Combinatorics of generalized exponents

Cedric Lecouvey, Cristian Lenart

Published 2017-07-11Version 1

We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to 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 to zero weight spaces in the irreducible representations of symplectic Lie algebras. We also prove a related conjecture of the first author as an application, and discuss some implications to relating two type C branching rules. Our methods are expected to extend to the orthogonal types.