3.I. In particular, irreducible representations of

~n

are in

one-to-one correspondence with irreducible representations of the groups

Gn_i, Gn. 2 ,°.., G O . By 9.4, they are parametrized

by the set

li

; 9 ),

145

the representation of

Pn' corresponding to a function ~

, is

I3.?. The next result will play the crucial role in our description of restriction of representations that of

Pn

to

equals

D-I

Gn

to

Pn

and

Gn. I .

Proposition. R o Res Gn Pn

of

(a) The composition

: R(Gn) ....

~::7

R (Gn_ k )

k=I (where

D

is defined in I2.2).

(b) The composition R o Ind Pn Gn_ I equals

:

R(Gn_ I)

> ~ k=I

R(Gn_ k )

D.

Proof.

(a) Let us compare the components of the operators

R • Res Gn and D-I, acting from R(G n) to R(Gn'k)'G By defiPn nition of R (see I3.2) the component of R ° Res n is the Pn composition rV

n-k+I 'I

o r¥

n-k+2'~

o rv

n-k+3'~

0..

"

o

rVn'~

On the other hand, the corresponding component of ~,

,

D-I, i.e.

equals the composition r t Uk,~

o

rUn.k,k,I

(see the proof of I2.2)

•

Bach of these compositions can be computed by means of 8.I (c); we obtain that they coincide, as desired. (b) First we compute the compositions

146

rVn,l

o IndPn

: R(Gn_ I) ~

R(Gn_ I )

~-I and rV

o

Ind

n~T

Pn

: R(Gn_I) -

>R(Pn.I).

Gn_ I

Applying the general theorem on the composition

r~1

(see 8.3),

we see that the first composition is the identity while the Gn. I second one equals Res ; as usual the details will be given Pn-I in Appendix 3 (see A3.7). Remembering the definition of

R, we see that the composi-

tion

R~ind

Pn

: R(Gn_I) ~

R(Gn_!)(~

Gn-I equals

id ~

~-4

(~ R(Gn.l_k)

L:~ (R~Res Gn'I

). It remains to apply (a).

Pn-I

Q.E.D.

13.4. As a first application of Prop.13.3, we give another proof of the S.I.Gelfand theorem, i.e. of the inequality g~ b (~) ~ 0 ( ~ ) ; see 9.4, II.4. Since ~ is primitive, the definition of

D

(see I2.2) implies that

But by I3.3 (a), (Dol)~5 Therefore,

~ 0

S (~)

for any non-zero

~ o, ~s desired.

147

I3.5. The next theorem gives a complete description of restriction of irreducible representations of of if

Pn

to

Gn. I . For ~

~I(~)__~

~(~)

~! ~

S(~

for all ~

~

Gn

~

to

Pn

and

) we write ~(-~

(see 4.3).

Theorem. (a) If y ~ Sn ( ~ ~ ~D) then the restriction of the irreducible representation ~ - ~

of

Gn

to

Pn

equals

~ ~'~, the sum is

over all ~'--~ ~6'

(b) If ~

S(~ ; ~

except

) and

~ o

~/=

deg~

tion of the irreducible representation

~

n

then the restric-

In S W ~

of Pn

to

%-i

equals

t~o

o~ ~o ovo~ ~-

~

.

~o ~ ,

~

soo~~

~

~-

In particular, the restriction of any irreducible representation of

Gn

(Pn) to

p!oof. By I2.3

Pn

(Gn-I) is multiplicity - free.

and 4.3p

T'~T ;~Ith the account of (x), each assertion to be proved follows from the corresponding part of Prop. 13.3 (to prove (b) one has to apply the Frobenius reciprocity). Q.E.D. I3.6. Corollary. (S.I.Gelfand ~I4J). The restriction of any cuspidal representation

~

and equals In ~ }

Pn = Ind ~ ? Un

of

Gn

to

Pn

is irreducible

148 in particular, it does not depend on ~ I3.7. Corollary.

.

An irreducible representation

remains ~rreducible, being restricted to it has the form

~qS~

of

Gn

Pn' if and only if

, where ~ 6 ~

and

n=k.deg~.

This corollary shows another way to define the isomorphisms ~(see

9.5) i.e. another approach to the classification of

irreducible representations of the groups ach for the

Gn . Such an appro-

~ -adic groups was developed in~2] .

I3.8. Consider the chain of groups Got_ PI ~ G I ~ P 2 ~ G2 ~

....

Theorem I3.5 says that the restriction of any irreducible representation of any group of the chain to the previous group has simple spectrumpand gives the explicit description of this speclrum. This is the branching rul 9 , similar to that for the chain of the groups Sn[G ~

where

G

Sn

(or, more generally, of the groups

is abelian; see 6.4 (2),7.9); of. also the

Gelfand-Cetlin Rule for the chain of

unitary groups,

Restricting an irreducible representation of and then to

Let ~ ~ S n ( ~ ~ ~ )

The multiplicity of ~ ' ~ ~

~

such that

to

Pn'

Gn_i, we obtaim.

Corollary.

~

Gn

(Gn)

to

T =d

6JL(G~_ I ) ~-

and ~ 1 6

Sn_I(~ ; ~

in the restriction of

Gn_i, equals the number of

~1/~ S ( ~

,~?

