Proceedings of the RIMS Research Project 9 on Innite Analysis c World Scientic Publishing Company DECOMPOSITION OF SYMMETRIC AND ETERIOR POWERS OF THE ADJOINT REPRESENTATION OF gl N. UNIMODALITY OF PRINCIPAL SPECIALIZATION OF THE INTERNAL PRODUCT OF THE SCHUR FUNCTIONS ANATOL N. KIRILLOV Steklov Mathematical Institute Fontanka 7, Leningrad 9, USSR Received October 5, 99 ABSTRACT The problem of decomposing the symmetric and exterior algebras of the adjoint representation of the Lie algebra gl N into sl N -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 congurations. The stable behavior of some polynomials is studied. Dierent examples are presented. x. Introduction The main subject of this note is the problem of decomposing into sl N -irreducible components the symmetric and exterior algebras of the adjoint representation of the Lie algebra gl N. The basic facts in this direction were obtained by B. Kostant in the early 96's, [,]. 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(gl N ) is equivalent to the problem of computating generalized exponents. This last problem is considered in the vast body of literature e.g. [{,,9,,7,4]. Note here particularly the work [] of Matsuzawa in which partial progress was obtained, using the classical isomorphism M Symm(gl N ) ' V V :
Anatol N. Kirillov However, the actual computation of generalized exponents has remained quite enigmatic. We repeat the words of R. Gupta []: \what has known [] and [5] 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 [{7], P. Hanlon [], R. Stanley [] 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 [] and D. Zeilberger [4] concerning the constructive proofs of the unimodality of q-gaussian coecients and also to try to connect their content with the theory of rigged congurations, [6{8]. The main result of this note is the construction of a bijection between some combinatorially dened 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 Kostka- Foulkes polynomials (see [8] 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 (.7) for Kostka polynomials may be considered as a generalization of the KOH identity (e.g. [5]) in the case of arbitrary partitions. Our approach is purely combinatorial and based on the theory of rigged congurations (e.g. [8]). 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 denitions, 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 denition of rigged congurations and also a formula for Kostka-Foulkes polynomials (see (.7)). The examples contain a denition and properties of maximal conguration and the relation between admissible matrices and reverse plane partitions. The last example (see x, Ex. ) is based on some part of our joint work with A. D. Berenstein. Also this section contains an example of computation of Kostka- Foulkes polynomials, using (.7). In the second section we dene, following R. Gupta [], the generalized exponents and recall some basic theorems from [,9]. As an example we compute generalized exponents for irreducible representations of sl (this is well-known) and for sl 4. Also we compute generalized exponents in some interesting cases. In the third section we recall a denition of the internal product of the Schur functions (e.g. []) and give, following J. Donin [8,9] (with some modications), 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 gl N the stable behavior of the following polynomials?? F V [;] N ; FN V V ; K[;] N;( N ) (q); s s (q; : : : ; q N? ): It is well-known (the theorem of Kato-Hessenlink-Peterson), that and (the theorem of Gupta-Stanley [6],[]) lim N! F? V [;] N F? V [;] N = K[;] N;( N ) (q); (:) = lim N! s s (q; : : : ; q N? ): (:) Using exact formula (4.) for F (V V ) it is easy to see that lim F N (V V ) = q?jj s s (q; q ): (:) N! There are a lot of explanations of (.), [,6,7],[],[],[8,9]. From the combinatorial point of view, the main reason for the validity of (.) and its generalizations is a stabilization of the number of admissible matrices of the type? (a n ); (a n ) when n!, [8]. This stabilization property may be used (see section 4) for proving the conjecture of R. Gupta, [4, Problem 9] K ; (q) K ; (:4) for any partition. Another proof of (.4) is given by G.-N. Han [6]. 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; : : : ; q N? ) as a generating function of some statistics (charge), dened on the set H N (; ) (see x5). In section 6 we present the main result of this note; it is the existence of a bijection between the set H N (; ) and some extension of the set of rigged congurations. 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 (.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 ( p ), the beautiful constructive proof is given by K. O'Hara [] and modied by D. Zeilberger [4]. 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 [] and rigged congurations is not clear. We also give a generalization of the KOH identity (e.g. [5]) on the case of arbitrary partitions. Let us remark that the proof of unimodality of generalized q-gaussian coecients, also based on works [6{8], was given by F. Goodman, K. O'Hara and D. Stanton [7].
4 Anatol N. Kirillov? In section 7 we consider the problem of computation of the polynomials E V [;] N. For the case =, J. Stembridge [9] obtained a compact expression for these polynomials and also proved recurrence relations between E(V ) in the general case. In preprint [7] A. D. Berenstein and A. V. Zelevinsky proved Kostant's conjecture, which characterizes? the partitions with E(V ) 6=. 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= = N 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; : : : ; q N? ); as sums over all admissible matrices of the type? [; ] N ; ( N ) of products of some q- binomial 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 99, 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 RIMS9 Project \Innite 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. x. The Kostka-Foulkes polynomials and rigged congurations. A partition P of n is a sequence = ( ; : : : ; l ) Z l + where the i are weakly decreasing and jj := i i = n. We use notation ` n when = ( ; : : : ; l ) is a partition of n. A composition of n is a sequence = ( ; : : : ; l ) Z l + 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 = ( ; ; : : : ; m) where i 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 [9]. Let ; be partitions. The Kostka-Foulkes polynomials K ; (q) are dened as elements of the transformation matrix between the Schur and Hall-Littlewood functions s (x) = K (q)p (x; q): (:)
Decomposition of symmetric and exterior powers of the adjoint representation of gl N 5 There exists a combinatorial description of the Kostka polynomials due to Lascoux and Schutzenberger [], as generating functions for the charge [,9] on the set of standard Young tableaux of the shape and content (abbreviation ST Y (; )): K (q) = T ST Y (;) q c(t ) (:) 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 congurations. For this aim we recall some necessary denitions. We denote by M(Z) f the set of all matrices m = (m ij ), m ij Z, i; j, 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 M(Z) f such that j m ij = i ; For each m M(; ) we dene the following matrices = ( kn ); kn = = ( kn ); i m ij = j: (:) jk+ m jn (:4) kn = m kj jn A matrix m M(; ) is said to be admissible of the type (; ) i all sequences (k) := ( kn ) n and (k) := ( kn ) n are in fact partitions, that is kn k;n+ ; kn k;n+ for all, k; n : We denote the set of all admissible matrices of the type (; ) by C(; ) and call the set of partitions f (k) g k (resp. f (k) g k ) s-congurations of the type (; ) (resp. c- congurations) (s and c are abbreviations of words shape and content). It is clear that j (k) j = j ; j (k) j = jk+ jk For each matrix m C(; ) we dene integers and polynomials P kn (m) = kn? k+;n Q kn (m) = kn? k;n+ K m (q) = Y k;n Pkn (m) + Q kn (m) P kn (m) j: q (:5) (:6) K ; (q) = m q c(m) K m (q); (:7) where the charge c(m) of any matrix m M f (Z) by denition is equal to = P i;j m ij(m ij?), and the summation in (.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.
