Character and dimension formulae for general linear superalgebra

TL;DR

This paper explicitly computes generalized Kazhdan-Lusztig polynomials for finite-dimensional irreducible representations of the general linear superalgebra, establishing a correspondence between composition factors and deriving a closed dimension formula.

Contribution

It provides explicit formulas for Kazhdan-Lusztig polynomials and simplifies a conjectural character formula into a Kac-Weyl form, leading to a dimension formula for irreducible representations.

Findings

Explicit computation of Kazhdan-Lusztig polynomials for gl(m|n)

Establishment of a correspondence between composition factors of different Kac-modules

Derivation of a closed formula for the dimension of irreducible representations

Abstract

The generalized Kazhdan-Lusztig polynomials for the finite dimensional irreducible representations of the general linear superalgebra are computed explicitly. Using the result we establish a one to one correspondence between the set of composition factors of an arbitrary $r$-fold atypical $gl_{m|n}$-Kac-module and the set of composition factors of some $r$-fold atypical $gl_{r|r}$-Kac-module. The result of Kazhdan-Lusztig polynomials is also applied to prove a conjectural character formula put forward by van der Jeugt et al in the late 80s. We simplify this character formula to cast it into the Kac-Weyl form, and derive from it a closed formula for the dimension of any finite dimensional irreducible representation of the general linear superalgebra.