Vanishing Results for Hall-Littlewood Polynomials

Abstract

It is well-known that if one integrates a Schur function indexed by a partition λ over the symplectic (resp. orthogonal) group, the integral vanishes unless all parts of λ have even multiplicity (resp. all parts of λ are even). In a recent work of Rains and Vazirani, Macdonald polynomial generalizations of these identities and several others were developed and proved using Hecke algebra techniques. However at q=0 (the Hall-Littlewood level), these approaches do not directly work; this obstruction was the motivation for this thesis. We investigate three related projects in chapters 2-4 (the first chapter consists of an introduction to the thesis). In the second chapter, we develop a combinatorial technique for proving the results of Rains and Vazirani at q=0. This approach allows us to generalize some of those results in interesting ways and leads us to a finite-dimensional analog of a recent result of Warnaar, involving the Rogers-Szego polynomials. In the third chapter, we provide a new construction for Koornwinder polynomials at q=0, allowing these polynomials to be viewed as Hall-Littlewood polynomials of type BC. This is a first step in building the analogy between the Macdonald and Koornwinder families at the q=0 limit. We use this construction in conjunction with the combinatorial technique of the previous chapter to prove some vanishing results of Rains and Vazirani for Koornwinder polynomials at q=0. In the fourth chapter, we provide an interpretation for vanishing results for Hall-Littlewood polynomials using p-adic representation theory; it is an analog of the Schur case. This p-adic approach allows us to generalize our original vanishing results. In particular, we exhibit a t-analog of a classical vanishing result for Schur functions due to Littlewood and Weyl; our vanishing condition is in terms of Hall polynomials and Littlewood-Richardson coefficients.