Lusztig’s t-Analogue of Weight Multiplicity via Crystals

Abstract

We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated with each classical root system. In finite type A _n−1, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type C _n, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type A _2n−1, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig t-analogues associated with zero weight spaces in the irreducible representations of symplectic Lie algebras. We then present three applications of our combinatorial formula. Our methods are expected to extend to the orthogonal types.

By a result of Lascoux, the type A Kostka–Foulkes polynomials also expand positively in terms of the so-called atomic polynomials. We define, in arbitrary type, a combinatorial version of the atomic decomposition, based on the connected components of a modified crystal graph. We prove this property in type A, as well as in types B, C, and D in a stable range for t = 1. We also discuss other cases, applications, and a geometric interpretation. Finally, in classical types, we state the atomic decomposition for stable 1-dimensional sums or, equivalently, for the stable Lusztig t-analogues.