Hall–Littlewood polynomials and characters of affine Lie algebras

Abstract

Abstract. The Weyl-Kac character formula gives a beautiful closed-form expression for the characters of integrable highest-weight modules of KacMoody 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 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.

A framework of Rogers–Ramanujan identities and their arithmetic properties

“…The approach of [5] does not in any obvious manner extend to A (1) n for all n, and this paper aims to give a more complete answer. By using a level-m Rogers-Selberg identity for the root system C n as recently obtain by Bartlett and the third author [8], we show that the Rogers-Ramanujan and Andrews-Gordon identities are special cases of a doubly-infinite family of q-identities arising from the Kac-Moody algebra A (2) 2n for arbitrary n. In their most compact form, the "sum-sides" are expressed in terms of Hall-Littlewood polynomials P λ (x; q) evaluated at infinite geometric progressions (see Section 2 for definitions and further details), and the "product-sides" are essentially products of modular theta functions. We shall present four pairs (a, b) such that for all m, n ≥ 1 we have an identity of the form λ λ 1 ≤m q a|λ| P 2λ (1, q, q 2 , .…”

“…It is indeed possible to extend some of our results to also include B (1) n and A (2) 2n−1 . However, the results of [8]-which are essential in the proofs of Theorems 1.1-1.3-are not strong enough to also deal with these two Kac-Moody algebras. In [66] Eric Rains and the third author present an alternative method for expressing characters of affine Lie algebras in terms of Hall-Littlewood polynomials to that of [8].…”

“…However, the results of [8]-which are essential in the proofs of Theorems 1.1-1.3-are not strong enough to also deal with these two Kac-Moody algebras. In [66] Eric Rains and the third author present an alternative method for expressing characters of affine Lie algebras in terms of Hall-Littlewood polynomials to that of [8]. Their method employs what are known as virtual Koornwinder integrals [64,65] instead of the C n Bailey lemma used in [8].…”

Exponents Associated with Y-Systems and their Relationship with q-Series

“…This type of q-series has been studied in various contexts, such as characters of conformal field theories, Rogers-Ramanujan type identities, algebraic K-theory, and 3-dimensional topology [1,2,3,8,9,10,19,21,23,27,28,32,33,34].…”

“…{(1, 0),(1,1),(1,2)}, e 2 = {(2, 0), (2, 1), (2, 2)}.The mutation network of γ is then given by Let γ be a mutation loop and N γ be its mutation network. For any given pair consisting of a black vertex labeled with e ∈ E and a square vertex labeled with t ∈ ∆, let N et 0 be the number of broken lines between e and t, N + et be the number of arrows from t to e, and N − et be the number of arrows from e to t:…”