#### 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.

#### Recommended Citation

Ly, Megan Danielle, "Schur--Weyl Duality for Unipotent Upper Triangular Matrices" (2018). *Mathematics Graduate Theses & Dissertations*. 59.

https://scholar.colorado.edu/math_gradetds/59