Crystal Graphs and $q$-Analogues of Weight Multiplicities for the Root System $A_n$

Abstract

We give an expression of the $q$-analogues of the multiplicities of weights in irreducible $\sl_{n+1}$-modules in terms of the geometry of the crystal graph attached to the corresponding $U_q(\sl_{n+1})$-modules. As an application, we describe multivariate polynomial analogues of the multiplicities of the zero weight, refining Kostant's generalized exponents.