Citation
Venkateswaran, Vidya (2012) Vanishing integrals for HallLittlewood polynomials. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:06082012201004802
Abstract
It is wellknown 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 HallLittlewood level), these approaches do not directly work; this obstruction was the motivation for this thesis. We investigate three related projects in chapters 24 (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 finitedimensional analog of a recent result of Warnaar, involving the RogersSzego polynomials. In the third chapter, we provide a new construction for Koornwinder polynomials at q=0, allowing these polynomials to be viewed as HallLittlewood 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 HallLittlewood polynomials using padic representation theory; it is an analog of the Schur case. This padic approach allows us to generalize our original vanishing results. In particular, we exhibit a tanalog of a classical vanishing result for Schur functions due to Littlewood and Weyl; our vanishing condition is in terms of Hall polynomials and LittlewoodRichardson 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): 

Thesis Committee: 

Defense Date:  25 May 2012 
Record Number:  CaltechTHESIS:06082012201004802 
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:06082012201004802 
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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