Date of Award

Spring 1-1-2018

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

First Advisor

Nathaniel Thiem

Second Advisor

Richard M. Green

Third Advisor

Martin Walter

Fourth Advisor

Amanda Schaeffer Fry

Fifth Advisor

Farid Aliniaeifard

Abstract

Schur--Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible representations of a group to the irreducible representations of its centralizer algebra. We investigate the analogue of Schur--Weyl duality for the group of unipotent upper triangular matrices over a finite field. In this case, the character theory of these upper triangular matrices is "wild" or unattainable. Thus we employ a generalization, known as supercharacter theory, that creates a striking variation on the character theory of the symmetric group with combinatorics built from set partitions. In this thesis, we present a combinatorial formula for calculating a restriction and induction of supercharacters based on statistics of set partitions and seashell inspired diagrams. We use these formulas to create a graph that encodes the decomposition of a tensor space, and develop an analogue of Young tableaux, known as shell tableaux, to index paths in this graph. These paths also help determine a basis for the maps that centralize the action of the group of unipotent upper triangular matrices. We construct a part of this basis by determining copies of certain modules inside a tensor space to construct projection maps onto supermodules that act on a standard basis.

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