/ 930499 March 1993 GENERALIZATION OF THE GALE - RYSER THEOREM

We prove an inequality for the Kostka Foulkes polynomials Kλ,μ(q). As a corollary, we obtain a nontrivial lower bound for the Kostka numbers and a new proof of the Berenstein Zelevinsky weight-multiplicity-one-criterium. The concept of Young tableau plays an important role in the representation theory of the symmetric and general linear groups. Based on the pioneering fundamental works of G. Frobenius, I. Schur, A. Young, H. Weyl and further developed in the works of C. Kostka, G. Robinson, A. Richardson, D. Littlewood, C. Schensted, H. Foulkes, J. Green, G. James, M.-P. Schutzenberger, R. Stanley, G .Thomas, A. Lascoux, C. Greene and many others, the theory of Young tableaux is now an important branch of representation theory and combinatorics with a large number of deep and beautiful constructions and results. A good introduction to the subject are the books of D. Littlewood [L], G. James [J], I. Macdonald [M], B. Sagan [S], W. Fulton [Fu]. An entirely new point of view on the Young tableaux and representation theory of general linear and symmetric groups comes from Mathemetical Physics, namely from the Bethe ansatz [Fa], [FT], [KR]. Bethe ansatz has an important role in the study of the exactly solvable models of Mathematical Physics [Fa]. From a representation theory point of view, the Bethe ansatz (for the glN invariant Heisenberg model) gives a very convenient constructive method for decomposing the tensor product of irreducible representations (irreps) of the Lie algebra glN into the irreducible parts. In fact, the Bethe vectors appear to be the highest weight vectors in the corresponding irreducible components. This observation allows to identify the tensor-product-multiplicities with the number of solutions of some special system of algebraic equations (Bethe’s equations). Finally, in some particular cases, the number of solutions of the corresponding Bethe equations admits a 2 Anatol N. Kirillov combinatorial interpretation in terms of rigged configurations [K1], [K3]. On the other hand, it is well-known (see e.g. [L], [M]), that the multiplicity of an irreducible representation of the Lie algebra glN in the tensor product of rectangular-shape-highest-weight irreps may be identified with the number of Young tableaux of some special kind (e.g. (semi)standard (super)tableaux, . . .). In this way one can identify a set of Young tableaux with a corresponding set of rigged configurations (see e.g. [K1]). This paper is devoted to the solution of the following problem: given the partitions λ and μ, when λ does only one configuration (see §1 below) of the type (λ, μ) exist? This problem may be reformulated in the following form. One can prove that for given partitions λ and μ there exist an inequality for the Kostka-Foulkes polynomial Kλ,μ(q) (see e.g. [K1], or §2 below): Kλ,μ(q) ≥ q c λ2 ∏ n=1 [∑ j≤n(μ ′ j − λ ′ j) + λ ′ n − λ ′ n+1 λ′n − λ ′ n+1 ]