On q-analogs of weight multiplicities for the Lie superalgebras gl(n,m) and spo(2n,M)

TL;DR

This paper extends Lusztig's q-analog of weight multiplicities to Lie superalgebras gl(n,m) and spo(2n,M), defining new polynomials with positivity properties and combinatorial interpretations.

Contribution

It introduces q-analogs of weight multiplicities for superalgebras, establishing their positivity and combinatorial interpretations, which were not previously known.

Findings

Polynomials have nonnegative integer coefficients for dominant weights.

Positivity holds when weights are sufficiently far from certain walls.

The q-analog relates to a super version of the charge statistic on semistandard hook-tableaux.

Abstract

The paper is devoted to the generalization of Lusztig's q-analog of weight multiplicities to the Lie superalgebras gl(n,m) and spo(2n,M). We define such q-analogs K_{lambda,mu}(q) for the typical modules and for the irreducible covariant tensor gl(n,m)-modules of highest weight lambda. For gl(n,m), the defined polynomials have nonnegative integer coefficients if the weight mu is dominant. For spo(2n,M), we show that the positivity property holds when mu is dominant and sufficiently far from a specific wall of the fundamental chamber. We also establish that the q-analog associated to an irreducible covariant tensor gl(n,m)-module of highest weight lambda and a dominant weight mu is the generating series of a simple statistic on the set of semistandard hook-tableaux of shape lambda and weight mu. This statistic can be regarded as a super analog of the charge statistic defined by Lascoux and Schutzenberger.