Title
An Overview of the Geometry and Combinatorics of the Macdonald Polynomial and q-t Catalan Number
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.
Recommended Citation
Liang, Tian, "An Overview of the Geometry and Combinatorics of the Macdonald Polynomial and q-t Catalan Number" (2010). Publicly Accessible Penn Dissertations. 114.
https://repository.upenn.edu/edissertations/114