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Proceedings of the RIMS Research Project 91 on In nite Analysis

c World Scienti c Publishing Company

DECOMPOSITION OF SYMMETRIC AND EXTERIOR POWERS OF THE ADJOINT REPRESENTATION OF glN

1. UNIMODALITY OF PRINCIPAL SPECIALIZATION OF THE INTERNAL PRODUCT OF THE SCHUR FUNCTIONS ANATOL N. KIRILLOV Steklov Mathematical Institute Fontanka 27, Leningrad 191011, USSR Received October 15, 1991

ABSTRACT The problem of decomposing the symmetric and exterior algebras of the adjoint representation of the Lie algebra glN into slN -irreducible components are considered. The exact formula for the principal specialization of the internal product of the Schur functions (similar to the formula for Kostka-Foulkes polynomials) is obtained by the purely combinatorial approach, based on the theory of rigged con gurations. The stable behavior of some polynomials is studied. Dierent examples are presented.

x0. Introduction

The main subject of this note is the problem of decomposing into slN -irreducible components the symmetric and exterior algebras of the adjoint representation of the Lie algebra glN . The basic facts in this direction were obtained by B. Kostant in the early 1960's, [1,2]. He introduced and studied some important invariants of nite-dimensional irreducible representations of compact Lie groups, the so-called generalized exponents. In fact the problem of decomposition into irreducible components of the symmetric algebra Symm(glN ) is equivalent to the problem of computating generalized exponents. This last problem is considered in the vast body of literature e.g. [1{10,12,19,20,27,41]. Note here particularly the work [20] of Matsuzawa in which partial progress was obtained, using the classical isomorphism M Symm(glN ) ' V V :

2 Anatol N. Kirillov

However, the actual computation of generalized exponents has remained quite enigmatic. We repeat the words of R. Gupta [3]: \what has known [22] and [25] suggested to us that their computation lies at the heart of a rich combinatorially avoured theory". One aspect of this note is an attempt to understand the series of beautiful papers of R. Gupta{R. Brylinski [3{7], P. Hanlon [12], R. Stanley [10] and J. Donin [8,9] concerning generalized exponents and their stable behavior and another aspect deals with the series of beautiful papers of K. O'Hara [13] and D. Zeilberger [14] concerning the constructive proofs of the unimodality of q-Gaussian coecients and also to try to connect their content with the theory of rigged con gurations, [16{18]. The main result of this note is the construction of a bijection between some combinatorially de ned sets (see section 5 and 6). The existence and properties of this bijection allows us to obtain an exact formula for the internal product of the Schur functions, which of the same kind as a formula for KostkaFoulkes polynomials (see [18] and section 6). Using this formula it is easy to see that the principal specialization of the internal product of the Schur functions is a symmetric and unimodal polynomial. In a particular case, we obtain that the same assertion is valid for generalized q-Gaussian coecients [ n ]q . In fact, in this case we obtain an expression of generalized q-Gaussian coecients as the Kostka-Foulkes polynomials of special kind, and the formula of type (1.7) for Kostka polynomials may be considered as a generalization of the KOH identity (e.g. [15]) in the case of arbitrary partitions. Our approach is purely combinatorial and based on the theory of rigged con gurations (e.g. [18]). However almost all our combinatorial assertions may be reformulated as a validity of some identities. It is an interesting open question to nd analytical proofs for these identities. Now let us say a few words about the content of this note. We decided to divide it into a few parts. In the rst part we give all necessary de nitions, exact formulations of the results, with proofs of some of them, and examples. The construction of the main bijection will be postponed to other parts. The rst section contains a de nition of rigged con gurations and also a formula for Kostka-Foulkes polynomials (see (1.7)). The examples contain a de nition and properties of maximal con guration and the relation between admissible matrices and reverse plane partitions. The last example (see x1, Ex. 2 ) is based on some part of our joint work with A. D. Berenstein. Also this section contains an example of computation of KostkaFoulkes polynomials, using (1.7). In the second section we de ne, following R. Gupta [3], the generalized exponents and recall some basic theorems from [3,19]. As an example we compute generalized exponents for irreducible representations of sl3 (this is well-known) and for sl4. Also we compute generalized exponents in some interesting cases. In the third section we recall a de nition of the internal product of the Schur functions (e.g. [10]) and give, following J. Donin [8,9] (with some modi cations), a combinatorial description for a decomposition of the internal product of the (super) Schur functions in terms of (product) monomial symmetric functions. Also we give some examples. In the section 4 we consider

Decomposition of symmetric and exterior powers of the adjoint representation of glN

3

the stable behavior of the following polynomials

?

