Combinatorics of generalized exponents

TL;DR

This paper provides combinatorial proofs for the positivity of generalized exponents in classical root systems, offering new descriptions and applications in types A and C, with potential extensions to orthogonal types.

Contribution

It introduces purely combinatorial proofs and descriptions of generalized exponents for classical root systems, avoiding traditional tableau and charge statistic methods.

Findings

Combinatorial proof of positivity for stabilized generalized exponents.

New crystal graph descriptions for types A_{n-1} and C_n.

Applications to Lusztig t-analogues and branching rules.

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 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 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.