6 Anatol N. Kirillov Proposition. ([8]). Assume that ; are partitions. Then K ; (q) = K ; (q) The proof of this proposition given in [8] is based on a study of properties of the bijection between the sets ST Y (; ) and Q (s) M(; ) (or Q (c) M(; )). Here Q (s) M(; ) denotes the set of all rigged s-congurations. By denition, a rigged s-conguration is some s-conguration f (k) g of the type (; ) together with a set of integers J (k) l;n, k; n, l s := Q kn (m), which satisfy the following inequalities J (k) ;n J (k) ;n J (k) s;n P kn (m): (:8) In the same way we dene a rigged c-conguration. It is a c-conguration f (k) g together with a set of integers I (k) l;n, k; n, l r := P kn(m), such that We ll the integers J (k) ;n I (k) ;n I(k) ;n I(k) r;n Q kn (m) (:9) (resp. I(k) ;n ) in the rst columns of the diagrams (k) (resp. (k) ) in ) for a rigged conguration the natural way (see examples). We use the notation ( (k) ; J (k) ;n corresponding to an s-conguration f (k) g and a set of quantum numbers J (k) ;n of a rigged conguration is dened as the sum of its quantum numbers:. The charge C(f (K) g; J (k) ;n ) = J (k) l;n (:) k;l;n It is easy to see that K m (q) = ( (k) ;J )Q (s) M(;) q c((k) ;J ) Remarks and examples.. It seems a very interesting task to study the numbers c(; ) := jc(; )j. For example c () = ; f () = 6 c () = 6; f () = 4 c (4) = 8; f (4) = 768 c () = ; f () = 5 c (54) = 58; f (54) = 9864 Here c := c(; jj ), f = jst Y ()j. It is clear that maxfc j ` ng! when n!. On the other hand it can be shown that for any partitions and the sequences fc? (a n ); (a n ) g and fc? + (n k ); + (n k ) g are bounded when n!. Here for
Decomposition of symmetric and exterior powers of the adjoint representation of gl N 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. [8], p.76) that if is a hook then c(; ). How can all partitions and which satisfy the condition c(; ) = be described?. In general an admissible matrix may have negative elements. However consider the case when m C(; ) \ M(Z f + ). 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 (abbreviation: rpp(; )) in the following way: let us ll in the i-th row of the diagram exactly m ij numbers equal to j. We obtain a tableau of the shape and content. 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 (.5) mean that rpp(; ) and word w() is a lattice sequence, and the correspondence m! denes a bijection C(; ) \ M(Z f + )?! f rpp(; )jw() is a lattice sequenceg: Then As an example, consider = (9; 5; 4; ; ), = (5; 5; 5; ; ; ; ; ) and = m = B @ C A 4 5 5 5 4 4 C(; ): rpp(; ); and w() = 444555 is a lattice sequence. Let us remark that if m C( ; ~ ~), where C(; ) \ M NM (Z) and l Z +, then matrix ~m = m + l E NM the matrix E NM has all elements is equal to, and ~ = +? (lm) N, ~ =? (M) ln ;. So we have an inclusion C(; ),! C( ~ ; ~), and consequently any m C(; ) may be considered as a lattice rpp for some l.. Consider a partition, a composition, and the set of rigged congurations QM(; ). Consider further the matrix m = (m ij ) with elements m ij = ( j? i) + i; ( j? j ), where (x) = ; if x ; (x) = ; if x < : (:) It is not dicult to see that m M(; ) (see (.)) and P kn (m) = P jn ( j? j ); if k = ; n; min(n; k )? min(n; k+ ); if k Q kn (m) = max(k; n)? max(k; n+); k :
8 Anatol N. Kirillov P So we see that if. with respect to the dominance order on partitions (e.g. [9]) then jn ( j? j ) for all n, and hence m C(; ) 6=. It can be shown that if the above matrix m = C(; ) then C(; ) =. We will call this matrix the maximal conguration and denote it by = ( ij ). It is easy to P see that the corresponding s- conguration (k) := ( (k) ; : : :), where (k) n := jk+ jn, is equal to ([k]), ; (k) where [k] := ( k+ ; k+ ; : : :), and for any s-conguration f (k) g of the type (; ) we have (k). (k), k. Furthermore if f (k) g k is as above then the set f (k) g k also is a s-conguration of the type? []; ( () ) and hence using a bijection (see [8]) we obtain a map namely ST Y (; )?!? QM(; ) (:) ST Y (; )?! ( () ; J () ;n ) ST Y? []; ( () ) ; (:) T ST Y (; )?!( (k) ; J (k) ;n ) k?!? ( () ; J () ;n ); ((k) ; J (k) ;n ) k?!? ( () ; J () ;n ); ~ T ; where ~ T ST Y? []; ( () ). However the inequalities on the quantum numbers J () ;n now depends on the tableau ~ T. Nevertheless using (.) we obtain an interpretation of the Morris identity for the Kostka-Foulkes polynomials [] in terms of rigged congurations. Note that from the existence of the maximal conguration there follows an inequality Y K ; (q) q c Qn ()? Q n () + n? n+ n n? ; (:4) n+ q where c = n() + n()? P k k ( k? ), Q n() := P jn j and n() := P i (i? ) i. The equality in (.4) is equivalent to the existence of only one conguration of the type (; ). In order to understand when the set C(; ) consists of only one element, let us dene for a partition and a composition the following set a D(; ) = D a;b (; ); a<b n D a;b (; ) = n where o jn( j? j) ; a n < b; a? > a b > max(; b+) here we assume that = +. Theorem.. D(; ) =. The set C(; ) consists of only one element i. and Corollary.. have in fact equality i the set D(; ) =. Assume that and are partitions,.. In the inequality (.4) we We may rewrite the condition D(; ) = as follows. Assume that = (l a ; la ; : : : ;
