| 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 1x1 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 statistics. |
