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Home > Vol 30, No 2 (2009)

Cédric Lecouvey and Cristian Lenart

The paper is devoted to the generalization of Lusztig's q-analog of weight multiplicities to the Lie superalgebras \mathfrak gl( n, m) \mathfrak{gl}(n,m) and \mathfrak spo(2 n, M). \mathfrak{spo(}2n,M). We define such q-analogs K _{λ , μ}  ( q) for the typical modules and for the irreducible covariant tensor \mathfrak gl( n, m) \mathfrak{gl}(n,m) -modules of highest weight λ . For \mathfrak gl( n, m), \mathfrak{gl}(n,m), the defined polynomials have nonnegative integer coefficients if the weight μ  is dominant. For \mathfrak spo(2 n, M) \mathfrak{spo(}2n,M) , we show that the positivity property holds when μ  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) \mathfrak{gl}(n,m) -module of highest weight λ  and a dominant weight μ  is the generating series of a simple statistic on the set of semistandard hook-tableaux of shape λ  and weight μ . This statistic can be regarded as a super analog of the charge statistic defined by Lascoux and Schützenberger.

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