Dissertation/Thesis Abstract

A combinatorial approach to the q; t-symmetry in Macdonald polynomials
by Gillespie, Maria Monks, Ph.D., University of California, Berkeley, 2016, 94; 10150833
Abstract (Summary)

Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation μ*(x; q,t) = μ(x; t,q). We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q = 0) when mu is a partition with at most three rows, and for the coefficients of the square-free monomials in X={x_1,x_2,...} for all shapes mu. We also provide a proof for the full relation in the case when mu is a hook shape, and for all shapes at the specialization t = 1. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.

Indexing (document details)
Advisor: Haiman, Mark D.
Commitee: Papadimitriou, Christos, Williams, Lauren
School: University of California, Berkeley
Department: Mathematics
School Location: United States -- California
Source: DAI-B 77/12(E), Dissertation Abstracts International
Source Type: DISSERTATION
Subjects: Mathematics
Keywords: Kostka polynomials, Macdonald polynomials, Mahonian statistics, Q-analogs, Symmetric functions, Young tableaux
Publication Number: 10150833
ISBN: 9781369055986
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