Decomposition of symmetric and exterior powers of the adjoint representation of gl N 9 l a r r ; a r+ ), where l > l > > l r >. Let us put n k := a + +a k, k r, n :=. Then the set D(; ) = i the following inequalities are satised ) if for some k, a k, k r, then min jn ( j? j) nk? < n < n k ; ) if for some k < r, we have a k = a k+ =, then jn k ( j? j) : 4. Now let us compute K ; (q) using (.7). a) = (6; ; ), = (; ; ; ; ; ). There exist 4 admissible matrices: m 4 f g max conguration rpp c(m) 5 5 5 6 Consequently, K ; (q) = q 5 + q 5 4 4 + q 5 + q 6. b) = (; ; ), = ( 8 ). At rst, we may compute K ; (q) using a well-known result (e.g. [9]) (put jj = N): K ;( N )(q) = q n( ) NY i= (? q i )H (q)? ; where H (q) = Q x (? qh(x) ) is a hook-length polynomial. So K ();( 8 )(q) = q 7 (? q6 )(? q 7 )(? q 8 ) (? q)(? q )(? q 4 ) : 4
Anatol N. Kirillov On the other hand let us use (.7). There exist 6 admissible matrices: m 4? 4? 5?? 5? f g c(m) 9 4 m 6?? max conguration f g Consequently, c(m) 9 7 K ;( 8 )(q) = q 7 + q 9 5 + q = q 7 (? q6 )(? q 7 )(? q 8 ) (? q)(? q )(? q 4 ) : In general, if = ( N ) we obtain an identity q n( ) NY i=(? q i )H (q)? = fmg 4 5 + q + q 4 4 + q 9 q c(m) K m (q); (:5) where the summation in (.5) is taken over all admissible matrices m C? ; ( N ). It is desirable to nd an analytical proof of (.5). c) For given partitions and we have the distinguished subset P in ST Y (; ), which corresponds to rigged s-congurations with zero values of quantum numbers J (k) ;n. It seems very interesting problem to give a direct denition of such tableaux. For example, take = (6; 5; 4; ), = (5; 4; 4; ; ). There exist admissible matrices.
Decomposition of symmetric and exterior powers of the adjoint representation of gl N m max conguration f g rpp 4 4 5 4 4 4 5 4 c(m) P m 5 4 4 4 4 5 4 4 f g rpp 4 5 4 4 c(m) P Consequently,. 4 5 4 4 K ; (q) = q + q + q
Anatol N. Kirillov x. Generalized exponents. Let g = sl(n; C) and G = SL(N; C). The adjoint action of G on the Lie algebra gl N = gl(n; C) extends to an action on symmetric algebra S (gl N ) = k S k (gl N ), where S k denotes the k-th symmetric power. By a theorem of Kostant [] S (gl N ) = I H (:) is a free module over G-invariants I generated by the harmonic polynomials H. Moreover, the ring of invariants I = S (gl N ) G = ff S (gl N ) f = f; 8 Gg is a polynomial ring in N variables f ; : : : ; f N, where f i S i (gl N ), and H = L k Hk is a graded, locally nite G-invariant subspace of S (gl N ) (so H k = H \ S k (gl N )). For each nite dimensional G-representation V let us consider polynomials F (V ) := k dim Hom g (V; H k )q k (:) E(V ) := k dim Hom g? V; k (gl N ) q k (:) where p denotes the p-th exterior power. Kostant also shows [] that F (V ) q= is equal to the dimension of the zero-weight subspace P V () of the representation V. Thus, F (V ) is really a polynomial in q, say s F (V ) = i= qd i, s = dim V (), and the integers d ; : : : ; d s 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 [,4], they describe a certain group cohomology [], and coincide with the Poincare polynomial for the ltration on the zero-weight subspace V () dened by the action of the principal nilpotent [5],[8]. It is interesting to note that the polynomials F (V ) coincide with the partition functions for some matrix models [8]. It is easy to see that if G has eigenvalues x ; : : : ; x N, then the adjoint action of on the space gl N has eigenvalues x i x? j ( i; j N). Therefore, the character of the exterior power k (gl N ) is the coecient of q k in the generating function Y i;jn So for each partition of length less than N we have (see (.)) Y i;jn ( + qx i x? j ) = ( + qx i x? j ) (:4) E(V )s (x ; : : : ; x N ) (:5)
Decomposition of symmetric and exterior powers of the adjoint representation of gl N Similarly, the character of the symmetric power S k (gl N ) is the coecient of q k in the generating function Y i;jn ; (:6)? qx i x? j and if we dene formal power series S N [](q) for each partition of length less than N via Y = S N [](q)s? qx i x? (x ; : : : ; x N ); j i;jn it follows that the coecient of q k in S N [](q) is the multiplicity of V in S k (gl N ). Using the theorem of Kostant (.) we see that S N [](q) = f(? q) (? q N )g? F (V ) (:7) Note also that we have the following decomposition from which it is easy to deduce that ;l()n S k (gl N ) = M `k l()n V V ; dim Hom g (V V ; V ) q jj = 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 V A be the representation of g = sl(n; C) corresponding to this diagram. Proposition. ([,,5]). KA;(l F (V A ) = n )(q); if jaj = l N; ; otherwise (:8) Proposition. ([,6]). Assume A and B are skew diagrams. Then F N (V A V B) = K A; (q)k B; (q) (? qn ) (? q N?l()+ ) ; (:9) b (q) where l() is the length of a partition, and b (q) = Y i(? q) (? q m i ) for = (i m i ); b (q) = : Here K A; (q) be the Kostka-Foulkes polynomial corresponding to a skew diagram A (e.g. [] or [7]).
