q-Identities and affinized projective varieties, II. Flag varieties

TL;DR

This paper extends the concept of affinized projective varieties to flag varieties, computing their Hilbert series and proposing a conjecture linking these series to Hall-Littlewood polynomials, with implications for affine Lie algebra representations.

Contribution

It introduces methods to compute Hilbert series for affinized flag varieties and conjectures a novel connection to Hall-Littlewood polynomials.

Findings

Hilbert series for affinized flag varieties computed

Conjectured correspondence with modified Hall-Littlewood polynomials

Potential applications to affine Lie algebra module characters

Abstract

In a previous paper we defined the concept of an affinized projective variety and its associated Hilbert series. We computed the Hilbert series for varieties associated to quadratic monomial ideals. In this paper we show how to apply these results to affinized flag varieties. We discuss various examples and conjecture a correspondence between the Hilbert series of an affinized flag variety and a modified Hall-Littlewood polynomial. We briefly discuss the application of these results to quasi-particle character formulas for affine Lie algebra modules.