T',

This result in another form and by a quite different method obtained by E.Thoma L183 .

)

Appendix I. Elements of the Hopf algebra theory AI.I. For convenience we collect together the general results on Hopf algebras, needed in this paper. We use the notation amd terminology from 1.3-1.5. Theorem. a field

K

(i)

Let ~

be a connected quasi-Hopf algebra over

of characteristic

I = P ~ 12

Then ~

O, satisfying

(see 1.5).

is the symmetric algebra of the subspace

is a polynomial algebra in any basis of

P, i.e.

P. In particular, J~ is

a Hopf algebra with commutative multiplication and comultiplication. Theorem is proved in AI.3-AI.5. AI.2. Remarks. of ~

(a) Suppose that any homogeneous component

is finite-dimensional, and that ~

has a non-degenerate

inner product satisfying the self-adjointness condition (S) from 1.4. According to 1.7, these assumptions imply (I), so Theorem AI.I

is applicable. In this work we use Theorem AI.I only in

such a situation. (b)

It is proved in § 4 of ~8] that (I$ and so the conclusi-

on of Theorem AI.I is equivalent to the fact that ~

is a Hopf

algebra with commutative multiplication and comultiplication. AI.3. Lemm~. Let J~

be a connected quasi-Hopf algebra over

arbitrary commutative ring, such that P ~ 12 = 0 . Then the multiplication in ~

is commutative and associative.

A

Pro of.

For

x,y ~ ~

put

~x,yJ = xy - yx.

Commutativity of the multiplication means that Since

~

is connected, the subspace

~O

~x,y~

O.

lies in the centre

150 of 4

, so we assume that

x ~ ,

y~ d~

where

k,l>0.

By

the Hopf axiom (H) we have me

x,

=

m x,m

=

x@l

+ Igx

+ m

(x),

y®I

+ Z~y

+

+ m+~(y)~ . Using induction on

k+l, we can assume that

each of the elements mutes with

sO E x , y j

x~T

y ~ I, I~ y, and

and

I~x.

m+~(x) commutes with

m+(y), while

m (y)

com-

Hence,

~ p. On the o t h e r hand,

~ x , y ] ~ I 2, hence ~ x , ~

= O, as

desired. The proof of associativity is similar (one has only to consider

x(yz) - (xy)z

instead of

~x,y] ).

Q.E.D. A1.4. Now let J~ satisfy the hypothesis of Theorem AI.I. Deh

note by

S(P)

lication on of

P

the symmetric algebra of S~P)

satisfying

P. Consider the comultip-

(H) and such that all elements

are primitive. Clearly, S(P)

becomes a Hopf algebra with

commutative mmltiplication and comultiplication. multiplication

in A

By AI.3, the

is commutative and associative,

so we ob-

tain the natural Hopf algebra morphism p

:

We must prove that The fact that

p p

is an isomorphism. is an epimorphism,

way from the fact that To prove that Lemma. ra

S(P)

p

P

follows in a standard

is mapped epimorphically onto

I/I 2.

is a monomorphism we use the next.

The subspace of primitive elements of the Hopf algebcoincides with

P.

151

Let us derive from this Lemm~ that pose that ment in u~P.

Ker p ~ 0

u

is monomorphic.

Sup-

be a non-zero homogeneous

ele-

Ker p of least possible degree. Clearly, By Lemma,

least degree in

m~(~) ~ 0. Since

Since

p

u

deg u > 0

and

is an element with the

Ker p,it follows that

(pep)

O.

(O

and

on ~ n

by f-I(u)

Clearly,

= -f(u)

f-I~G(~

- moCf®f

)

-I

)o

for

u

is well-defined and is inverse to f.

Part (b) follows at once from definitions. Q.E .Do AI.9. In terms of AI.8 the conjugation of a Hopf algebra is the element of id : k

~ ~

G(~v4

), inverse to the identity morphism

. Therefore, AI.6 (a) follows from AI.8 (a). The

next proposition follows at once from definitions.

>

154

Proposition. If jx, and

j)

are connected Hopf algebras

then

is the conjugation of AI.IO. Proof of AI.6 (b).

First we prove that if the co

multiplication in yjt is commutative then

D =1 ..4 is a coalgebra morphism, i.e. the diagram

is commutative. Since

m * is an algebra morphism, by AI.8 (b)

we have the group morphism

G(id,m* ) t a(JiyJl ) It follows that the elements

> G(Jl,A&Jt). m * and

m*o T

of

are inverse to each other. It remains to verify that also is inverse to identity element of (T^T)° m

where

m * , i.e. that G ( ^ Jb®Jt

m* x f (T&T) ° m

] is the

B

)• 7 definition, m*K

j is the composition

ia is the multiplication in Ji,®Jt

the comultiplication in jt- implies that

. Commutativity of

(m~ where

m

m ~ )~m

~

= -m- o~ m

,

is the co~ltiplication in

~

. Hence,

m

4 By AI.8 (b), i d ~ @ ~ G ( ~ ~

A

~ (T~T)

g~

)

identity element of One c ~

G(%~@~£

)

is the

similar way, that if the ~ l t i p l i T

is ~

algebra

morphism.

