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

The multiplicities a�,µ of simple modules Lµ in the composition series of Kac modules Vfor 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�,µ are determined for a given �; this algorithm turns out to be rather complicated. In this Letter a simple rule is conjec- tured to find all nonzero a�,µ for any given weight µ. In particular, we claim that for an r-fold atypical weight µ there are 2 r distinct weightssuch that a�,µ = 1, and a�,µ = 0 for all other weights �. Some related properties on the multiplicities a�,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.