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

Marty Walter

Fourth Advisor

Katherine Stange

Fifth Advisor

James Wilson

Abstract

The character theory for semisimple Hopf algebras with a commutative representation ring has many similarities to the character theory of finite groups. We extend the notion of supercharacter theory to this context, and define a corresponding algebraic object that generalizes the Schur rings of the group algebra of a finite group. We show the existence of Hopf-algebraic analogues for the most common supercharacter theory constructions, specifically the wedge product and supercharacter theories arising from the action of a finite group. In regards to the action of the Galois group of the field generated by the entries of the character table, we show the existence of a unique finest supercharacter theory with integer entries, and describe the superclasses for abelian groups and the family GL2(q).

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