4 Anatol N. Kirillov 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 [,6], we dene a representation V ; := V [N] ; as the Cartan piece in V V, i.e. the irreducible g-component generated by the tensor product of the highest weight vectors in each factor. It follows that V [N] ; ' V [;]N, where For example V [N] ();() {z } N?r?t [; ] N = ( + ; : : : ; r + ; ; : : : ; ;? t ; : : : ;? ; ): ' C, V [N] ();() ' g. As the rst example let us compute the generalized exponents for g = sl() and g = sl(4). a) Given an irreducible representation V of sl() with highest weight = ( ), + = l, = (l; l; l), consider the set C(; ). It is clear that there exists only one s-conguration f g = f( )g, and P = min(? ; ); c = c(m) =?? min? Consequently F (V ) = K ;(l )(q) = q c min(? ; ) + : ; : b) Assume V is an irreducible sl(4)-module with highest weight = ( ), + + = 4l, = (l 4 ). We have the following collection of s-congurations f(? k; + k); ( )g; where k? : It is easy to see that P =, Consequently, P ; +k = min( +?? 4k; ) P ;?k = min(? ;?? 4k) c k = 8 >< >: K ;(l 4 )(q) = k<??? l + k; if? k l? + l? k; if + k l? k +? l + k; if l + k q c k min( +?? 4k; ) + min(? ;?? 4k) + + "( + )q l? + min(? ; ) + ; (:)
Decomposition of symmetric and exterior powers of the adjoint representation of gl N 5 where "(n) =, if n is odd, and "(n) = for even n. Now consider the particular case when l: Then +?? 4k and??? 4k. It is not dicult to compute the sum (.) in this region, and obtain K ;(l 4 )(q) = q +?l (? q? + )(? q? + )(? q? + ) : (? q) (? q) (? q ) If we dene a partition = (? l;? l;? l), whence it is clear that K ;(l 4 )(q) = q n()+l This result is a particular case of corollary 6.. As the second example we consider the computation of the Kostka polynomials K ;, when = [; ] N and = ( N ), using the formula (.7). A) Let = = (; ), N 4. In this case there are only two admissible matrices (with (N? ) rows) m : B @ C A and m : B @? It follows that P (m ) =, P (m ) =, P (m ) =, P (m ) =, P (m ) = ; all others vacancies are equal to zero. Consequently, n? n? n? K [;] N;( N ) (q) = q + q 5 : (:) It is easy to see that lim K [;]N;( N! N ) (q) = q + q 5 (? q) (? q ) : B) = (4; ), = (; ; ; ), N 6. In this case there are eight admissible matrices. The corresponding s-congurations f (k) g are (N? ; N? 5; ); P = ; P = 6; P = ; c = 7 (N? ; N? 5; ; ); P = ; P = 6; P = ; c = 5 (N? ; N? 4; ); P = ; P = 4; P = ; P = ; c = (N? ; N? ); P = ; P = ; c = 9 (N? ; N? 6; ); P = 6; P = ; P = ; c = (N? ; N? 6; ; ); P = 6; P 4 = ; P = ; c = 9 (N? ; N? 5; ); P = 4; P = ; P = ; P = ; c = (N? ; N? 4); P = ; P = ; P = ; c = 9 C A
6 Anatol N. Kirillov Here we represent only () diagrams of fg; the others are the same in all cases and equal to (k) =? N?? k; max(n? 4? k; ) ; k N? : Consequently, N? K [;] N;( N ) (q) = q 9 + q N? 4 N? N? 5 4 N + q 5 4 + q N? Remark. N some of the s-congurations vanish. + q 4 4 4 + q 7 6 6 : 6 N? + q 9 N? N? 4 N? N? 5 4 + q 9 N? 6 + + (:) Formally, jc? [; ] N ; ( N ) j = 8 in our case only if N 8. For smaller values x. Internal product of Schur functions. It is well known that the irreducible characters of the symmetric group S p are parametrized by partitions of p. If w S p, then dene (w) to be the partition of p whose parts are the cycle lengths of w. For any partition of p P of length l, dene the powersum symmetric function p = p p p l, where p n (x) = i xn i, n, p (x) =. We use notation p w := p (w). The Schur functions and power-sums are related by a result of Frobenius (e.g. [9]): + s = (w)p w : (:) p! ws p Given three partitions ; ; of p, consider the following numbers g = (w) (w) (w): (:) p! ws p It is clear that = g ; (:) So each g is a nonnegative integer. D. E. Littlewood [5] dened an associative, commutative product f g on symmetric functions by s s = g s ; (:4)
Decomposition of symmetric and exterior powers of the adjoint representation of gl N 7 and extending to all ring of symmetric functions by bilinearity. Following R. P. Stanley we will call f g the internal product of f and g. Note that s s (p) = s and s s ( p ) = s ; (:5) where denotes the conjugate partition to. In terms of the power-sums we have the expansion s s = (w) (w)p w (:6) p! ws p Now let A be a skew diagram, jaj = p, S A be the representation of the symmetric group S p corresponding to A and A its character. By bilinearity we extend formulas (.), (.) and (.5) to the case of skew diagrams. In the same way we dene the internal product of the super-schur functions (e.g. [9]): s A s B (xjy) = A (w) B (w)p w (xjy); (:7) p! ws p where for a partition = ( ; ; : : :), the super power-sum functions