Abacus-Tournament Models of Hall-Littlewood Polynomials

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dc.contributor.authorWills, Andrew Johanen
dc.contributor.committeechairLoehr, Nicholas A.en
dc.contributor.committeememberLinnell, Peter A.en
dc.contributor.committeememberFloyd, William J.en
dc.contributor.committeememberBrown, Ezra A.en
dc.contributor.departmentMathematicsen
dc.date.accessioned2016-01-09T09:01:14Zen
dc.date.available2016-01-09T09:01:14Zen
dc.date.issued2016-01-08en
dc.description.abstractIn 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.en
dc.description.degreePh. D.en
dc.format.mediumETDen
dc.identifier.othervt_gsexam:7062en
dc.identifier.urihttp://hdl.handle.net/10919/64427en
dc.publisherVirginia Techen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectSymmetric polynomialsen
dc.subjectHall-Littlewood polynomialsen
dc.subjectabacus-tournamentsen
dc.subjectPieri rulesen
dc.titleAbacus-Tournament Models of Hall-Littlewood 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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