T 2 = id. It suffices to v e r i ~ that

is inverse to TxT

m

)) as desired.

is commutative then

It r e ~ i n s to prove that T26 G ( A ~ %

so its composition with

prove in a ~ i t e

cation in ~

is the identity element of

T, i.e. that

2 = eoe ~

.

By definition,

Since

T

is an algebra morphlsm, m~(TgT)

= Tom

we

have

,

hence T ~ T2 = T o

~id~

x T U - T o e oe m

= eoe ~ . Q.E.D.

Appendix 2. A combinatorial proposition. A2.1. In this Appendix we complete the proof of Prop. 4.I8. Let us give another combinatorial definition of the coefficient ;/

g~v

(see 4.18). Define the linear order relation, ~j as

on

oZlows:

(i,j) ~j (it ,j~)

if and only if either and

j> jl .

i < i g , or

i = i~

I56 Let ~

be a

of ~ X ~

subset

N X N . We say that

f

and

f

satisfies (J) if

ordered sets ( ~ ) ~ ) - - - - - ~ C ~ × ~ ) ~ j ) if

x, x t 6 ~

a mapping from ~

and

X~p

xt

(see 4.I2). We call a picture

f

is a morphism of i.e.

then

f ( x ) ~ j f(x I) 0

any biJection

ween two skew diagrams such that

f

both satisfy (J); denote by ~ ( , ~ 4 ~ 2 ) t h e

res

f :~

set of pictu-

o

It turns out that f :~,-~~

f :~4~z~2bet-

and the inverse biJection

f-I

.~"~-~

to

g2Y~Y

equals the number of pictures

. The proof of this fact will be sketched in

A2.3(b). Note that this is not significant for us: we include the present formulation of 4.I8 only since it is classical. Now, in principle, the reader may forget it; we shall prove that the inner product ~

~>equals

the number of pictures

f:~--~

Remembering 4.I8 (mm), we see that this follows at once from the next combinatorial. Proposition.

Let

ram such that IWI
~ I

by

(j = 1,2,...,1)

there exists x; =

162 Evidently ,

f

is a bijection of

to the reader to verify that f

\ ~ a~ onto ~ S .

~

bo

We leave

is a regular point o f ~ ,

and

is a picture; this can be done in a quite similar way as for

Algorithm E (cf. the next item). By definitions, Algorithms E anf R are inverse to each other in the obvious sense. In particular, this proves Prop. AY,I in the case when

[~I = I ~ + I. in the general case, one

has to use the iteration of these algorithms. More precisely, let

f:V~~

I be a picture, ~ l ,...,b

=

3g'

•

_ U ~b(I),b (2),.

to see that each ~ point of ~ b =

0,...,k)

define

..,b (i)}

" Let us define the pictures

for

i~I fi-I

one can easily

verify that

i

and

~=

the correspondence inverse mapping

f ~

.

(0~$gk). b (i)

It is

easy

is a regular

fi:~-~~(i=

as follows. We se~

b (i) . Let ~ k ~ _ 1 :

pr I a ( I ) ~ p r l a ( 2 ) ~ and

=

fo=f, and

to be the picture obtained by applying

Algorithm E to

Set~=~

,~__~\-~!

...~ Prib(k)

is a skew diagram and

by induction on

fi

, and

, where

Prib (I)L Prib(2)

Put ~

~

fk, i.e. f ~

~ f

...~ Pria(k) ~I~C~).

~ ( ~ :~ /(I~i~k);

. Clearly, V - - ~

is just that of Prep. A2,1. The

can be constructed in a quite similar

way, by means of Algorithm R° This proves Prop. AY,I and hence Prop. 4.I8. A2.5. It remains to verify that Algorithm E is well-defined, i.e. that while

f

~

constructed in its description is a Young diagram,

is a picture. We proceed in a series of steps.

163

(I) Let us prove that Prlal > I P r l a 2 ~

-.- >/Prlal+ I

in particular, this implies that ~ al+ I

is a Young diagram and

is a regular point of V • Suppose

Priaj. I ~ p r I a j

= (Priaj_i,J)~Cj. that of

;

f(x) j ~ f

Since

for some

J. Consider

x =

a j _ l ~ P x, the condition (J) implies . This contradicts the choise

(aj_ I) = bj_ I

aj . (2) Let us prove that

x r, x" ~

and

sume that

x w and

either If

x

f

x'~px" x"

satisfies (J), i.e. that if then

~(x s) j ~ ( x " ) .

We can as# are neighbours with respect to , ~ p ,i,e.

x r = (PrlX" + l,Pr2x")

and

x"

or

x t = (Prlx",Pr2x"+I) •

both don't lie in ~ al,..,al+l~ then

f(x' ) -- f(x' ) 5> f(x") = f(x"). If

x e and x"

and

x"= aj

both lie in for some

~ai,...,al+i~

j>bj_ I

Finally, let exactly one of Since

~j

f

x1= a~+ I

J, therefore

f(x p) = bj

al+i~.

then

= f(x") x r and

on ~ a i , . . , a l + i ~ ,

x"

• lie in

~ai,...,

the case when

ai,...,al+i~ is obvious. It remains to prove that if = (i,j)

and bj_ I

aj_ I ~ (i,j-I) j~f(i-I,j)

and

x"~ aj =

then bj_ I j> f (i,j-I)

.

