Date of Award

1-1-2020

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

College/School/Department

Department of Mathematics and Statistics

Content Description

1 online resource (v, 104 pages) : illustrations.

Dissertation/Thesis Chair

Cristian Lenart

Committee Members

Alexandre Tchernev, Antun Milas, Changlong Zhong

Keywords

Polynomials, Orthogonal polynomials, Coulomb functions, Combinatorial analysis, Lie algebras

Subject Categories

Physical Sciences and Mathematics

Abstract

In this work we study the relationship between several combinatorial formulas for type $A$ Hall-Littlewood polynomials and Whittaker functions. The former are spherical functions on $p$-adic groups, while the latter arise in the theory of automorphic forms. Both depend on a parameter $t$, are specializations of Macdonald polynomials, and specialize to Schur polynomials upon setting $t=0$. There are three types of formulas for these polynomials. The first formula is in terms of so-called alcove walks, works in arbitrary Lie type, and is derived from the Ram-Yip formula for Macdonald polynomials. The second one is in terms of certain fillings of Young diagrams, and is derived from, or is analogous to the Haglund-Haiman-Loehr formula for Macdonald polynomials. The third formula is in terms of the classical semistandard Young tableaux. We study the way in which each such formula is obtained from the previous one by combining terms $-$ a phenomenon called compression. In the case of Hall-Littlewood polynomials, we complete the picture given by existing work, while no such results existed in the case of Whittaker functions.

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