?

F V[; ]N ; FN V V ; K[; ]N ;( N ) (q); s s (q; : : :; qN ?1 ): 1

It is well-known (the theorem of Kato-Hessenlink-Peterson), that

?

F V[; ]N = K[; ]N ;( N ) (q); 1

(0:1)

and (the theorem of Gupta-Stanley [6],[10])

?

s s (q; : : :; qN ?1 ): lim F V[; ]N = Nlim !1 N !1

(0:2)

Using exact formula (4.10) for F (V V ) it is easy to see that lim F (V V ) = q?jj s s (q; q2 ): N !1 N

(0:3)

There are a lot of explanations of (0.2), [3,6,7],[10],[12],[8,9]. From the combinatorial point of view, the main reason for the validity of (0.2) and its generalizations is a stabilization ? n n of the number of admissible matrices of the type (a ); (a ) when n ! 1, [18]. This stabilization property may be used (see section 4) for proving the conjecture of R. Gupta, [4, Problem 9]

K; (q) K;

(0:4)

for any partition . Another proof of (0.4) is given by G.-N. Han [26]. In section 5 we give, based on the works [8,9] of J. Donin, a combinatorial description for the principal specialization of the Schur functions s s (q; : : :; qN ?1 ) as a generating function of some statistics (charge), de ned on the set HN (; ) (see x5). In section 6 we present the main result of this note; it is the existence of a bijection between the set HN (; ) and some extension of the set of rigged con gurations. The exact construction is postponed to other parts of this note. As a corollary we obtain a formula for the principal specialization of the Schur functions, which is similar to the formula (1.7) for Kostka polynomials. As a corollary we prove the symmetry and unimodality of the principal specialization of the internal product (of any number) of Schur functions. In a particular case we obtain that the same assertion is valid for generalized q-Gaussian coecients, corresponding to an arbitrary partition . If is equal to (p) or (1p), the beautiful constructive proof is given by K. O'Hara [13] and modi ed by D. Zeilberger [14]. Our proof reduces the general case to the case studied by K. O'Hara and in fact is purely combinatorial. However, at this moment the relation between the constructions in [13] and rigged con gurations is not clear. We also give a generalization of the KOH identity (e.g. [15]) on the case of arbitrary partitions. Let us remark that the proof of unimodality of generalized q-Gaussian coecients, also based on works [16{18], was given by F. Goodman, K. O'Hara and D. Stanton [37].


?

In section 7 we consider the problem of computation of the polynomials E V[; ]N . For the case 1 = 1, J. Stembridge [19] obtained a compact expression for these polynomials and also proved recurrence relations between E (V) in the general case. In preprint [27] A. D. Berenstein and A. V. Zelevinsky proved Kostant's conjecture, which characterizes ? the partitions with E (V) 6= 0. As it follows from examples the polynomial E V[; ]N can not be in general represented as a ratio of the product of cyclotomic polynomials and the structure of E (V) seems to be very mysterious. In one particular case, when E (V )q=1 = 2N we give a conjecture about E (V ). As we mentioned above, this note is a very imperfect attempt to understand the richest and mysterious combinatorial sentence of the symmetric and exterior algebras. In this note we obtain exact formulas for the following polynomials ? F V[; ]N ; F (V V ) and s s (q; : : :; qN ?1 ); ? as sums over all admissible matrices of the type [; ]N ; ( 1N ) of products of some qbinomial coecients. But the relations between admissible matrices as above and the hook numbers of the diagram , as predicted by R. Gupta's conjecture, see [4, Problem 8] are still unclear. Acknowledgement. It is my pleasure to thank many people who encouraged me in this work. I am obliged to L. Faddeev, M.-P. Schutzenberger, A. Lascoux, A. Berenstein, J. Donin, G.-N. Han, S. Kerov, M. Kashiwara and T. Miwa for discussions. This note was started at LOMI, Leningrad in the fall of 1990, and was written during my stay at RIMS, Kyoto University. I thank my colleagues at Kyoto University and RIMS for the invitation, hospitality and possibility to nish this work. I would like to acknowledge my special indebtedness to T. Miwa and express my appreciation to the organizers of the RIMS91 Project \In nite Analysis" and the secretaries of RIMS for their assistance and help in preparing the manuscript for publication and Dr. D. S. McAnally for his help in translation of this note into English.

