Modified Hall-Littlewood polynomials and characters of affine Lie algebras

Abstract

In the late 60's and early 70's V. Kac and R. Moody developed a theory of generalised Lie algebras which now bears their name. As part of this theory, Kac gave a beautiful generalisation of the famous Weyl character formula for the characters of integrable highest weight modules, raising the classical result to the level of Kac-Moody algebras. The WeylKac character formula, as it is now known, is a powerful statement that preserves all of the desireable properties of Weyl's formula. However, there is one drawback that also remains. Kac's result formulates the characters of Kac-Moody algebras as an alternating sum over the Weyl group of the underlying affine root system. This inclusion-exclusion type representation obscures the natural positivity of these characters.The purpose of this thesis is to provide manifestly positive (that is, combinatorial) representations for the characters of affine Kac-Moody algebras. In our pursuit of this task, we have been partially successful. All of our original work towards Littlewood-type character formulas is contained in Part II. This work is broken down into four chapters.In the first chapter, we use Milne and Lilly's Bailey lemma for the C n root system to derive a C n analogue of Andrews' celebrated q-series transformation. It is from this transformation that we will ultimately extract our character formulas.In the second chapter we develop a substantial amount of new material for the modified Hall-Littlewood polynomials Q λ . In order to transform one side of our C n Andrews transformation into Littlewood-type combinatorial sums, we need to prove a novel qhypergeometric series identity involving these polynomials. We (partially) achieve this by i first proving a new closed-form formula for the Q λ . For this proof in turn we rely heavily on earlier work by Jing and Garsia.The highlight of our work is the third chapter, where we bring together all of our prior results to prove our new combinatorial character formulas. The most interesting part of the calculations carried out in this section is a bilateralisation procedure which transforms unilateral basic hypergeometric series on C n into bilateral series which exhibit the full affine Weyl group symmetry of the Weyl-Kac character formula.The fourth and final chapter explores specialisations of our character formulas, resulting in many generalisations of Macdonald's classical eta-function identities. Some of our formulas also generalise famous identities from partition theory due to Andrews, Bressoud, Göllnitz and Gordon.ii

Hypergeometric and basic hypergeometric series and integrals associated with root systems

“…This identity was utilized in [4] and in [5] to obtain identities for characters of affine Lie algebras.…”

“…Basic hypergeometric integrals associated to root systems were used in the construction of BC n orthogonal polynomials and BC n biorthogonal rational functions that generalize the Macdonald polynomials, see [62,98]. Watson transformations (and related transformations) associated to root systems were used in [4,5,12,30,136] to derive multiple Rogers-Ramanujan identities and characters for affine Lie algebras. For applications to quantum groups, see [109].…”