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 Weyl--Kac 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. For 1-parameter families of weights, we derive combinatorial formulas of so-called Littlewood type for the characters of affine Kac--Moody algebras of typesA^{(2)}_{2n}$ and $\rmC^{(1)}_{n}$. Furthermore we obtain a similar result for $\rmD^{(2)}_{n+1}$, although this relies on an as-yet-unproven case of the key combinatorial q-hypergeometric identity underlying all of our character formulas. Our approach employs the machinery of basic hypergeometric series to construct q-series identities on root systems. Upon specialisation, one side of these identities yields the above-mentioned characters of affine Kac--Moody algebras in their representation provided by the Weyl--Kac formula. The other side, however, leads to combinatorial sums of Littlewood type involving the modified Hall--Littlewood polynomials. These polynomials form an important family of Schur-positive symmetric functions. This thesis is divided into two parts. The first part contains three chapters, each delivering a brief survey of essential classical material. The first of these chapters treats the theory of symmetric functions, with special emphasis on the modified Hall--Littlewood polynomials. The second chapter provides a short introduction to root sytems and the Weyl--Kac formula. The introductory sequence concludes with a chapter on basic hypergeometric series, highlighting the Bailey lemma. 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 Cn root system to derive a Cn 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'l. In order to transform one side of our Cn Andrews transformation into Littlewood-type combinatorial sums, we need to prove a novel q-hypergeometric series identity involving these polynomials. We (partially) achieve this by first proving a new closed-form formula for the Q'l. 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 Cn 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, Gollnitz and Gordon.n