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

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

1998-11-04

arxiv.org/abs/math/9811015v1

Mathematics

Representation Theory

LaTeX2e, 15 pages. Submitted to Lett. Math. Phys

Scientific paper

The multiplicities a_{lambda,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 a_{lambda,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 a_{lambda,mu} for any given weight mu. In particular, we claim that for an r-fold atypical weight mu there are 2^r distinct weights lambda such that a_{lambda,mu}=1, and a_{lambda,mu}=0 for all other weights lambda. Some related properties on the multiplicities a_{lambda,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.

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