The first inequality follows at once from the definition of furthermore by (I),

aj_ i p~(i,j-I)

hence

a j;

164

bj_ I = f(aj_ I) j>f(i,j-I) , as desired. -

f" I : ~

(3) It remains to prove that

~

satisfies (J).

As in the previous step, it suffices ¢o prove that ~-I (yt) j>~-I (y.)

whenever

yI~y., and

yl and

bouts with respect to tJ ~ J ' • The case when don't lie in ~ bo,...,bl} when

yl and

y"

y"

are neigh-

yl and

y"

both

, is obvious. Let us treat the case

both lie in ~bo,...,bl~ ~-z (bj) = aj+ I

. We have

.

Since ai~ J a2~J

...~j

al+ I

we must prove that the inequality 0~j~j~

1. For

j = 0

a regular point of ~

(see (I)

),

bj~2 bj

cannot hold for

this follows from the fact that

, while for

j~ 0

bo

from the fact that

is f-Z

satisfies ~J) since f'I(bj) = aj j~aje = f-I(bj!) Combining this assertion on the bo ~ J b I ~ j

bj's

-

with the inequalities

...~j b I ,

we obtain that bj(W,SW) bj_ I

for

j = I,...,1

.

(4) It remains to treat the case when exactly one of y"

lies in

the case when

bo,...,b I y" ~

. Since

~ bo,..°,bl~

f - ~ j f-I

on

yt and

bo,.°.,b I

, is obvious. Thus the only

case to be considered is the following:

,

165

y¢ = bj , y" ~ y"

ibo,...,bl~

is either

, and

(Pribj-I , Pr2b j)

we must prove that

or

f-I(y,)~ aj+i" Put

Let us prove that the inequality

j = l, since

point of ~ ;

then

inequality

jaj+ I

Indeed, this is clear for but if

(Pribj,Pr2bj-I);

cannot hold.

al+ I

x~+ I & ~ /

is a regular

, hence by (J) the

would imply that

y" = f(x")>j f(aj+ I) = bj+ I This contradicts the condition

. (see (3)

bj+I(W,SW)b j

).

(5) We have proved that x" (NE,N,NW,W,SW) aj+ I

.

It remains to verify that the relation

x"(W,SW) a~+ I

cannot

hold. Assume the contrary, i.e. that x" (W,SW) aj+ I

.

First, we see that y"L2 bj

and

f-I

implies

x"(N,NW) aj

it follows that

Pr2x" ~ j; in particular,

satisfies (J), we see that hence

x'~Lp aj. Since

j > O. Since

x"~ f

aj. This

satisfies (J),

y"~j bj. Therefore, only one possibility for

the relative position of

y"

and

bj

can ~old

(see (4)), name[

ly y"= (PrIbj-I , Pr2b j ) Denote this point

y"

by

.

b~, and

f-I(y,) = x"

by

xj .

(6) This is just the time to apply Lemma A2.2 : Applying (2) of A2.2 to the picture that

f-I

and the pair

b~(N)bj, we obtain

166

=j (NE,N) aj

.

Comparing this with the relation

xj(N,NW) aj

obtained in (5),

we conclude that xj (N) aj

o

Thus, the desired contradiction follows from the next assertion (x)

For

Suppose

j)~ 0

the relation

xj(N) aj

xj(N) aj. By defihition of

b7 = f(xj)

bj_

cannot hold. aj, we have

.

Combining this with the relation

bj_I(E,NE) bj

proved in (3)

we see that bj_l(E) bj NOW consider the point b~Lp b ~ _ i ~ bj, I

o b~_ I

and ~

A2.2 to the picture

f-I

(it lies in ~ \

~l

' since

is a skew diagram). Applying (I) of and the pair

b~_l(E) b~, we see

that

xj_1(w,sw) xj In particular,

.

Pr2xj_i~Pr2x j = j, so

Applying (2) of A2.2 to the picture b~_l(N) bj_i, we obtain xj_I(NE,N) aj_ I But Pr2xj_ 1~j-I = Pr2aj. I

•

hence

xj-I(N) aj_ I •

j~I.

f-I

Therefore,

and the pair

Thus, under the assumption that j ~I

and

on

xj(N)aj

we have proved that

xj_i(N) aj_ I. The assertion (~) follows by induction

J. ~' . E .

A2.6.Remark.

D .

Our algorithms E and R are closely connected

with the algorithms of insertion and deletion of a number into a Young tableau, playing the crucial role in the proof of the well-known Robinson-Schensted

correspondence

(see ~30~, 5.I.4).

