### Title page for ETD etd-04082010-090925

Type of Document Dissertation
Author Niese, Elizabeth M
Author's Email Address eniese@vt.edu
URN etd-04082010-090925
Title Combinatorial Properties of the Hilbert Series of Macdonald Polynomials
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Loehr, Nicholas A. Committee Chair
Brown, Ezra A. Committee Member
Green, Edward L. Committee Member
Haskell, Peter E. Committee Member
Keywords
• permutation statistics
• tableaux
• symmetric functions
• Macdonald polynomials
Date of Defense 2010-03-30
Availability unrestricted
Abstract
The original Macdonald polynomials $P_\mu$ form a basis for the vector space of symmetric

functions which specializes to several of the common bases such as the monomial, Schur, and

elementary bases. There are a number of di fferent types of Macdonald polynomials obtained

from the original $P_\mu$ through a combination of algebraic and plethystic transformations one

of which is the modi fied Macdonald polynomial $\widetilde{H}_\mu$. In this dissertation, we study a certain

specialization $\widetilde{F}_\mu(q,t)$ which is the coefficient of $x_1x_2 ... x_N$ in $\widetilde{H}_\mu$ and also the Hilbert series

of the Garsia-Haiman module $M_\mu$. Haglund found a combinatorial formula expressing $\widetilde{F}_\mu$ as

a sum of $n!$ objects weighted by two statistics. Using this formula we prove a $q,t$-analogue of

the hook-length formula for hook shapes. We establish several new combinatorial operations

on the fillings which generate $\widetilde{F}_\mu$. These operations are used to prove a series of recursions

and divisibility properties for $\widetilde{F}_\mu$.

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