{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/64427"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/64427","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Abacus-Tournament Models of Hall-Littlewood Polynomials","abstract":"In this dissertation, we introduce combinatorial interpretations for three types of HallLittlewood polynomials (denoted Rλ, Pλ, and Qλ) by using weighted combinatorial objects called abacus-tournaments. We then apply these models to give combinatorial proofs of properties of Hall-Littlewood polynomials. For example, we show why various specializations of Hall-Littlewood polynomials produce the Schur symmetric polynomials, the elementary symmetric polynomials, or the t-analogue of factorials. With the abacus-tournament model, we give a bijective proof of a Pieri rule for Hall-Littlewood polynomials that gives the Pλ-expansion of the product of a Hall-Littlewood polynomial Pµ with an elementary symmetric polynomial ek. We also give a bijective proof of certain cases of a second Pieri rule that gives the Pλ-expansion of the product of a Hall-Littlewood polynomial Pµ with another Hall-Littlewood polynomial Q(r) . In general, proofs using abacus-tournaments focus on canceling abacus-tournaments and then finding weight-preserving bijections between the sets of uncanceled abacus-tournaments.","abstract_html":"In this dissertation, we introduce combinatorial interpretations for three types of HallLittlewood polynomials (denoted Rλ, Pλ, and Qλ) by using weighted combinatorial objects called abacus-tournaments. We then apply these models to give combinatorial proofs of properties of Hall-Littlewood polynomials. For example, we show why various specializations of Hall-Littlewood polynomials produce the Schur symmetric polynomials, the elementary symmetric polynomials, or the t-analogue of factorials. With the abacus-tournament model, we give a bijective proof of a Pieri rule for Hall-Littlewood polynomials that gives the Pλ-expansion of the product of a Hall-Littlewood polynomial Pµ with an elementary symmetric polynomial ek. We also give a bijective proof of certain cases of a second Pieri rule that gives the Pλ-expansion of the product of a Hall-Littlewood polynomial Pµ with another Hall-Littlewood polynomial Q(r) . In general, proofs using abacus-tournaments focus on canceling abacus-tournaments and then finding weight-preserving bijections between the sets of uncanceled abacus-tournaments.","abstract_has_math":false,"creators":["Wills, Andrew Johan"],"institution":"Virginia Tech","degree_name":"Ph. 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We then apply these models to give combinatorial proofs of properties of Hall-Littlewood polynomials. For example, we show why various specializations of Hall-Littlewood polynomials produce the Schur symmetric polynomials, the elementary symmetric polynomials, or the t-analogue of factorials. With the abacus-tournament model, we give a bijective proof of a Pieri rule for Hall-Littlewood polynomials that gives the Pλ-expansion of the product of a Hall-Littlewood polynomial Pµ with an elementary symmetric polynomial ek. We also give a bijective proof of certain cases of a second Pieri rule that gives the Pλ-expansion of the product of a Hall-Littlewood polynomial Pµ with another Hall-Littlewood polynomial Q(r) . In general, proofs using abacus-tournaments focus on canceling abacus-tournaments and then finding weight-preserving bijections between the sets of uncanceled abacus-tournaments."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["ETD"]},{"key":"dc:title","label":"Title","values":["Abacus-Tournament Models of Hall-Littlewood Polynomials"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Loehr, Nicholas A."],"dc:contributor.committeemember":["Linnell, Peter A.","Floyd, William J.","Brown, Ezra A."],"dc:contributor.department":["Mathematics"],"dc:creator":["Wills, Andrew Johan"],"dc:date.accessioned":["2016-01-09T09:01:14Z"],"dc:date.available":["2016-01-09T09:01:14Z"],"dc:date.issued":["2016-01-08"],"dc:description.abstract":["In this dissertation, we introduce combinatorial interpretations for three types of HallLittlewood polynomials (denoted Rλ, Pλ, and Qλ) by using weighted combinatorial objects called abacus-tournaments. We then apply these models to give combinatorial proofs of properties of Hall-Littlewood polynomials. For example, we show why various specializations of Hall-Littlewood polynomials produce the Schur symmetric polynomials, the elementary symmetric polynomials, or the t-analogue of factorials. With the abacus-tournament model, we give a bijective proof of a Pieri rule for Hall-Littlewood polynomials that gives the Pλ-expansion of the product of a Hall-Littlewood polynomial Pµ with an elementary symmetric polynomial ek. We also give a bijective proof of certain cases of a second Pieri rule that gives the Pλ-expansion of the product of a Hall-Littlewood polynomial Pµ with another Hall-Littlewood polynomial Q(r) . In general, proofs using abacus-tournaments focus on canceling abacus-tournaments and then finding weight-preserving bijections between the sets of uncanceled abacus-tournaments."],"dc:description.degree":["Ph. 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