x1. The Kostka-Foulkes polynomials and rigged con gurations. A partition of n is a sequence = (1 ; : : :; l ) 2 Zl+ where the i are weakly decreasing P and jj := i i = n. We use notation ` n when = (1 ; : : :; l ) is a partition of n. A composition of n is a sequence = (1; : : :; l ) 2 Zl+ such that jj = n. The integers i

are called the parts of the partition (or composition). The number l() of nonzero parts of a partition is called the length of the partition. Let be a partition; then the conjugate of is the partition 0 = (01 ; 02; : : :; 0m) where 0i is the length of the i-th column of . As for other backgrounds of the theory of symmetric functions, we will adhere to the notation and terminology of [29]. Let ; be partitions. The Kostka-Foulkes polynomials K;(q) are de ned as elements of the transformation matrix between the Schur and Hall-Littlewood functions X s (x) = K (q)P (x; q): (1:1)


5

There exists a combinatorial description of the Kostka polynomials due to Lascoux and Schutzenberger [31], as generating functions for the charge [31,29] on the set of standard Young tableaux of the shape and content (abbreviation STY (; )): X c(T ) K (q) = q (1:2) T 2STY (;) In this section we give another expression for K (q) as a generating function for the sum

of \quantum numbers" on the set of rigged con gurations. For this aim we recall some necessary de nitions. We denote by M1f (Z) the set of all matrices m = (mij ), mij 2 Z, i; j 1, which contain only a nite number of nonzero elements. Let us x a partition and a composition and consider the set M (; ) of matrices m 2 M1f (Z) such that X X mij = 0j : (1:3) mij = i ; i

j

For each m 2 M (; ) we de ne the following matrices X = (kn ); kn = mjn

= ( kn ); kn =

j k+1 X

j n

(1:4)

mkj

A matrix m 2 M (; ) is said to be admissible of the type (; ) i all sequences (k) := (kn )n1 and (k) := ( kn )n1 are in fact partitions, that is kn k;n+1 ; kn k;n+1 for all, k; n 1: We denote the set of all admissible matrices of the type (; ) by C(; ) and call the set of partitions f(k) gk1 (resp. f (k) gk1 ) s-con gurations of the type (; ) (resp. ccon gurations) (s and c are abbreviations of words shape and content). It is clear that X X j(k) j = j ; j (k) j = 0j : j k+1

j k

For each matrix m 2 C(; ) we de ne integers Pkn (m) = kn ? k+1;n 0 Qkn (m) = kn ? k;n+1 0 and polynomials Y Pkn (m) + Qkn(m) Km(q) = P (m) k;n

K; (q) =

X m

kn

qc(m)Km(q);

q

(1:5) (1:6) (1:7)

where the charge c(m) of any matrix m 2 M1f (Z) by de nition is equal to P 1=2 i;j mij (mij ? 1), and the summation in (1.7) is taken through all admissible matrices of the type (; ). The following result gives a very fast method for computation of the Kostka- Foulkes polynomials.


