Date of Award

1-1-2021

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

College/School/Department

Department of Mathematics and Statistics

Content Description

1 online resource (vi, 115 pages) : illustrations (some color)

Dissertation/Thesis Chair

Cristian Lenart

Committee Members

Antun Milas, Alexandre Tchernev, Changlong Zhong

Keywords

alcove model, charge statistic, Kashiwara-Nakashima column, Kirillov-Reshetikhin crystal, Kostant partition, Kostka-Foulkes polynomial, Representations of Lie algebras, Lie algebras, Polynomials, Combinatorial analysis, Crystals

Subject Categories

Mathematics

Abstract

Part I: Koskta-Foulkes polynomials are Lusztig's q-analogues of weight multiplicities for irreducible representations of semisimple Lie algebras. It has long been known that these polynomials can be written with all non-negative coefficients. A statistic on semistandard Young tableaux with partition content, called \textit{charge}, was used to give a combinatorial formula exhibiting this fact in type $A$. Defining a charge statistic beyond type $A$ has been a long-standing problem. In the first part of this thesis, we take a completely new approach based on the definition of Kostka-Foulkes polynomials as an alternating sum over Kostant partitions, which can be thought of as formal sums of positive roots. We use the structure of crystals $B(\infty)$ and $B^*(\infty)$, which are certain infinite dimensional crystals of highest weight, to construct a collection of modified crystal operators on these Kostant partitions. We then provide a matching on the resulting crystal graph that corresponds to a sign-reversing involution which cancels out all of the negative terms in the Kostka-Foulkes polynomial. This positive expansion in terms of Kostant partitions gives way to a statistic which is simply read by counting the number of parts in the Kostant partitions. The hope is that the simplicity of this new crystal-like model will naturally extend to other classical types.

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