Irreducible Characters of General Linear Superalgebra and Super Duality

TL;DR

This paper introduces a new method linking the irreducible character problem for general linear superalgebras to classical Kazhdan-Lusztig theory, confirming super duality and a conjecture by Brundan.

Contribution

It develops a novel approach to compute irreducible characters for a broad class of modules over general linear superalgebras, including finite-dimensional cases, and verifies key conjectures.

Findings

Established a direct relation to Kazhdan-Lusztig theory.

Verified a parabolic version of Brundan's conjecture.

Proved the super duality conjecture.

Abstract

We develop a new method to solve the irreducible character problem for a wide class of modules over the general linear superalgebra, including all the finite-dimensional modules, by directly relating the problem to the classical Kazhdan-Lusztig theory. We further verify a parabolic version of a conjecture of Brundan on the irreducible characters in the BGG category $\mc{O}$ of the general linear superalgebra. We also prove the super duality conjecture.