Hall–Littlewood polynomials and characters of affine Lie algebras

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 Cn(1), A2n(2) and Dn+1(2). Through specialisation this yields generalisations for Bn(1), Cn(1), A2n−1(2), A2n(2) and Dn+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 Göllnitz–Gordon q-series as special, low-rank cases.