Hall-Littlewood polynomials and characters of affine Lie algebras

TL;DR

This paper employs Hall-Littlewood polynomials to derive combinatorial formulas for affine Lie algebra characters, generalizing Macdonald identities and connecting to classical q-series identities like Rogers-Ramanujan.

Contribution

It introduces a new combinatorial approach using Hall-Littlewood polynomials to express affine Lie algebra characters, extending Macdonald identities to broader cases.

Findings

Derived combinatorial formulas for affine Lie algebra characters.

Generalized Macdonald identities for multiple affine types.

Connected identities to classical q-series like Rogers-Ramanujan.

Abstract

The Weyl-Kac character formula gives a beautiful closed-form expression for the characters of integrable highest-weight modules of Kac-Moody algebras. It is not, however, a formula that is combinatorial in nature, obscuring positivity. In this paper we show that the theory of Hall-Littlewood polynomials may be employed to prove Littlewood-type combinatorial formulas for the characters of certain highest weight modules of the affine Lie algebras C_n^{(1)}, A_{2n}^{(2)} and D_{n+1}^{(2)}. Through specialisation this yields generalisations for B_n^{(1)}, C_n^{(1)}, A_{2n-1}^{(2)}, A_{2n}^{(2)} and D_{n+1}^{(2)} of Macdonald's identities for powers of the Dedekind eta-function. These generalised eta-function identities include the Rogers-Ramanujan, Andrews-Gordon and G\"ollnitz-Gordon q-series as special, low-rank cases.