The original Macdonald polynomials P<sub>μ</sub> 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<sub>μ</sub> through a combination of algebraic and plethystic transformations one of which is the modified Macdonald polynomial H̃<sub>μ</sub>. In this dissertation, we study a certain specialization F̃<sub>μ</sub>(q,t) which is the coefficient of x₁x₂…x<sub>N</sub> in H̃<sub>μ</sub> and also the Hilbert series of the Garsia-Haiman module M<sub>μ</sub>. Haglund found a combinatorial formula expressing F̃<sub>μ</sub> 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 F̃<sub>μ</sub>. These operations are used to prove a series of recursions and divisibility properties for F̃<sub>μ</sub>. / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/26702 |
| Date | 27 April 2010 |
| Creators | Niese, Elizabeth M. |
| Contributors | Mathematics, Loehr, Nicholas A., Haskell, Peter E., Green, Edward L., Brown, Ezra A. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Dissertation |
| Format | application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | Niese_EM_D_2010.pdf |
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