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 AbstractThe original Macdonald polynomials $P_\mu$ form a basis for the vector space of symmetricfunctions which specializes to several of the common bases such as the monomial, Schur, and

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

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

of which is the modified 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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