Date of Award

Summer 2024

Language

English

Embargo Period

7-25-2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

College/School/Department

Department of Mathematics and Statistics

Program

Mathematics

First Advisor

Cristian Lenart

Committee Members

Antus Milas, Alexandre Tchernev, Changlong Zhong

Keywords

affine crystals, alcove model, combinatorial representation theory

Subject Categories

Discrete Mathematics and Combinatorics

Abstract

The symmetric and non-symmetric Macdonald polynomials are special families of orthogonal polynomials with parameters q and t. They are indexed by dominant, (resp. arbitrary) weights associated to a root system and generalize several well-known polynomials such as the Schur polynomials, Jack polynomials, Hall-Littlewood polynomials, etc. There are two well-known combinatorial models for computing these polynomials: a tableau model in type A, due to Haglund, Haiman and Loehr, and a type-independent model due to Ram and Yip, based on alcove walks.

Crystals bases are an important construction encoding information about Lie algebra representations. It turns out that there is an interesting connection between crystals and Macdonald polynomials. Certain affine crystals, known as Kirillov-Reshetikhin (KR) crystals, were realized in a uniform way (for all untwisted affine types) in terms of the quantum alcove model. Their graded characters were shown to coincide with the special- ization of symmetric Macdonald polynomials at t = 0. There are analogous results that the non-symmetric Macdonald polynomials at t = 0 coincide with the graded character of Demazure-type subcrystals of tensor products of single-column KR crystals. As discussed in the mentioned paper, the former can be thought of as a Demazure submodule of a certain quotient of a level zero extremal weight module. The corresponding crystal is called a DARK crystal (Kirillov-Reshetikhin Affine Demazure). Thus, we expect a combinatorial model for the mentioned DARK crystal based on the alcove walks in the non-symmetric Ram-Yip formula specialized at t = 0. This is the first main result of the paper, and is based on a “non-symmetric version” of the quantum alcove model.

On another hand, in type A, non-symmetric Macdonald polynomials at t = 0 are expressed in terms of semistandard key tabloids. These objects were given an affine crystal structure, which was shown to realize the crystals of certain level one Demazure modules for the affine Lie algebra of sln. The second main result of our paper is the construction of an affine crystal isomorphism between the non-symmetric quantum alcove model in type A and the corresponding semistandard key tabloid model.

License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

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