{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/26702"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/26702","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Combinatorial Properties of the Hilbert Series of Macdonald Polynomials","abstract":"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>.","abstract_html":"The original Macdonald polynomials P&lt;sub&gt;μ&lt;/sub&gt; 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&lt;sub&gt;μ&lt;/sub&gt; through a combination of algebraic and plethystic transformations one of which is the modified Macdonald polynomial H̃&lt;sub&gt;μ&lt;/sub&gt;. In this dissertation, we study a certain specialization F̃&lt;sub&gt;μ&lt;/sub&gt;(q,t) which is the coefficient of x₁x₂…x&lt;sub&gt;N&lt;/sub&gt; in H̃&lt;sub&gt;μ&lt;/sub&gt; and also the Hilbert series of the Garsia-Haiman module M&lt;sub&gt;μ&lt;/sub&gt;. Haglund found a combinatorial formula expressing F̃&lt;sub&gt;μ&lt;/sub&gt; 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&lt;sub&gt;μ&lt;/sub&gt;. These operations are used to prove a series of recursions and divisibility properties for F̃&lt;sub&gt;μ&lt;/sub&gt;.","abstract_has_math":false,"creators":["Niese, Elizabeth M."],"institution":"Virginia Tech","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Loehr, Nicholas A."],"committee_members":["Haskell, Peter E.","Green, Edward L.","Brown, Ezra A."],"year":2010,"date_issued":"2010-03-30","date_published":"2010-03-30","updated_at":"2026-07-22T22:19:04Z","subjects":["permutation statistics","tableaux","symmetric functions","Macdonald polynomials"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-04082010-090925"],"render_values":[{"text":"etd-04082010-090925","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/26702","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Loehr, Nicholas A."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Haskell, Peter E.","Green, Edward L.","Brown, Ezra A."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Niese, Elizabeth M."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T20:09:07Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T20:09:07Z","2010-04-27"]},{"key":"dc:date.issued","label":"Date","values":["2010-03-30"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. 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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>."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:title","label":"Title","values":["Combinatorial Properties of the Hilbert Series of Macdonald Polynomials"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Loehr, Nicholas A."],"dc:contributor.committeemember":["Haskell, Peter E.","Green, Edward L.","Brown, Ezra A."],"dc:contributor.department":["Mathematics"],"dc:creator":["Niese, Elizabeth M."],"dc:date.accessioned":["2014-03-14T20:09:07Z"],"dc:date.available":["2014-03-14T20:09:07Z","2010-04-27"],"dc:date.issued":["2010-03-30"],"dc:description.abstract":["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>."],"dc:description.degree":["Ph. D."],"dc:identifier.other":["etd-04082010-090925"],"dc:identifier.uri":["http://hdl.handle.net/10919/26702"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["permutation statistics","tableaux","symmetric functions","Macdonald polynomials"],"dc:title":["Combinatorial Properties of the Hilbert Series of Macdonald Polynomials"],"dc:type":["Dissertation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:04Z"}