Characters and Composition Factor Multiplicities for the Lie Superalgebra Gl(m/n)

The multiplicities a λ,µ of simple modules L µ in the composition series of Kac modules V λ for the Lie superalgebra gl(m/n) were described by Serganova, leading to her solution of the character problem for gl(m/n). In Serganova's algorithm all µ with nonzero a λ,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 a λ,µ for any given weight µ. In particular, we claim that for an r-fold atypical weight µ there are 2 r distinct weights λ such that a λ,µ = 1, and a λ,µ = 0 for all other weights λ. Some related properties on the multiplicities a λ,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.