Combinatorics of Macdonald polynomials and extensions

The theory of symmetric functions is ubiquitous throughout mathematics. They arise naturally in combinatorics, algebra, and geometry, and as a result have been studied intensively for many years. This classical area was revitalized in 1988, with Ian Macdonald's description of what are now known as the Macdonald polynomials. These are a two parameter basis for the space of symmetric functions, which specialize to many of the well-known one parameter and classical bases. Macdonald conjectured that when a certain normalization of these polynomials were expanded in terms of the classical Schur functions, the coefficients would always be polynomials in N[q,t]. He called these coefficients q, t-Kostka functions, and the conjecture became known as the Macdonald positivity conjecture. It was proved in 2001 by Mark Haiman. While attempting to prove the positivity conjecture, Garsia and Haiman conjectured the existence of a larger class of symmetric functions, satisfying certain properties and indexed by finite subsets of N x N (usually thought of as a collection of 1 x 1 squares in the first quadrant of the plane). Computer generated data strongly suggested the existence of these polynomials in general. In 2003, Jim Haglund proposed a purely combinatorial description of the Macdonald polynomials. This description, the generating function for a particular pair of statistics, was soon proved correct by Haiman, Haglund and Nick Loehr. In this dissertation, we show that the statistics of Haglund allow us to construct the polynomials of Garsia and Haiman for a particular class of diagrams; namely, skew shapes with no column of height greater than two. The proof of this fact involves a new and careful analysis of these statistic