Proposition 1.1 ([18]). Assume that ; are partitions. Then K; (q) = K; (q) The proof of this proposition given in [18] is based on a study of properties of the bijection between the sets STY (; ) and Q(s) M (; ) (or Q(c) M (; )). Here Q(s) M (; ) denotes the set of all rigged s-con gurations. By de nition, a rigged s-con guration is some (k) , k; n 1, s-con guration f(k) g of the type (; ) together with a set of integers Jl;n 1 l s := Qkn (m), which satisfy the following inequalities (k) Pkn (m): 0 J1(;nk) J2(;nk) Js;n

(1:8)

In the same way we de ne a rigged c-con guration. It is a c-con guration f (k) g together (k) , k; n 1, 1 l r := P (m), such that with a set of integers Il;n kn (k) Qkn (m) 0 I1(k;n) I2(k;n) Ir;n

(1:9)

We ll the integers J(;nk) (resp. I(;nk) ) in the rst columns of the diagrams (k) (resp. (k) ) in the natural way (see examples). We use the notation ((k) ; J(;nk) ) for a rigged con guration corresponding to an s-con guration f(k) g and a set of quantum numbers J(;nk) . The charge of a rigged con guration is de ned as the sum of its quantum numbers:

C (f(K ) g; J(;nk) ) = It is easy to see that

Km(q) =

Remarks and examples.

X

k;l;n

X ((k) ;J )2Q(s) M (;)

(k) Jl;n

(1:10)

qc( k ;J ) ( )

1 . It seems a very interesting task to study the numbers c(; ) := jC(; )j. For example c(321) = 2; f (321) = 16 (332) c = 6; f (332) = 42 c(4321) = 8; f (4321) = 768 c(3322) = 12; f (3322) = 252 c(54321) = 58; f (54321) = 292864 Here c := c(; 1jj), f = jSTY ()j. It is clear that maxfc j ` ng ! 1 when n! 1. On the other hand it can be shown that for any partitions and the sequences ? ? n n k k fc (a ); (a ) g and fc + (n ); + (n ) g are bounded when n ! 1. Here for


7

partitions ; we denote by the partition corresponding to a composition compiling from the parts of and . It is natural to ask what values the numbers maxfc j ` ng and minfc =f j ` ng may take? It is easy to see (e.g. [18], p.376) that if is a hook then c(; ) 1. How can all partitions and which satisfy the condition c(; ) = 1 be described? 2 . In general an admissible matrix may have negative elements. However consider the case when m 2 C(; ) \ M1f (Z+ ). Then we may construct a reverse plane partition (i.e. a tableau whose rows and columns are weakly increasing sequences) of the shape and content 0 (abbreviation: rpp(; 0)) in the following way: let us ll in the i-th row of the diagram exactly mij numbers equal to j . We obtain a tableau of the shape and content 0 . After this consider the word w() which is obtained by reading in sequence the entries of from the left to the right starting from the bottom row. It is easy to check that the conditions (1.5) mean that 2 rpp(; 0) and word w() is a lattice sequence, and the correspondence m ! de nes a bijection f 2 rpp(; 0 )jw( ) is a lattice sequenceg: C(; ) \ M1f (Z+ ) ?! As an example, consider = (9; 5; 4; 3; 2), = (5; 5; 5; 3; 2; 1; 1; 1) and 02 1 2 1 31 BB 2 1 1 1 0 CC m=B @ 21 11 01 10 00 CA 2 C(; ): 1 1 0 0 0 Then 1 1 2 3 3 4 5 5 5 1 1 2 3 4 = 1 1 2 4 2 rpp(; 0); and 1 2 3 1 2 w() = 12123112411234112334555 is a lattice sequence. Let us remark that if m 2 C(; ) \ MN M (Z) and l 2 Z+ , then matrix m~ = m + l ?EN M 2 C(~?; ~), where the matrix EN M has all elements is equal to 1, and ~ = + (lM )N , ~ = (M )lN ; . So we have an inclusion C(; ) ,! C(~; ~), and consequently any m 2 C(; ) may be considered as a lattice rpp for some l. 3 . Consider a partition , a composition , and the set of rigged con gurations QM (; ). Consider further the matrix m = (mij ) with elements mij = (0j ? i) + i;1 (0j ? 0j ), where (x) = 1; if x 0; (x) = 0; if x < 0: (1:11) It is not dicult to see that m 2 M (; ) (see (1.3)) and P (0 ? 0 ); if k = 1; 2 n; j j n j Pkn (m) = min(n; k ) ? min(n; k+1); if k 2 Qkn (m) = max(k; 0n) ? max(k; 0n+1); k 1:


