A Combinatorially Explicit Relative Möbius Function on Affine Grassmannians and a Proposal for an Affine Infinite Symmetric Group

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dc.contributor.authorLugo, Michael Rubenen
dc.contributor.committeechairShimozono, Mark M.en
dc.contributor.committeememberOrr, Daniel D.en
dc.contributor.committeememberMihalcea, Constantin Leonardoen
dc.contributor.committeememberLoehr, Nicholas A.en
dc.contributor.departmentMathematicsen
dc.date.accessioned2019-05-10T08:00:41Zen
dc.date.available2019-05-10T08:00:41Zen
dc.date.issued2019-05-09en
dc.description.abstractFor an affine Weyl group W, we explicitly determine the elements for which the Möbius function of the subposet of affine Grassmannians under the Bruhat order is non-zero by utilizing the quantum Bruhat graph of the classical Weyl group associated to W . Then we examine embedding stable and consistent statistics on the affine Weyl group of type A which permit the definition of an affine infinite symmetric group.en
dc.description.abstractgeneralSimilar to the integers, there are groups that have both an infinite number of elements and also a way to partially order those elements. With a partial ordering, we can consider the interval between two elements. When we make a function that sums over an interval of elements, then we can invert the function by using something called the Mӧbius function. For many groups, the Mӧbius function is extremely unpredictable and calculating the inverse may require us to consider an infinite number of elements. In this paper, we focus on groups called affine Weyl groups, which are very useful in algebraic geometry. It turns out that most elements in these groups have a very predictable pattern in their Mӧbius functions which only considers a finite number of elements. The first part of this paper gives very simple rules for calculating it. The second part of this paper focuses on a special type of affine Weyl group: the affine symmetric groups. We provide an attempt at defining a large parent group, which we call the affine infinite symmetric group, that contains all the other affine symmetric groups.en
dc.description.degreeDoctor of Philosophyen
dc.format.mediumETDen
dc.identifier.othervt_gsexam:19683en
dc.identifier.urihttp://hdl.handle.net/10919/89477en
dc.publisherVirginia Techen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectcombinatoricsen
dc.subjectaffine Weyl groupen
dc.subjectaffine Grassmannianen
dc.subjectquantum Bruhat graphen
dc.subjectMöbius functionen
dc.subjectinfinite affine Weyl groupen
dc.titleA Combinatorially Explicit Relative Möbius Function on Affine Grassmannians and a Proposal for an Affine Infinite Symmetric Groupen
dc.typeDissertationen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.leveldoctoralen
thesis.degree.nameDoctor of Philosophyen

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