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Vanishing integrals for Hall-Littlewood polynomials

Citation

Venkateswaran, Vidya (2012) Vanishing integrals for Hall-Littlewood polynomials. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:06082012-201004802

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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Representation Theory, Combinatorics, Special Functions.
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2012
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Rains, Eric M.
Thesis Committee:
  • Ramakrishnan, Dinakar
  • Rains, Eric M. (chair)
  • Wales, David B.
  • Vazirani, Monica
Defense Date:25 May 2012
Record Number:CaltechTHESIS:06082012-201004802
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:06082012-201004802
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7153
Collection:CaltechTHESIS
Deposited By: Vidya Venkateswaran
Deposited On:26 Jun 2012 21:46
Last Modified:26 Dec 2012 04:44

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