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

Lecouvey, Cédric ; Lenart, Cristian

Journal of Algebraic Combinatorics, Tome 30 (2009) no. 2, pp. 141-163.

Résumé

Summary: The paper is devoted to the generalization of Lusztig's $q$-analog of weight multiplicities to the Lie superalgebras $\mathfrak gl( n, m)$ mathfrakgl(n,m) and $\mathfrak s$po($2 n, M$). mathfrak${spo(}2n,M)$. We define such $q$-analogs $K _{ \lambda , \mu }( q)$ for the typical modules and for the irreducible covariant tensor $\mathfrak gl( n, m)$ mathfrakgl(n,m) -modules of highest weight $\lambda $. For $\mathfrak gl( n, m)$, mathfrakgl(n,m), the defined polynomials have nonnegative integer coefficients if the weight $\mu $ is dominant. For $\mathfrak s$po($2 n, M$) mathfrak${spo(}2n,M)$ , 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 $\mathfrak gl( n, m)$ mathfrakgl(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 Schützenberger.

Détail

Keywords: keywords general linear superalgebras, orthosymplectic superalgebras, typical modules, irreducible covariant tensor modules, Lusztig's $q$-analog of weight multiplicity, semistandard hook-tableaux, charge statistic