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

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 conngurations. The stable behavior of some polynomials is studied. Diierent examples are presented. x0. 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 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(gl N) is equivalent to the problem of computating generalized exponents. This last problem is considered in the vast body of literature e. 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. 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 coeecients and also to try to connect their content with the theory of rigged conngurations, 16{18]. The main result of this note is the construction of a bijection between some combinatorially deened 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 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 coeecients n ] q. In fact, in this case we obtain an expression of generalized q-Gaussian …