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.

LaTeX file with epic, eepic pictures, 17 pages, November 1994, to appear in Lett. Math. Phys

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