Control over the KKR bijection with respect to the nesting structure on rigged configurations and a CSP instance involving Motzkin numbers

dc.contributor.authorChan, William
dc.date.accessioned2024-08-27T19:29:03Z
dc.date.available2024-08-27T19:29:03Z
dc.date.issued2024-08-27
dc.date.submitted2024-08-22
dc.description.abstractThere are two disjoint main projects that this thesis covers. The Motzkin numbers are a sort of “re- laxed version” of the Catalan numbers. For example, Catalan numbers count perfect non-crossing matchings, while Motzkin numbers count not necessarily perfect non-crossing matchings. The first project deals with instances of the cyclic sieving phenomenon involving Motzkin numbers and their standard q−analogue. We also show that the standard q−analogue for Motzkin numbers satisfy a similar generating series interpretation to that of the q−Catalan numbers. The second project deals with understanding the Kerov-Kirillov-Reshetikhin (KKR) bijection between semistandard tableaux and rigged configurations with a particular emphasis on the standard case. In partic- ular, we understand inducing perturbations on the corresponding rigged configuration via direct operations on the tableau. We develop a technique to take a standard tableau T, and output a new standard tableau T′ which has the same corresponding rigged configuration up to a rigging on any desired row of the first rigged partition. We also develop an alternate technique to Kuniba, Okado, Sakamoto, Takagi, and Yamada that “unwraps” the natural nesting structure on rigged configurations. The primary operation from which all the above follows from is “raise” which we introduce and give various combinatorial models for. The operation raise induces a very simple, controlled perturbation on the corresponding rigged configuration. Our results are formulated in terms of paths or (classically) highest weight elements of tensor products of Kirillov-Reshetikhin crystals.
dc.identifier.urihttps://hdl.handle.net/10012/20885
dc.language.isoen
dc.pendingfalse
dc.publisherUniversity of Waterlooen
dc.subjectrigged configurations
dc.subjectbox-ball system
dc.subjectsoliton decomposition
dc.subjectCatalan numbers
dc.subjectMotzkin numbers
dc.subjectcyclic sieving phenomenon
dc.titleControl over the KKR bijection with respect to the nesting structure on rigged configurations and a CSP instance involving Motzkin numbers
dc.typeMaster Thesis
uws-etd.degreeMaster of Mathematics
uws-etd.degree.departmentCombinatorics and Optimization
uws-etd.degree.disciplineCombinatorics and Optimization
uws-etd.degree.grantorUniversity of Waterlooen
uws-etd.embargo.terms0
uws.comment.hiddenThe thesis contains two projects, but neither of them were previously published. Comments on the revisions: ----------------------------------- There is no preprint available of either. I had a (much smaller) version of Chapter 1 submitted to FPSAC (and nowhere else), but it was not accepted. The work done in Chapter 2 was not submitted anywhere. I deleted the old submission because they have the same name, so they don't get confused. I hope this is not an issue.
uws.contributor.advisorMandelshtam, Olya
uws.contributor.affiliation1Faculty of Mathematics
uws.peerReviewStatusUnrevieweden
uws.published.cityWaterlooen
uws.published.countryCanadaen
uws.published.provinceOntarioen
uws.scholarLevelGraduateen
uws.typeOfResourceTexten

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