may be dened as follows: p (xjy) = p (xjy)p (xjy) ; and p n (xjy) := P i xn i + (?)n+ P i yn i, n, p (xjy) =. It is easy to see that s s (xjy) = g s (xjy): Note that it is an important open problem to obtain a nice combinatorial interpretation of g. Now we dene a new family of polynomials from the decomposition of the internal product of the Schur functions in terms of the Hall-Littlewood ones: s A s B = `p b AB (q)p (x; q) (:8) From (.) it follows that b AB (q) = g AB K ; (q); (:9) so b AB (q) are polynomials with nonnegative integer coecients, and b AB () = g AB. The integers (A; B) := b AB () admit a nice combinatorial interpretation due to J. Donin [8,9]. It is an interesting problem to nd a nice combinatorial denition of some statistics on the set (A; B) (see below) such that b AB (q) = J (A;B) q (J) (:)
8 Anatol N. Kirillov The solution of this problem gives a rule for computation of the numbers g AB, namely g AB = fj M (A; B)j (J) = g: (:) We emphasize the similarity between (.) and the classical Littlewood{Richardson rule c = ft ST Y ( n ; )jc(t ) = g; (:) where c(t ) is the charge of the tableau T. Now let us return to a combinatorial denition of the integers (A; B). First, following [8], we give a combinatorial description for the scalar product of the skew Schur functions (e.g. [9]): (A; B) := hs A ; s B i (:) in terms of B-tableaux of the shape A. Note that there exist other combinatorial characterizations of the scalar product due to A. M. Garsia and J. Remmel [5,6], and A. V. Zelevinsky []. Both of these approaches are based on the properties of the (generalized) Robinson-Schensted correspondence. So let A and B are the skew Young diagrams. We consider the standard lling ~ B of the shape B = (b ; b ; ) which contains b ones in the rst row of B, b twos in the second row etc. Let C ; : : : ; C l be the columns of ~ B, writing from left to right. A standard tableau T of the shape A is called to be a B-tableau i the tableau T may be presented as disjoint union of a standard subtaleaux T j, j = ; : : : ; l, which up to permutation of the subindices j = ; : : : ; l, are isomorphic to the columns C j. The set of all B-tableaux of the shape A is denoted by (A; B) and the number of its elements by (A; B) := j (A; B)j. This numbers possesses the following properties: ) (A; B) = (B; A) = (A ; B ); ) (A ; B) = (A; B ); ) if A and B are an ordinary diagrams, then (A; B) = AB ; 4) if B is a diagram, then (A; B) is equal to the \Littlewood-Richardson number", i.e. the number of a standard Young tableaux T of the shape A and content B such that corresponding word w(t ) is a lattice sequence (compare with (.)). It is possible to dene similarly the set (A; B) for an arbitrary diagram B as follows. Assume B = n and let us put d k = min(b k ; k?? k ), k. A sequence w of positive integers is called B-lattice, i for any xed k!, k, the dierence between the number of elements equal to k preceding it and the number of elements equal to k? preceding it is greater than or equal to (?) d k. Using the notion of a B-lattice sequence we may reformulate the denition of the set (A; B) of a B-tableaux of the shape A as follows. (A; B) = ft ST Y (A; B)j!(A) is a B-lattice sequenceg: Let us consider an explanatory example. Assume A = (4; ; ; ) n (; ; ), B = (5; ; ; ) n (; ). Then K A;(;;;) (q) = + q + q + q, (A; B) = 8. It easy to see that only the
Decomposition of symmetric and exterior powers of the adjoint representation of gl N 9 following tableaux, from ST Y (A; B) 4 do not appear in (A; B). For example, and 4 T = 4 (A; B); because w(t ) = 4 is a B-lattice sequence. Let us continue. Assume is a skew diagram corresponding to the disjoint union of a one-row or one-column diagrams, A and B are skew diagrams such that jaj = jbj = jj = p. We want to obtain a combinatorial description for the following intertwining number of representations of the symmetric group S p (A; B) = dim Hom C[S p ](S ; S A S B ): (:4) Proposition. ([8]). Assume = `i i, where either i = ( i ) or i = ( i ). Then (A; B) = ` A= i A i B= i B i ja i j=jb i j=jij Y i i (A i ; B i ); (:5) where i (A; B) = (A; B), if i = ( i ) or i (A; B) = (A; B ), if i = ( i ). Here symbol ` means a disjoint union of sets. Using proposition. one can obtain the expression for the internal product of the super-schur functions in terms of the product of monomial symmetric functions. Theorem.. Assume A and B are skew diagrams, jaj = jbj. Then where (; ) = ` i s A s B (xjy) = ; ( i )``j : ( j ) (;) (A; B)m (x) (y); (:6) In the particular case when there are no odd variables y, this result is proved in [9].
