Date of Award

Spring 1-1-2014

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Nathaniel Thiem

Second Advisor

Richard M. Green

Third Advisor

Sebastian Casalaina-Martin

Fourth Advisor

Martin Walter

Fifth Advisor

James Wilson

Abstract

Supercharacter theories are a relatively new tool in studying the representation theory of unipotent groups over finite fields. In this thesis I present two new approaches to constructing supercharacter theories of finite unipotent groups. The first method utilizes group actions to construct supercharacter theories of the unipotent orthogonal, symplectic and unitary groups. The second technique is via the method of little groups, which describes the irreducible characters of a semidirect product with an abelian normal subgroup in terms of the irreducible characters of the factor groups. Motivated by these constructions, I produce supercharacter theories for a large collection of unipotent matrix groups and construct a Hopf monoid on the supercharacters.

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