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Normal Supercharacter Theories

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dc.contributor.advisorBergeron, Nantel
dc.creatorAliniaeifard, Farid
dc.date.accessioned2018-03-01T13:56:07Z
dc.date.available2018-03-01T13:56:07Z
dc.date.copyright2017-06-26
dc.date.issued2018-03-01
dc.date.updated2018-03-01T13:56:07Z
dc.degree.disciplineMathematics & Statistics
dc.degree.levelDoctoral
dc.degree.namePhD - Doctor of Philosophy
dc.description.abstractClassification of irreducible characters of some families of groups, for example, the family of the groups of unipotent upper-triangular matrices, is a "wild" problem. To have a tame and tractable theory for the groups of unipotent-upper triangular matrices Andr and Yan introduced the notion of supercharacter theory. Diaconis and Issacs axiomatized the concept of supercharacter theory for any group. In this thesis, for an arbitrary group G, by using sublattices of the lattice of normal subgroups containing the trivial subgroup and G, we build a family of integral supercharacter theories, called normal supercharacter theories (abbreviated NSCT). We present a recursive formula for supercharacters in an NSCT. The finest NSCT is constructed from the whole lattice of normal subgroups of G, and is a mechanism to study the behavior of conjugacy classes by the lattice of normal subgroups. We will uncover a relation between the finest NSCT, faithful irreducible characters, and primitive central idempotents. We argue that NSCT cannot be obtained by previous known supercharacter theory constructions, but it is related to *-products of some certain supercharacter theories. We also construct an NSCT for the family of groups of unipotent upper-triangular matrices. These groups are crucial to the supercharacter theory. The supercharacters of the resulting NSCT are indexed by Dyck paths, which are combinatorial objects that are central to several areas of algebraic combinatorics. Finally, we show that this supercharacter construction is identical to Scott Andrews' construction after gluing the superclasses and the supercharacters by the action of the torus group.
dc.identifier.urihttp://hdl.handle.net/10315/34307
dc.language.isoen
dc.rightsAuthor owns copyright, except where explicitly noted. Please contact the author directly with licensing requests.
dc.subjectMathematics
dc.subject.keywordsCharacter Theory
dc.subject.keywordsSupercharacter Theory
dc.subject.keywordsLattice Theory
dc.subject.keywordsNormal Subgroups
dc.subject.keywordsGroup Theory
dc.subject.keywordsLattice of Normal Subgroups
dc.subject.keywordsPrimitive Central Idempotents
dc.titleNormal Supercharacter Theories
dc.typeElectronic Thesis or Dissertation

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