Anatol N. Kirillov Note some consequences of the theorem... g AB = b AB (q)(k? (q)) ; where K? (q) is the inverse to the Kostka matrix K(q) = (K (q)).. \Determinantal formula" for g ABC ([8,9]). Let A; B; C be a skew diagrams. Assume C = n, l() n. Then g ABC = ws n (?) l(w) (??w(?)) (A; B); where = (n? ; n? ; : : : ; ; ). Now let us continue a consideration of the examples (.) and (.) A) Take = = (; ). Then s s = fp + p g = m () + m () + 4m ( ) = = s () + s () + s = P () + ( + t)p () + ( + t + t + t )P ( ): Recall that P = P (x; t) are the Hall-Littlewood polynomials. We see that b () =, b () = + t, ) b( = ( + t)( + t ). Further, ( s s (q; : : : ; q N? ) = q? q N ) + q? q N? q? q N : Note that it is possible to transform this expression to the following form: " #" # " #" #" s s (q; : : : ; q N? ) = q N? N? + q 5 N? # : (:7) It is interesting to compare this formula with (.). B) Consider now = (4; ), = (; ; ; ). Then s s = 8p 6? 5p 4 7 p + 45p p? 9p p 4 + 44p p 5? 5p + 9p p 4 = m (4) + m () + 4m () + 9m () + 6m () + 6m () + 6m ( 5 ) + 8m ( 6 ) = s (4) + s () + s () + s () + s () + s ( 4 ) + s ( 6 ): After some simple but extensive computations, we may transform a formula for s s (q; : : : ; q N? ) to the following form: " #" " #" #" # s s (q; : : : ; q N? )=q 9 N? N?4 #+q 9 N? N?4 (:8) " #" #" #" " #" #" #" " #" #" # +q N? N?5 #+q N? N?5 #+q 5 4 N 4 4 6 " #" #" " #" #" " #" #" # +q 7 4 4 N? #+q 9 N? #+q 4 N? : 6 6 6
Decomposition of symmetric and exterior powers of the adjoint representation of gl N The partial explanation of this striking analogy with formula (.) for K [;] N;( N ) (q) will be given in Section 6. Remark. Using a result of P. Hanlon [, sect. 4B] and a formula for decomposition of a tensor product of irreducible representations of gl N, corresponding to rectangular diagrams [8] it is not dicult to see that (p) (; ) = dim Hom g V [;] N ; glp N = f f ; (:9) where f := jst Y (; ( p ))j. As above, we assume that jj = jj = p and l() + l() N. x4. The stable behavior of the Kostka-Foulkes polynomials. R. K. Gupta [,6] has studied the decomposition (.6) into Schur functions in the limit as N tends to innity. A certain amount of delicacy is requied to do this, since it is not clear how to pass to a limit in the rst place.? Shesuggested the correct way of passing to the limit based on a study of polynomials F V [N] as N!, and showed that the limit F (q) := lim F? V [N] N! ; (4:) exists as a formal power series. Also she conjectured that the F 's satisfy a number of remakable properties. In [] R. P. Stanley proved the following results Proposition 4.. ) We have F (q) = s s (q; q ; : : :) := lim N! s s (q; : : : ; q N? ) (4:) ) There is a polynomial P (q) Z [q] for which F (q) = P (q) H (q)? ; where H (q) = Q x (? qh(x) ) is the hooklength polynomial. ) P () = (), the number of standard Young tableaux of shape (and content ( jj )). A comparison of the results of R. Stanley [] and I. Macdonald [9] gives the equality P (q) = q jj K ; (q; q); (4:) where K ; (q; t) is the double Kostka polynomial constructed by I. Macdonald [9]. From (4.) and (.8) we see that F (q) = lim N! K [;]N;( N ) (q) (4:4) and consequently, q jj K ; (q; q) = H (q) lim N! K [;]N( N ) (q): (4:5)
Anatol N. Kirillov It follows from (.8) that the problem of computation of the generalized exponents is a particular case of a similar problem for Kostka polynomials. The last problem is settled in [8], namely, in section one we presented the exact formula (.7) for Kostka-Foulkes polynomials in terms of a sum of products of q-binomial coecients. The last sum is taken over the set of all admissible matrices of the type (; ). The essential observation [8] is that for any partitions and and positive integer a there exist only a nite (independent on N) number of admissible matrices of the type ( (a N ); (a N )). Here, for partitions and we denote by the partition which corresponds to a composition (; ), at the same time + denotes the partition whose parts equals to i + i. As a consequence we obtain Corollary 4. (e.g. [4]). the limit For any partitions and and positive integer a, there exists lim K (a N! N );(a N )(q) := F ;;a (q) (4:6) Similarly, for any vectors = ( l ) Z l and = ( l ) Z l and positive integer k, the number of the admissible matrices of the type ( + (N k ); + (N k )) is also stabilized when N!. In this case we have even more: the Kostka-Foulkes polynomials K +(N k );+(N k )(q) in fact does not depend on N when N is suciently big. We intend to give an interpretation for this stable values of the Kostka polynomials elesewhere. Note that from the result concerning stabilization of the admissible matrices it is possible to deduce the following assertion, which was conjectured by R. Gupta [4] and proved by G.-N. Han [6]. Theorem 4.. Assume that ; ; are partitions. Then K ; (q) K ; (q); (4:7) where the inequality between polynomials P (q) P (q) means that the dierence P (q)? P (q) is a polynomial with nonnegative coecients. Sketch of the proof. It is sucient to prove (4.7) when the partition consists of only one part, namely = (a). Consider the set C n := C( (a n ); (a n )). Now assume that the row (a) appears at rst in the diagram (a n ) in a position with number r, and m C n. We dene a new matrix em in the following way: em ij = m ij, if i < r, em rj = (a?j) (see (.)) and em ij = m i?;j, if i > r. The main observation is the following: the matrix em C n+, and in this way we obtain bijection C n! Cn+ as soon as n is suciently big. From the formula (.7) it follows that the polynomial K (a n );(a n )(q) is a nite sum of products of q-binomial coecients of the type [ n+c c ] and others, which do not depend on n. This gives (4.7) in the case = (a). In fact we see that it is more generally valid namely, the inequality (4.7) is correct on the level of congurations: there exists an imbedding C(; ),! C( ; ) (m! m ; c(m) = c(m )) such that (see (.6)) K m (q) K m (q):
Decomposition of symmetric and exterior powers of the adjoint representation of gl N In [6] G.-N. Han constructed an imbedding ST Y (; ),! ST Y ( ; ) which preserves the charge. It is interesting to understand whether or not the G.-N. Han's imbedding preserves the congurations. As another application of the theorem [8] about stabilization of the number of admissible matrices of the type ( (a N ); (a N )) when N!, we obtain the following result, which was suggested by A. Lascoux [4]. Theorem 4.. Let ; ; a be as above. Then the following sum n z n K (a n );(a n )(q) is a rational function of q and z. Now let us consider the stable behavior of the polynomials F N (V V ) (see (.9), or []) when N!. Our approach is based on the results obtained by R. Brylinski [5,6]. Proposition 4.4 [6]. following identities Let ; be partitions, jj = jj = p, l() + l() N. We have the F N? V V F? V [;] N? = ( n ; n )F V [;] N ; (4:8) ; = (?) jj F N V n Vn ; (4:9) where for skew diagrams A and B the number (A; B) is dened in (.). For the beginning, consider the example A) with = = (; ). Using formula (.9) it easy to see that F N (V V ) = " #" N # " N? + q #" #" # N : Considering other examples we again reveal a striking analogy with the formula (.7) for Kostka polynomials. Theorem 4.4. We have Y q jj F N (V V ) = q c(m) fmg k;n " # P kn (m) + Q kn (m) + k n; (r? k) P kn (m) ; (4:) where the summation is taken over all admissible matrices m of the type ([; ] N, ( N )), r = l().
