Virginia Tech
Combinatorial Properties of the Hilbert Series of Macdonald Polynomials
Abstract
dc:description.abstractThe 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>.
Degree
thesis:*- Name thesis:degree_name
- Ph. D.
- Level thesis:degree_level
- doctoral
- Discipline thesis:degree_discipline
- Mathematics
- Department dc:contributor.department
- Mathematics
- Grantor dc:publisher
- Virginia Tech
- Year dc:date.issued
- 2010
Author and committee
dc:creator, dc:contributor.*- Author dc:creator
-
- Niese, Elizabeth M.
- Chair dc:contributor.committeechair
-
- Loehr, Nicholas A.
- Committee members dc:contributor.committeemember
-
- Haskell, Peter E.
- Green, Edward L.
- Brown, Ezra A.
Subjects
dc:subject × 4Rights
dc:rights- Statement dc:rights
-
- In Copyright
- Licence dc:rights.uri
Identifiers
dc:identifier.*- Dc Identifier Other
- etd-04082010-090925
- OAI identifier oai:identifier
- oai:vtechworks.lib.vt.edu:10919/26702