Date of Award

Spring 2010

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Mathematics

First Advisor

James Haglund

Abstract

We will explore the combinatorial and geometric properties related to the Macdonald polynomials and the diagonal harmonics. We have the combinatorial Macdonald polynomial formula that ts the de ning conditions directly. The shue conjecture gives an elegant expression of the Frobenius series of the diagonal harmonics. While the geometric properties of the Hilbert scheme and schemes over it provides explanations from a di erent perspective. We use examples to show that these two approaches arrive at the same goal.

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