4 Anatol N. Kirillov Corollary 4.6. We have a) lim N! F N(V V ) = q?jj F (q); (4:) where F (q) is dened by (4.) and (4.). b) q jj F N (V V ) K [;] N;( N ) (q): (4:) It seems plausible that one can also have q jj F N (V V ) s s (q; : : : ; q N? ): x5 Generalized exponents and the principal specialization of the Schur functions. Let e ij ; i; j N be the standard basis in gl N. Denote by SN k := Sk (gl N ) the k-th symmetric power of gl N. Dene a linear map k : gl N! SN k for all positive integers k: e ij = e ij ; k e ij = k? N e i e e k? j; k : (5:) It is clear that k 's are g-module homomorphisms. Highest weights of irreducible components of SN k are of the form f = (f f f N ) Z N such that jfj = ; so {z } N?r?s f = ( ; ; r ; ; ;? s ;? s? ; ;? ;? ): We denote the corresponding partitions by and, thus f + ( N ) = [; ] N ; and jj = jj := p: Given in addition a composition ; e() N? ; jj = p; consider the reverse plane partitions (rpp) of the shape and content ; rpp of the shape and the same content, and bijection ' :! such that ' = : Using this data we construct some monomial from SN k : For this let us enumerate all cells of from the left to the right and from top to bottom and assign to the m-th cell of the following integers: i m is the number of the row where this cell is situated, j m is the number of the row where the cell '(m) is situated and k m is the number which belongs to the m-th cell of : Consider further the product py m= km e im j m S k N ; where k = P p m= k m = P j j j: The weight of this element is equal to f. Finally we form the sum [8] py v( ; ; ') := (?) e()+e() km e i(m) j (m) ; (5:) ; m=
Decomposition of symmetric and exterior powers of the adjoint representation of gl N 5 where (resp.) runs all elements of the column stabilizer of (resp. ). It easy to see that the element (5.) is a primitive vector in S k N of weight, f + (N ) with respect to the g-action, i.e. E ij v( ; ; ') = ; if i < j N; H i v( ; ; ') = (f i + )v( ; ; '); i N? ; where E ij ( i 6= j N) and H i ( i N? ) are the standard generators of g. Hence, any non zero vector v( ; ; ') generates an irreducible subspace with highest weight f + ( N ) in S k N : Let us denote by Hk N;f (; ; ) the space generated by primitive vectors v( ; ; ') of the form (5.) with xed ; ; ; but dierent '. Now for any i with i 6= ; consider a skew diagram A i (resp. B i ) which corresponds to all cells of (resp. ) occupied by the number i. The map '; by assumption, induces a bijections ' i : A i! B i : It easy to see that H k N;f (; ; ) ' O i S(A i ; B i ); where = `i A i (resp. = `i B i) denotes the disjoint union of skew diagrams A i ; ja i j = jb i j = i ; and for any skew diagrams A and B the space S(A; B) is generated by primitive vectors of the form (5.) for i rpp(a; jaj ); rpp(b; jbj ); but dierent choices of bijection ' : A! B: Proposition 5. ([8]). We have dim S(A; B) = (A; B); where (A; B) := hs A ; s B i is the scalar product of skew Schur functions, and may be computed by an analogue of the Littlewood-Richardson rule (see Section ). Now we dene the space H k N;f = M C(k;p;N?) M rpp(;) rpp(;) H k N;f (; ; ); where C(k; P p; N? ) is the set of compositions = ( ; ; N? ) such that jj = p; i i i = k: It easy to see that dim H k N;f = C(k;p;N?) Therefore the generating function for dim H k f;n k dim H k f;n q k = k = where in the last step we use (.6). C(k;p;N?) (; ): (5:) is equal to (; )q k (5: ) (; )m (q; ; q N? ) = s s (q; ; q N? );
6 Anatol N. Kirillov Proposition 5. ([8,9]). Let H k = H \ S k (gl N ) be the space of harmonic polynomials of the degree k, and for any irreducible g-representation V with highest weight ; H k denotes the isotopic component of H k corresponding to : Then we have ) for all k, H k [;]N Hk N;f ) if k < N; then H k [;]N = Hk N;f ; where [; ] N = f + ( N ): Corollary 5.. We have K [;] N;( N ) (q) s s (q; ; q N? )(mod deg N): (5:) In particular we see that if k < N then C(k;p;N?) where c(t ) is the charge of the tableau T (e.g. [9]): (; ) = jft ST Y ([; ] N ; ( N ))jc(t ) = kgj; (5:4) Both sides of equality (5:4) admit pure combinatorial interpretation in terms of Young tableaux of special kind. So it is desirable to nd a bijective proof (5:4). In Section we introduced the polynomials b (t) (see (.8)) such that It is easy to see that k b ; () = (; ); C(k;p;N?) b (t)qk = b () = g : b (t)m (q; ; q N? ): (5:5) When t is equal to, the expression (5.5) reduces to (5: ). In connection with (4.) and (4.) it seems interesting to nd a good expression for the limit lim N! m (q; ; q N? ): Note that from (.) it follows that on the left side of (5.4) there exists an additional statistic : It seems interesting to understand its value on the right side of (5.4). Now we consider the simplest nontrivial example which probably helps to clarify the dierence between the left and right sides of (5.). Consider k = 4; N = 4; = = (; ): It is known (e.g. [,4]) that K [;]4 ;( 4 )(q) = q + q 4 + q 5 + q 6 + q 7 s s (q; q ; q ) = q + q 4 + 4q 5 + 5q 6 + 4q 7 + q 8 + q 9 : Now the set C(k; p; N? ) contains only one composition = (; ); and (; ) = ; according to the following decompositions into skew diagrams of the pair (; ) J J or J J
