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

TL;DR

This paper proposes a simple conjectural rule for determining the composition multiplicities of simple modules within Kac modules for the Lie superalgebra gl(m/n), simplifying the complex existing algorithms and providing insights into their structure.

Contribution

It introduces a conjecture that predicts the nonzero multiplicities for any weight, offering a simpler alternative to Serganova's complex algorithm.

Findings

Conjecture that for an r-fold atypical weight mu, there are 2^r weights lambda with a_{lambda,mu}=1.

Some properties of the multiplicities a_{lambda,mu} are proved.

Discussion of the inverse matrix of Kazhdan-Lusztig polynomials related to the conjecture.

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.