Our algorithms allow us to obtain a combinatorial generalization of the Robinson-Schensted

correspondence and derive from it the

following generalization of Prop. 4.I8: for any two skew diagrams ~ ~d~

~8~)equals

and ~

the inner product

the number of pictures

f:~4~,Pe~

,

These questions are treated in detail in [20J. A~ppendix ). The composition of functors A3.I. Let in

G , @

pose that N

G

normalizes

= N~V V

and

be a finite group, M,U,N, and

a character of M~U

r

and ~

U, and

= {e} ,

~ M

V

i. be subgroups

a character of

normalizes

U

and

V. Sup@, while

, i.e. there are defined the functors

Under the extra assumption we shall compute the composition

Put

P = MU, Q = NV

tives of double cosets

Q\

and choose a set

W

of representa-

168

For any

wg W

morphism of

denote also by G, i.e.

w

the corresponding inner auto-

w(g) = wgw -I ; write

w(@)

for the charac-

ter x~ We say that a subgroup to

(M,U)

) @ (w-I(x) H

of

G

)

of

w(U).

is decomposable with respect

if

H~Gz) = (HnM),(~nu)

.

Let us make the following assumption: (D) 9or any w(U)

w~

W

each of the groups

is decomposable with respect to

w-I(Q), w'I(N), and

w'I(v)

is

(N,V)

w(P), w(M), and while each of

decomposable with respect to

(M,u). Now for any

w~W

we defime the functor

Consider the condition. (~) The characters

w(@)

and

~

coincide on

w(U)~ ¥.

!

If #I) does not hold,we put ~ w

= O. If (m) holds then

~w

defined as follows. Pu~ M # = M ~ w'I(N),

N'= w~M') = w(M)~ ~,

V t = Mt%w'I(v),

Ut= Nnw(U),

~! = restriction of

w'I(~)

@i = restriction of

w(e)

to

V', and

to u t.

By (D)~there are defined the functors

w- ~ (M ~) -

~

(N')

(tr~nsfe~ of structure by w),

is

169

iu' @' : ~

and

(N #) -

~ ~(N)

.

We set

~ w : i~' e'° w°rv',~': i ( M ) ~ Theorem.

i (~)

The functor

F = rv, ~ e iu, @ : ~ ( M ) ~ i (N) l is isomorphic to the direct sum of the functors ~ w ' w 6. W . This theorem even in a more general setting (for locally

compact 0-dimensiona~ groups) was proved in ~I ~, ~ 5. The more elementary proof of the fact that mappings from

R(M)

to

F

and ~

~

coincide as

R(N) (this is sufficient for the purpo-

ses of this work) can be obtained by a computation of characters via the formulas from 8.2; we leave this to the reader. ~Wnen

U = V = ~e~ , our Theorem is just the well-known Mac-

key theorem ( ~ 9 ~ w~W

, 7.4); the conditions (D) and (x) for any

hold tautologically.

A3.2. Let us apply Theorem A3.I the algebra

R(S)

(see 6.2). We have to compute the composition

R(S)k,@R(S)I ' (here

to verify the axiom (H) for

m

> R(Sn ) _ m~,l > R(S) k @

R(S) 1

n = k+l=k~+ 1 I ). By definition, this composition can be

rewritten as R(SkS x Sle) - - ~

R(S n)

re

II> R(S k x S l) .

So we apply Theorem A3.I in the case when N = Sk x SI, and choose a set

W

U = V =~e}

G = Sn, M = SklX SlS ,

. According to A3.I, we must

of representatives ~f double cosets Sk x S I \

Sn / SklX S11 .

Let us give the more general result. For any ordered partition

170

= (aI,...,a r)

of

n

let

S~ C

Sn

of permutations preserving blocks o f ~ I~ = ~ , . . . , a ~ } ,

be the subgroup consisting i.e. subsets

12 =~a~+~,...,a[+a2}

,..., Ir=~al

+ ...+

+ ar_1+~,... ,n} . Proposition.

Let

~

= (aW,,,~z) ~ud

be two ordered partitions of J~'''''Js

n

= (bi,...,b s)

with blocks

I~,...,I r

and

S~ \ Sn/SJ

respectively. Then double cosets

are

parametrized by matrices K = (kij)i ~ i ~ r, I~ j ~ s such that

kij ~ ,

= bj

I~j~s

for

~

~jkij

= ai

w~ Sn

I~i~r

(cf. 3.17 (c)). Namely, the

corresponding to a matrix

tions

for

K = (kij)

, and

~kij=

double coset

consists of permuta-

such that

I w (Ii)a Jjl = kij

for all

One can choose as a representative of J ~ K

(i,j)

.

the permutation

w~

which acts on each block of the partition (kII,kI2,..,kls, k2i,..., k2s,...,kri,...krs) by a shift, sending the block

~roof. For any s~s~ ,

kij

to the

s0 E S

and

sews (Ii)/1 Jj = s;w (Ii)~ Jj = : o' [

=

o'

Hence

Is'ws (q)aJjl

: l~ (q)aJjl,

i-th place in w~S n

we have

Jj •

171

i.e. the number

I w(Ii)~ Jj I does not change when

in its double coset

S~ k Sn/SoL

w

varies

. Conversely, it is easy to v ~ i

fy that if

lw(q)n Jjl = lwqq)n Jjl then

w

and

wt

i,J

lie in the same double aoset. Finally, the in-

WE ~ ] 4

oluslon

for all

is evident. Q,E.D,

In particular, double cosets

Sk x el\ Sn/SklX S ~

are para-

metrized by matrices

K = ( kll

k12

k21

k22

1

such that kii + ki2

= kI

, k2i + k22

= 11

, kii + k 2i = k

and

ki2 + k22 = 1 ; so we choose Pu~

W

consisting of all

w~ .

w = w K . Then under the notation of AI.I we have

f M

! = S(kii,ki2,k2i,k22 )

while the functor

and

w : A ( M I) ~ to

(here

~ij

~ O~(SI

K

N

= S(kiI,k2i,ki2,k22 ) ,

J4 (N I)

~l~II@ /52I• ~I2 4~ ]522

)). Therefore

= mkii,ki2

sends

mk2i,k22

172

Adding these expressions together for all (k,1) with

k+l=n,

K

and then for all

we see that m (JO

(/5'),

as desired.