So that if . with respect to the dominance order on partitions (e.g. [29]) then P we(see 0j ? 0j ) 0 for all n 1, and hence m 2 C(; ) 6= . It can be shown that if j n the above matrix m 2= C(; ) then C(; ) = . We will call this matrix the maximal con guration and denote it by = (ij ). It is easy to see that the corresponding sP ( k ) ( k ) ( k ) ( k ) con guration := (1 ; 2 ; : : :), where n := jk+1 jn , is equal to ([k])0, where [k] := (k+1 ; k+2; : : :), and for any s-con guration f(k) g of the type (; ) we have (k) . (k) , k 1. Furthermore if f(k) gk1 is as above then the set f(k) gk2 also ? is a s-con guration of the type [1]; ((1))0 and hence using a bijection (see [18]) we obtain a map

STY (; )?! ?QM (; )

(1:12)

?

(1) 0 STY (; ) ?! ((1) ; J(1) ;n ) STY [1]; ( ) ;

(1:13)

namely

? ? (1) (1) (k) (k) T 2 STY (; ) ?! ((k) ; J(;nk) )k1 ?! ((1) ; J(1) ;n ); ( ; J;n )k2 ?! ( ; J;n ); T~ ; ? where T~ 2 STY [1]; ((1))0 . However the inequalities on the quantum numbers J(1) ;n now depends on the tableau T~. Nevertheless using (1.13) we obtain an interpretation of the Morris identity for the Kostka-Foulkes polynomials [30] in terms of rigged con gurations. Note that from the existence of the maximal con guration there follows an inequality Y Qn() ? Qn() + 0n ? 0n+1 c K; (q) q ; (1:14) 0n ? 0n+1 q n P P P where c = n() + n() ? k 0k (0k ? 1), Qn () := jn 0j and n() := i (i ? 1)i . The 2

equality in (1.14) is equivalent to the existence of only one con guration of the type (; ). In order to understand when the set C(; ) consists of only one element, let us de ne for a partition and a composition the following set

D(; ) =

a

1a 0a 0b > max(1; 0b+1)

j n here we assume that 0 = +1.

0

Theorem 1.2. The set C(; ) consists of only one element i . and D(; ) = . Corollary 1.3. Assume that and are partitions, . . In the inequality (1.14) we have in fact equality i the set D(; ) = . We may rewrite the condition D(; ) = as follows. Assume that 0 = (l1a ; l2a ; : : :; 1

2


9

lrar ; 1ar ), where l1 > l2 > > lr > 1. Let us put nk := a1 + + ak , 1 k r, n0 := 1. Then the set D(; ) = i the following inequalities are satis ed 1) if for some k, ak 2, 1 k r, then +1

X (0 ? 0 ) n < n < n 1; k?1 k j j

min

1j n

2) if for some 1 k < r, we have ak = ak+1 = 1, then 0

X

1j nk

(0j ? 0j ) 1:

4 . Now let us compute K; (q) using (1.7). a) = (6; 3; 2), = (3; 3; 2; 1; 1; 1). There exist 4 admissible matrices:

m

3 1 2 2 1 0 1 1 0

f0 g

rpp

1

3 2 1 2 0 1 1 1 0

1

1

1

0

4 1 1 1 1 1 1 1 0 3

3

0

max con guration

111233 112 12

c(m)

3

2 2 2 2 1 0 2 0 0 1 0 0

111223 113 12

5

112233 112 11

5

Consequently,

5

111123 123 12 6

K; (q) = q5 3 2 2 + q5 3 4 + q5 2 + q6 4 4 1

.

1

1

1

1

1

1

1

b) = (3; 3; 2), = (18 ). At rst, we may compute K; (q) using a well-known result (e.g. [29]) (put jj = N ):

Q

where H (q) = x2

N 0) Y n ( K;(1N ) (q) = q (1 ? qi )H (q)?1; i=1 (1 ? qh(x) ) is a hook-length polynomial. So

6 )(1 ? q 7 )(1 ? q 8 ) K(332);(1 ) (q) = q7 (1(1??qq)(1 ? q2 )(1 ? q4 ) : 8


On the other hand let us use (1.7). There exist 6 admissible matrices:

m

4 ?1 3 0 1 1 0

f0 g

1

c(m)

