Characters and composition factor multiplicities for the Lie superalgebra gl(m/n)

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The multiplicities alambda,mu of simple modules L(mu) in the composition series of Kac modules V(lambda) 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 mu with nonzero alambda,mu are determined for a given lambda; this algorithm turns out to be rather complicated. In this Letter a simple rule is conjectured to find all nonzero alambda,mu for any given weight mu. In particular, we claim that for an r-fold atypical weight mu there are 2ʳ distinct weights lambda such that alambda,mu=1, and alambda,mu=0 for all other weights lambda. Some related properties on the multiplicities alambda,mu 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.