A3.3. The axiom (H) for the algebra

R(S [G~ ) (see 7.2) can

be verified in a quite similar way as in A3.2. It suffices to observe

that

Sn[G3=

Sn. Gn

and

SkEG~x

S1 [G~=

S(k,1)'G n

Sk gG~ x S 1 EG]~Sn- [ G]/SkI[-G ] ~

x SleEG 3

= S(k '

(see 7.I), so

and one can choose the same

W

Sn/S (kl,1~

as in A3.2.

A3,4. Now we prove that the restriction R : R(S [G~ ) ~ R ( S ) is a ring homomorphism. If and %@~

k+l=n, them

R(~'~

~

6

(see 7.I0) R(Sk£G ~ ), ~ e

R(SI[G~) ,

) is by definition the image of

R(S(k,1 ) [G3 ) under the action of the composition R(S(k,I ) ~ G ~ )

ie

,I

~R(Sn[G~

)

re'I;

R(S n)

This composition can be computed by means of Theorem A3.I. We have

W = {e~

so our composition equals °

R(S(k,I)[ G7 relic- R(S(k,I ) ) le, I k i.e. it sends ~

~

to

R(Sn),

R(~).R( ~ ), as desired.

173 A3.5. Now we verify the axiom (H) for the algebra

R(q)

(see 9.I). As in A3.2, the compositiom R(q)kl @ R(q)it m >

R(q) n

mk,1 >

R(q) k ~

R(q) 1

can be rewritten as

(Gk,x

IUk;i",I

R(Gn )

ru ,I ,,,k,l

,>

R(GkXGI)

(see 9.I). So we apply Theorem A3.I in the next situation: G=Gn, M-- ~ , x

Gl~, U = Uke,l,, e = I, N = GkxG l, V = Uk, l, and

~ = I. We have to choose a set

W

ef representatives of

double cosets (Gk-XGI) Uk,l \

Gn / ( ~ k , X GI,) Uk1,1'

As in A3.2, let us give the more general result. For any ordered partition

~

group of

Gn

-- (ai,...,ar)

denote by

P~

ai,...,a r

the sub-

(from up to down).

(Bruhat Decomposition). Let ~

ordered partitions of (w I

n

consisting of all cellular upper triangular mat-

rices with cells of lengths ~osition

of

> (Bi,w(j))

and

n. The natural embedding )

~

Sn ~

be two Gn

induces the biJection

Thus,the representatives of double c o s e t s

~\~-'Vll1~ !

be chosen in accordance to Prop. A3.2. For the proof see [29~ , ch. IV, § 2, items 2 and 3. Q.E.D,

can

174

The fUrther arguments are similar to those in A3.2. Note that A3.1 (x) holds for any

w~W,

since each of

equals I; the condition (D) for any

wgW

@

and

can be verified di-

rectly. A3.~. Let us verify that the form 9.4) is multiplicative. Zet J~5~

~:

R(Gk) ,

R(q) - - - - ~ ~ ~

k+l=n. By de~inltion (see 9.1, 9.4), ~ ( ~ f ;

(see

R(G1 )' and

is obtaiued b,

applying the composition

R(Gk~GI) iUk,l'Z~ R(,n) rUn'T ~ R(Go) ~ ~(GkxG1).

To compute this compositiom we apply Theorem

A3.I in the next situation; G = Gn, M = GkXG I, U ~ Uk, 1 , @ = I, N = G O = ~ e ~ F = Un, and

~

,

is defined in 9.4.

We have to choose a set

W

of representatives of double cow~

sets

Un\Qn/P(k,1 ) Let

B = Bn

in

Gn, and

B = P(in)

(see A3.5).

be the subgroup of all upper triangular matrices D = Dn

be the diagonal subgroup in

Gn. We have

(see A3.5), so the Bruhat decomposition implies that

B\Gn/Pk,I)

=

Sn/s(k,l)

On the other hand, B = UD, and tation matrix, hence

u \Gn/(k,1

D

is normalized by any permu-

t75

Thus, we can choose

W

as the set of representatives of

S~S(k,1 ) ; we choose it in accordance to Prop. A3.2. It is easy to see that

W

consists of permutations

w ( I ) < w ( 2 ) < ...<w(k) The condition

and

w

such that

w(k+I)<w(k+2)< ...<w(n).

