Characters and composition factor multiplicities for the Lie
superalgebra gl(m/n)
release_kz3k36qs6vbalj5rqsxtm53sxa
by
J. Van der Jeugt, R.B. Zhang
1998
Abstract
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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