##### 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 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 $\widetilde{F}_\mu$.