Decomposition of symmetric and exterior powers of the adjoint representation of gl N 7 In all cases we have ' = id: Using formula (5.), which is due to J. Donin, we may nd the corresponding highest weight vectors. They are equals to a = ( e 4 e 4 e? e 4 e 4 e ); a = e 4 e 4 e? e 4 e 4 e? e 4 e e 4? e 4 e e 4 : It is easy to see that a and a are linearly independent and dene dierent representations in S 4 (gl 4 ): However, if we write in detail the denition of (say e = e e + e e + e e + e 4 e 4 ; see (5.)) and substitute in a ; then after some cancellation, we obtain the following result a = b ; where b = e 4 e 4 e? e 4 e 4 e ; = e + e + e + e 44 : It easy to see that b is a highest weight vector in S (gl 4 ) and is an invariant. This means that only a H 4 [;] 4 and H 4 [;] 4 6 H4;f 4 ; where f = (; ;?;?): x6. Unimodality of the principal specialization of the internal product of the Schur functions. In the section we gave formula (.7) computing the Kostka-Foulkes polynomials in terms of the sum over admissible matrices of some product of the q-binomial coecients. The proof (see [8]) is based on the study of the properties of the coecients bijection [7,8] ST Y (; )! QM (s) (; ) (6:) Let us now x diagrams and ; jj = jj = p, and a positive integer N; N l() + l(). Consider the set H N (; ), whose elements are the collections J = ft i g N? of i= B i-tableaux T i of the shape A i (see section ) for some subdivisions fa i g and fb i g of the diagrams and into nonoverlapping skew diagrams A i and B i. Any element J H N (; ) denes a composition P = ( ; ; ), where i = ja i j = jb i j. We dene the charge of J as c(j) := i i i. It is clear that jh N (; )j q = s s (q; ; q N? ) (6:) Another interpretation of the right side of (6.) was obtained by R. Brylinski [7]. Now we want to construct a bijiection between the set H N (; ) and some extension QM ~ N (; ) of the set of rigged c-congurations QM (c) ([; ] N ; ( N )). For this aim we consider the set of all c-congurations of the type ([; ] N ; ( N )) and introduce new upper bounds for quantum numbers I (k) ;n. Namely, we will assume that for a given matrix m C([; ] N ; ( N )) the following inequalities are satised: I (k) ;n I(k) ;n I(k) l;n Q kn(m) + (k? )( n? n+)(r? k); (6:)
8 Anatol N. Kirillov where = ( N ); (x) is the step-function (.), r = l() and l = P kn (m). We denote by QM ~ N (; ), the set of pairs (fmg; I) where fmg is a c-conguration of the type ([; ] N ; ( N )) and I = fi (k) ;n g is the set of an integers, which satisfy the conditions (6.). Our main observation is the following: the bijection (6.) with = [; ] N ; = ( N ) may be extended to a bijection H N (; )! ~ QM N (; ) (6:4) P such that c(j) = c(m) + k;j;n I(k) jn, if J $ (fmg; I): We do not give here the construction of such bijection, because it is a rather lengthy. Nevertheless, using the existence of such a bijection, we obtain the following expression for the principal specialization of the Schur functions Theorem 6.. We have s s (q; ; q N? ) Y = q c(m) fmg k;n " P kn (m) + Q kn (m) + N(k? ) n; (r? k) P kn (m) where the summation is taken over all admissible matrices m of the type ([; ] N ; ( N )); r = l(). # q ; (6:5) In the case when the diagram consists of only one part (p), we obtain the following result Corollary 6.. We have " # q jj+n() N? q = s (q; ; q N? ) = K [(p);] N;( N ) (q); (6:6) where [ N ] q is the generalized q-gaussian coecient, corresponding to a diagram (e.g. [9]), and n() = P i (i? ) i. Before we consider another application of the Theorem 6., let us recall that a polynomial P (q) = a k q k + + a n q n Z[q] is called symmetric and unimodal i a i = a j when i + j = k + n, and the following inequalities hold. a k a k+ a [ k+n ] Theorem 6.. The polynomial s s (q; q N? ) is symmetric and unimodal.
Decomposition of symmetric and exterior powers of the adjoint representation of gl N 9 Proof. First, it is well known (e.g. []) that the product of symmetric and unimodal polynomials again is symmetric and unimodal. Second, we use a well known fact (e.g. []), that the ordinary q-gaussian coecient [ m+n n ] q is a symmetric and unimodal polynomial of degree mn. So in order to prove Theorem 6., it is sucient to show that the sum c(m) + P k;n (m)[q kn (m) + N(k? ) n; (r? k)] (6:7) k;n is the same for all admissible matrices m of the type ([; ] N ; ( N )). In fact, it is not dicult to see that the sum (6.7) is equal to N jj. This concludes the proof. Let us distinguish the following particular case of Theorem 6., when is a one-row diagram. Corollary 6.4. The generalized q-gaussian coecient " N is a symmetric and unimodal polynomial of degree (N? )jj? n( ) and coincides (up to the power of q) with the following Kostka-Foulkes polynomial q jj+n() " # N? q # q = K [(p);] N;( N ) (q): (6:8) If we rewrite the right side of (6.8) using the formula (.7) for Kostka polynomials, then we obtain some identity, which may be considered as a generalization of the KOH (e.g. [5]) on the case of an arbitrary partition. The KOH identity corresponds to the case when the partition (or ) consists of only one part. Note that in the proof of the Theorem 6. we use symmetry and unimodality of the ordinary q-gaussian coecient [ m+n n ] q. However, it is possible to obtain the proof of the unimodality of [ m+n n ] q by induction starting directly from the identity (6.8) when = ( p ). See details in []. Note nally that it seems a very attractive task to understand the excellent constructive proof of the unimodality of the q-gaussian coecients [ m+n n ] q given by K. O'Hara [] from the point of view presented here. Corollary 6.5. Let ; ; n be partitions of p and s s s n (x) = (w) (w) n (w)p p! (w) (x) W S p be the corresponding internal product. Then the polynomial is symmetric and unimodal. s s s n (q; ; q N? )