(D) from A3.I for all these

w

can be verified

directly. Now consiger the condition (x) from A3.1. Clearly, w(U) consists of matrices entry

uij

(uij) such that

can be non-zero only if

lows that if w(U) ~ V

w'I(i) ~ k < w-I(i+I)

uii=I, and for

i~J

w'I(i)~k<w'I(J). for some

i

then

~

the

It fol~ I

on

i.e. (z) does not hold. Thus, (~) holds only if

There exists the unique such

w ~ W:

~l+i

if ig k

[i-k

if

w(i) i> k

The corresponding subgroup

V # = M ~ w'I(v)

(see At.I) equ-

. By Theorem AI.I

and 8.1 (c),(d):

as desired. A3.7. In conclusion we compute the induction of representations of

Gn_ I

to

Pn

(see I3.3). We have to compute the

compositions rv,i o ie, I : ~ ( G n . I) ~

~

rv,~

-~ ~ ( P n . i ) .

ie,I : ~

(Gn_ I)

(Gn_ I)

and

176

Let us apply Theorem A3.I in the next two situations: I. G =Pn,M=Gn_I, U = ~e~ , ~ = Gn_ I , V = Un_i, I II. G=Pn,M=Gn_I, U = ~ e ~ ,

N =

and ~

Pn-l' V = Un_I,I, and

=I.

~

is

defined in 9.4. In each case

W = ~e~ , and (D) and (x) from A3.I hold auto-

matically. Theorem A3.I implies, that the first composition is the identity fUnctor, while the second one equals completes the proof of Prop. 13.3.

rv, ~ I

. This

References

I. l.N.Bernstein, A6V.Zelevinsky. Induced representations / of reductive ~-adic groups I. Anm. Sc. Ec. Norm. Sup., t. IO, mo. 4, 441-472, 1977. 2. A.V.Zelevinsky. Induced

representations of reductive

adic groups II. On irreducible representations of f Sc. Ec. Norm. Sup., t. I3, no. 2, 165-210,1980. ~

~-

GL(n). Ann.

AoV.Zelevinsky. On representations of the general linear

and a~fine groups over a finite field. Uspekhi Mat. Nauk, 32, no. 3, I59-I60, 1977. 4. J.A.Green. The characters of the finite general linear groups. Trams. Amer. Math. Soc., 80, no. 2, 402-447, I955. 5. L.Geissinger. Hopf algebras ef symmetric functions and / class functions, in "Combinatoire et representation du groupe / symetrique", Lecture Notes in Math., 579, I68-I8I, Springer-Vetlag, I977. 6. A.V.ZeleVlnsky. The representation ring of the groups GZ(n) over a

~-adic

field. Funkc. Anal., If, no. 3, 78-79,

I977. 7. D.K.Faddeev. The complex representations of the general linear group over a finite field. Zapiski Nauchn. Semin. LOMI, 46, 64-88, I974. 8. J.W.Milnor, J.C.Moore. On the

structure of Hopf algebras.

Annals. of Math., 81, no. 2, 2II-264, I965. I / 9. J.-P.Serre.Representations lineaires des groupes finis. Hermann, Paris, I967. I0. D.Knutson.

~

-rings and the representation theory of the

symmetric group. ~ecture Notes in Math., 308, Springer-Verlag, I972.

178 II. J.S.Frame,

G. de B.Robinson,

R.M.Thrall.

The hook graphs

of the symmetric group. Canad, J. Math., 6, 316-324, I954. I2. A.Kerber. Representations

of permutation groups I. Lec-

ture Notes in Math., 240, Springer-Verlag,

I971.

I3. I.M.Gelfand, M.I.Graev. The construction of irreducible representations

of simple algebraic groups over a finite field,

Dokl. Akad. Nauk U.8.S.R., 14. S.I.Gelfand.

I47, no. 3, 529-532, 1962.

Representations

of the general linear

group over a finite field. Mat. $bornlk, 83, no. I, I5-4I,I970. I5. T.A.Springer.

Characters of special groups, in "Seminar

on algebraic groups and related finite groups~ Lecbure Notes in Math.,~3~, Springer-¥erlag,

1970.

I6. Z.Ohmori. On the Schur indices of

GL(n,q)

and

SL(2n+l,q). J.Math. Soc. Japan, 29, no.3, 693-707, I977. 17. D.K.Faddeev.

On the complex representations

of the

general affine group over a finite field. Dokl. Akad. Nauk U.S.S.R.,

230, no. 2, 295-297, I976.

I8. E.Thoma. Die Einschrankung der Charactere yon auf

GL(n,q)

G~(n-I,q). Math. Z., Il~, 321-338, I971. I9. G.D.James, M.H.Peel. Specht series for skew represemta-

tions of symmetric groups. J. Alg., 56, 343-364, I979. 20. A.V.Zelevinsky.

A generalization of the Littlewood-

Richardson rule and the Robinson-Schensted-Knuth

correspon-

dence. To appear in J. of Algebra. 2I. R°A.Liebler, M.R.Vitale.

Ordering the partition charac-

ter of the symmetric group. J. Alg., 25, no.3, 487-489, I973. 22. D.E.Littlewood. matrix representations 1950.

