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 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 Göllnitz-Gordon q-series as special, low-rank cases.

33 pages, proofs of several conjectures from the earlier version have been included

References in corpus (1)

• A framework of Rogers-Ramanujan identities and their arithmetic properties

Cited by in corpus (8)

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• A Nekrasov-Okounkov Type formula for $\tilde{C}$
• The A_{2n}^{(2)} Rogers-Ramanujan identities
• Exponents Associated with $Y$-Systems and their Relationship with $q$-Series
• Affine Pieri rule for periodic Macdonald spherical functions and fusion rings