10

m

4 ?1 2 1 2 0 1

2

c(m)

19

Consequently,

0

3

0

5 ?2 2 1 1 1 0

3

13

0

1

14

3 0 3 0 2 0

max con guration

2

0

9

6 ?2 ?1 1 1 1 1 1 0

f0 g

0

5 ?1 ?1 2 0 1 1 1 0

0

0

1

7

K;(18 ) (q) = q7 32 + q9 53 + q10 21 41 + q13 52 + q14 21 41 + q19 31 6 )(1 ? q 7 )(1 ? q 8 ) = q7 (1(1??qq)(1 ? q2 )(1 ? q4 ) : In general, if = (1N ) we obtain an identity N X 0) Y n ( (1 ? qi )H (q)?1 = qc(m) Km(q); q i=1 fmg

(1:15)

?

where the summation in (1.15) is taken over all admissible matrices m 2 C ; (1N ) . It is desirable to nd an analytical proof of (1.15). P c) For given partitions and we have the distinguished subset in STY (; ), which corresponds to rigged s-con gurations with zero values of quantum numbers J(;nk) . It seems very interesting problem to give a direct de nition of such tableaux. For example, take = (6; 5; 4; 2), = (5; 4; 4; 3; 1). There exist 3 admissible matrices.


m

2 1 1 1

1 1 1 1

2 1 1 0

1 1 1 0

max con guration

f0 g rpp

2

P m

1 2 3 4 2 1 1 1

1 2 3 4 1 1 1 1

1 1 1 1

Consequently, .

1 1 1 1

0

1 2 1 0

0 1 1 1 1

2 0 1 0 1

0 1 0 0

0

1 2 3 4 4 2 3 3 5 2 3 4 2 3

1 1 1 5 2 2 4 3 3 1 2 1 0 0

c(m)

P

0

0

2

f0 g rpp

1

2 1 1 1

1 1 2 3 3 4 2 3 4 5 2 3 4 2

1 1 1 1

c(m)

0 1 0 0

0

1 1 1 0

1 2 3 4

1 1 1 1 4 2 2 2 3 3 3 5 4

1 0 0 0 0

1

0

1 2 3 4 5 2 3 3 4 2 3 4 2 2

1 2 3 4

1 1 1 1 4 2 2 2 5 3 3 3 4

1

1

K; (q) = q2 2 2 3 + q2 2 + q3 2 1

1

1

0

0

11


x2. Generalized exponents. Let g = sl(N; C) and G = SL(N; C). The adjoint action of G on the Lie algebra glN = gl(N; C) extends to an action on symmetric algebra S (glN ) = k0 S k (glN ), where S k denotes the k-th symmetric power. By a theorem of Kostant [2] S (glN ) = I H

(2:1)

is a free module over G-invariants I generated by the harmonic polynomials H . Moreover, the ring of invariants

I = S (glN )G = ff 2 S (glN ) X f = f; 8X 2 Gg

L

is a polynomial ring in N variables f1; : : : ; fN , where fi 2 S i (glN ), and H = k0 H k is a graded, locally nite G-invariant subspace of S (glN ) (so H k = H \ S k (glN )). For each nite dimensional G-representation V let us consider polynomials

F (V ) := E (V ) :=

X

k0

X

k0

dim Homg (V; H k )qk

? dim Homg V; k (glN ) qk

(2:2) (2:3)

where p denotes the p-th exterior power. Kostant also shows [2] that F (V )q=1 is equal to the dimension of the zero-weight subspace P V (0) of the representation V . Thus, F (V ) is really a polynomial in q, say F (V ) = si=1 qdi , s = dim V (0), and the integers d1 ; : : :; ds are called the generalized exponents of V . The polynomial F (V ) turns out to be a rather deep invariant of the representation V . For instance, the F (V ) are certain Kazdan-Lusztig polynomials for the ane Weyl group [23,24], they describe a certain group cohomology [21], and coincide with the Poincare polynomial for the ltration on the zero-weight subspace V (0) de ned by the action of the principal nilpotent [5],[28]. It is interesting to note that the polynomials F (V ) coincide with the partition functions for some matrix models [38]. It is easy to see that if X 2 G has eigenvalues x1; : : : ; xN , then the adjoint action of X on the space glN has eigenvalues xi x?j 1 (1 i; j N ). Therefore, the character of the exterior power k (glN ) is the coecient of qk in the generating function

