, Volume 47, Issue 1, pp 49-61

En">Characters and Composition Factor Multiplicities for the Lie Superalgebra \({\mathfrak{g}}{\mathfrak{l}}\) ( m / n )

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Abstract

The multiplicities aλ μ of simple modules Lμ in the composition series of Kac modules V lambda for the Lie superalgebra \({\mathfrak{g}}{\mathfrak{l}}\) (m/n ) were described by Serganova, leading to her solution of the character problem for \({\mathfrak{g}}{\mathfrak{l}}\) (m/n ). In Serganova's algorithm all μ with nonzero aλ μ are determined for a given λ this algorithm, turns out to be rather complicated. In this Letter, a simple rule is conjectured to find all nonzero aλ μ for any given weight μ. In particular, we claim that for an r-fold atypical weight μ there are 2r distinct weights λ such that aλ μ = 1, and aλ μ = 0 for all other weights λ. Some related properties on the multiplicities aλ μ are proved, and arguments in favour of our main conjecture are given. Finally, an extension of the conjecture describing the inverse of the matrix of Kazhdan–Lusztig polynomials is discussed.