The theory of group characters and ef groups, 2-nd ed. Oxford, Clarendon,

179

23. R.Stanley. Theory and applications of plane partitions. Part I. Studies in Applied Math., 50, 167-I88, I97I. 24. T.A.Springer. Cusp forms for finite groups, in "Seminar on algebraic groups and related finite groups", Zecture lores in ~lath°, 131, Springer-Verlag, X970. 25. R.Steinberg. A geometric approach to the representations of the full linear group over a Galois field. Trans. Amer. Math. Soc., 7I, no. 2, 274-282, I95Io 26. C.W.Curtis. Truncation and duality in the character ring of a finite group of Lie type. J. Alg. 62, 320-332 (~980). 27. D.Alvis. The duality operation in the character ring of a finite Chevalley group. Bull. Amer. Math. Soc,, New Series, I, 907-9II (~979). 28° R.Gow. Schur indices of some groups of Lie Type. J. Alg., 42, No.I, I02-I20, I976. l 29. N.Bourbaki. Groupes et algebres de Lie, Chaps. IV,V, FI, Hermann, Paris, I968. 30. D.Knuth. The art of computer programming, v. 3, Sorting and Searching. Addison-Wesley, I973.

~ D E X OF NOTATION

Sn

, ik, 1

, rk, 1

I2

R(S)

I3

/I(R)

I4

R+

t ~

>

m,

m~ , e , em

c~

48

s (~)

64

g~)]]

67

I5 I6

I , m~+(x) , P

18

X~

I9

Xn ~ Yn

27

S(X;~)

t

28

Sn(X; ~)

~ , r~ ,~

Ill , ~ n

, ~y

' O.fo(l ) , ~ %

3~

, deg~

6S

KiO~

69

34

~

u~ ~ , ?~

72

T

39

c , p(N), p

73

Zn

42

~-,

~,

~,

~[~]], x(~ ) , ~(~ ), z(~ ) 4~

.A x ~ , y~ M~,~

~ , ~

, z~

45 46

~

, g ~,

~/

D~,

D~

80 S , 81 86

181

c(G) , R(s)

87

~,s~ , s~, c~

88

h(a)

92

s~[a] , s~gQ]

93

m s [ ~] )

94

~C

99

~ K(G)

Pf

128 129

g,

132

~ k l , o. o,k r ' Dk I , • o o ,kr

138

D,D k

I39

iu,e , rU,@

I07

Pn ' Vn

I43

Gn , R(q) , Uk,l, m~, I

II0

R n , In

I44 147

II2 II3

G(A,B)

I52

II4

fxg

I53

II5

gj

I55 156

P~

II6

b (X,Y, oo.) a

157

II7

PrIX , Pr2x

16o

II9

q

12I

dq

I25

Q(~

,U.}

i26

INDEX

Aitken's Theorem

76

Branching rule, for symmetric groups

90

, for wreath products

104

, for general linear and affine groups Bruhat decomposition

I73

Conjugation of a Hopf algebra Cuspidal representation

I52

Iii

Decomposition Theorem

22

Degenerate representation

II3

Degenerate Gelfand-Graev model

I42

Degenerate Gelfand-Graev modules Frobenius Character Formula

~unctors iu, 8 and ru, ~

I38 91

107 , Composition Theorem

Gelfand- Graev Theorem

II3, I3I

S.Gelfand's Theorem Graded set

I69

II3, I3I, I46

68

Green formula

I35

Green polynomials

126 , orthogonality relations

Hall algebra

I16

Hall polynomials

II7

Hook Formula

92

Hopf algebra

i6 , cocommutative

I7

127

I48

183

Hopf algebra, commntative

17

, connected

17

Inner product on a T-group Irreducible elements

I5

Jacobi-Trudi Formula

76

15

Littlewood-Richardson Rule Littlewood-Roe Theorem

77, 90 77

Macdonald conjecture

I37

Murnaghan- Nakayama Character Formula Naegelsbach-Kostka Formula Newton formnlas

76

76

Non-degenerate representation Numbering of a skew diagram

II3 59

.

.

, column-strict

-

-

, row-strict , type of

Partition .

-

-

9I

6O 6O

59

33 , canonical form of

Picture

34

156 , extension of

I60

, reduction of

I60

Positive element of a T-group

I5

Positive morphism (= T-group morphism ) Power sums

75

Primary representation Primitive element PSH-algebra

I5

II2 I8

18 , of representations of symmetric groups

13, 87

184

PSH-algebra, of representations of wreath products 9

III

of representations of the groups GZ(n,Fq)

q-binomial formula

II9

Quasi-Hopf algebra

I7

Regular subset of a skew diagram

57

Schur functions (= S-functions )

76

Schur index

142

Skew diagram

57

Skew-hook

94

63, 64

Skew Schur functions

76

Steinberg representation Symmetric group

II5

I2

Symmetric polynomials

7I

Symmetrized monomials

72

T-group (= trivialized group) T-subgroup

I4

I5

Tensor product of PSH-algebras Thoma's Theorem

2I

I48

Transposition of a Young diagram Unipotent classes

34

II6

The universal PSH-algebra R

27, 28 , irreducible elements of

m

, primitive elements of

m

, various bases of

The wreath products Sn~G~

, irreducible characters

i

Young diagram

34

42 49

93 , conjugacy classes

m

49

I00

Io3