Y

1i;j N

(1 + qxi x?j 1)

(2:4)

So for each partition of length less than N we have (see (2.3))

Y

1i;j N

(1 + qxi x?j 1 ) =

X

E (V )s(x1 ; : : :; xN )

(2:5)


13

Similarly, the character of the symmetric power S k (glN ) is the coecient of qk in the generating function Y 1 (2:6) ?1 ; 1 ? qx x i j 1i;j N and if we de ne formal power series S N [](q) for each partition of length less than N via X N Y 1 = S [](q)s(x1; : : :; xN ); ?1 1 ? qx x i j 1i;j N

it follows that the coecient of qk in S N [](q) is the multiplicity of V in S k (glN ). Using the theorem of Kostant (2.1) we see that

S N [](q) = f(1 ? q) (1 ? qN )g?1 F (V )

(2:7)

Note also that we have the following decomposition

S k (glN ) =

M

`k l()N

V V ;

from which it is easy to deduce that

X

;l()N

dim Homg (V V ; V ) qjj = S N [](q):

Our approach to the problem of a computation of the generalized exponents based on the following facts. Let A be a skew diagram and VA be the representation of g = sl(N; C) corresponding to this diagram. Proposition 2.1 ([3,22,25]). KA;(ln)(q); if jAj = l N; F (VA ) = (2:8) 0; otherwise Proposition 2.2 ([3,6]). Assume A and B are skew diagrams. Then

X

N N ?l()+1 ) KA;(q)KB; (q) (1 ? q ) b(1(q?) q ; where l() is the length of a partition , and

FN (VA VB ) = b (q) =

Y

i1

(2:9)

(1 ? q) (1 ? qmi ) for = (imi ); b0(q) = 1:

Here KA; (q) be the Kostka-Foulkes polynomial corresponding to a skew diagram A (e.g. [3] or [17]).


Now we consider the class of irreducible representations of g which play an important role in our combinatorial constructions. Given any two partitions and of lengths respectively r and t, of the same integer p such that l() + l( ) N . Following R. Gupta [N ] as the Cartan piece in V V , i.e. the [3,6], we de ne a representation V; := V; irreducible g-component generated by the tensor product of the highest weight vectors in [N ] ' V each factor. It follows that V; [; ]N , where [; ]N = (1 + 1 ; : : :; r + 1 ; | 1 ; :{z: :; 1}; 1 ? t ; : : :; 1 ? 2 ; 0): N ?r?t

[N ] ' C, V [N ] ' g. For example V(0) ;(0) (1);(1) As the rst example let us compute the generalized exponents for g = sl(3) and g = sl(4). a) Given an irreducible representation V of sl(3) with highest weight = (1 2 0), 1 + 2 = 3l, = (l; l; l), consider the set C (; ). It is clear that there exists only one s-con guration f0 g = f(2 )g, and 1 ? 22 P1 = min(1 ? 2; 2); c = c(m) = 1 ? 2 ? 3 min 3 ;0 : Consequently min( ? ; ) + 1 1 2 2 c : F (V ) = K;(l ) (q) = q 1 2

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b) Assume V is an irreducible sl(4)-module with highest weight = (1 2 3 0), 1 + 2 + 3 = 4l, = (l4). We have the following collection of s-con gurations f(2 ? k; 3 + k); (3)g; where 0 k 2 ?2 3 : It is easy to see that P2 = 0, P1; +k = min(1 + 2 ? 23 ? 4k; 3) P1; ?8k = min(1 ? 2 ; 22 ? 3 ? 4k) > < 21 ? 2 ? 2l + 2k; if 2 ? k l ck = > 2 ? 23 + 2l ? 2k; if 3 + k l 2 ? k : 2 + 23 ? 2l + 2k; if l 3 + k Consequently, K;(l ) (q) = X ck min(1 + 2 ? 23 ? 4k; 3) + 1 q 1 ? 3

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