Combinatorial Properties of the Hilbert Series of Macdonald Polynomials

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dc.contributor.authorNiese, Elizabeth M.en
dc.contributor.committeechairLoehr, Nicholas A.en
dc.contributor.committeememberHaskell, Peter E.en
dc.contributor.committeememberGreen, Edward L.en
dc.contributor.committeememberBrown, Ezra A.en
dc.contributor.departmentMathematicsen
dc.date.accessioned2014-03-14T20:09:07Zen
dc.date.adate2010-04-27en
dc.date.available2014-03-14T20:09:07Zen
dc.date.issued2010-03-30en
dc.date.rdate2010-04-27en
dc.date.sdate2010-04-08en
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>.en
dc.description.degreePh. D.en
dc.identifier.otheretd-04082010-090925en
dc.identifier.sourceurlhttp://scholar.lib.vt.edu/theses/available/etd-04082010-090925/en
dc.identifier.urihttp://hdl.handle.net/10919/26702en
dc.publisherVirginia Techen
dc.relation.haspartNiese_EM_D_2010.pdfen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectpermutation statisticsen
dc.subjecttableauxen
dc.subjectsymmetric functionsen
dc.subjectMacdonald polynomialsen
dc.titleCombinatorial Properties of the Hilbert Series of Macdonald Polynomialsen
dc.typeDissertationen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.leveldoctoralen
thesis.degree.namePh. D.en

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