First Advisor

John Caughman

Term of Graduation

Spring 2024

Date of Publication

5-23-2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.) in Mathematical Sciences

Department

Mathematics and Statistics

Language

English

DOI

10.15760/etd.3764

Physical Description

1 online resource (vii, 92 pages)

Abstract

The notion of a Leonard pair was introduced by Terwilliger in 2001 to simplify Leonard's theorem, which classifies the orthogonal polynomials in the terminating branch of the Askey-Wilson scheme. In the same year, Kresch and Tamvakis made a conjecture about a certain 4F3 hypergeometric series while studying the arithmetic analogues of the standard conjectures for the Grassmanian G(2,n). The 4F3 series appearing in their conjecture is closely related to a family of orthogonal polynomials in the Askey-Wilson scheme. Consequently, the theory of Leonard pairs provides a useful framework for understanding their conjecture.

In this dissertation, we present our proof of the Kresch-Tamvakis conjecture. To do so, we construct a specific Leonard pair A, A* and a related sequence of matrices Bi. We identify the hypergeometric series in question with the eigenvalues of these matrices. We then use a result from mathematical physics known as the Biedenharn-Elliot identity to prove that the entries of the Bi are nonnegative, and, from this, we obtain the conjectured bound from the Perron-Frobenius theorem.

The Leonard pair studied here has many special properties related to spin models and strongly regular graphs. We formulate a number of results exploring these connections, and we prove a generalization that holds for a larger family of Leonard pairs.

Rights

© 2024 Taiyo Summers Terada

In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).

Persistent Identifier

https://archives.pdx.edu/ds/psu/42251

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