Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra q(n)

TL;DR

This paper introduces a novel approach to computing characters of finite and infinite dimensional irreducible representations of the Lie superalgebra q(n) by linking it to canonical bases of a quantum algebra, and proposes a new conjecture.

Contribution

It relates the character problem for q(n) to canonical bases of quantum groups and formulates a conjecture for infinite dimensional cases in an extended category O.

Findings

Established a new connection between character formulas and canonical bases.

Proposed a conjecture for characters of infinite dimensional irreducible representations.

Provided an alternative method to previous character computations.

Abstract

The problem of computing the characters of the finite dimensional irreducible representations of the Lie superalgebra $\mathfrak q(n)$ over $\C$ was solved in 1996 by I. Penkov and V. Serganova. In this article, we give a different approach relating the character problem to canonical bases of the quantized enveloping algebra $U_q(\mathfrak b_{\infty})$. We also formulate for the first time a conjecture for the characters of the infinite dimensional irreducible representations in the analogue of category $\mathcal O$ for the Lie superalgebra $\mathfrak{q}(n)$.