Combinatorics of generalized exponents
Abstract
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 An-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 Cn, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type A2n-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 present three applications of our combinatorial formula, and discuss some implications to relating two type C branching rules. Our methods are expected to extend to the orthogonal types.