Letters in Mathematical Physics

, Volume 47, Issue 1, pp 49-61

First online:

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

  • J. Van der Jeugt
  • , R. B. Zhang

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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.

character formula Lie superalgebra \({\mathfrak{g}}{\mathfrak{l}}\) (m/n ) Kac module composition factors Kazhdan–Lusztig polynomial.