Type of Document Dissertation Author Niese, Elizabeth M Author's Email Address eniese@vt.edu URN etd-04082010-090925 Title Combinatorial Properties of the Hilbert Series of Macdonald Polynomials Degree PhD Department Mathematics Advisory Committee
Advisor Name Title Loehr, Nicholas A. Committee Chair Brown, Ezra A. Committee Member Green, Edward L. Committee Member Haskell, Peter E. Committee Member Keywords
- permutation statistics
- tableaux
- symmetric functions
- Macdonald polynomials
Date of Defense 2010-03-30 Availability unrestricted Abstract The original Macdonald polynomials $P_\mu$ form a basis for the vector space of symmetricfunctions which specializes to several of the common bases such as the monomial, Schur, and
elementary bases. There are a number of di fferent types of Macdonald polynomials obtained
from the original $P_\mu$ through a combination of algebraic and plethystic transformations one
of which is the modi fied Macdonald polynomial $\widetilde{H}_\mu$. In this dissertation, we study a certain
specialization $\widetilde{F}_\mu(q,t)$ which is the coefficient of $x_1x_2 ... x_N$ in $\widetilde{H}_\mu$ and also the Hilbert series
of the Garsia-Haiman module $M_\mu$. Haglund found a combinatorial formula expressing $\widetilde{F}_\mu$ 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 $\widetilde{F}_\mu$. These operations are used to prove a series of recursions
and divisibility properties for $\widetilde{F}_\mu$.
Files
Filename Size Approximate Download Time (Hours:Minutes:Seconds)
28.8 Modem 56K Modem ISDN (64 Kb) ISDN (128 Kb) Higher-speed Access Niese_EM_D_2010.pdf 422.76 Kb 00:01:57 00:01:00 00:00:52 00:00:26 00:00:02
|
|
If you have questions or technical problems, please Contact DLA.
