Date of Award

Fall 2024

Project Type

Dissertation

Program or Major

Mathematics

Degree Name

Doctor of Philosophy

First Advisor

Edward Hinson

Second Advisor

Maria Basterra

Third Advisor

Junaho Shen

Abstract

A doubly stochastic matrix over the real numbers is a matrix $A=[a_{ij}]$ such that $0\leq a_{ij}\leq1$ for all $i,j$ and all rows and columns sum to 1. The classical Birkhoff-von Neumann theorem states that every doubly stochastic matrix over $\R$ can be expressed as an affine combination of permutation matrices, matrices with a single 1 in each row and column and 0's elsewhere. Over a general commutative ring with identity $R$ we drop the condition that $0\leq a_{ij}\leq1$ and consider matrices with only the row and column sum condition; we refer to such matrices as \textit{algebraically doubly stochastic} and denote the set of these matrices by $\Estar$. We begin by showing that the set of invertible $\Estar$ matrices, denoted $\EstarGL$, forms a subgroup of $GL(n,R)$ and explore some basic properties of this group, along with two closely related groups $\ErhoGL$ and $\EchiGL$. Next we generalize the Birkhoff-von Neumann theorem to $\Estar$ matrices over arbitrary commutative rings in dimensions 2, 3, and 4, and to principal ideal domains for all dimensions $n$. Furthermore, we use this Birkhoff-style representation of $\Estar$ matrices to introduce a monoid structure on certain subsets of $Um(n,R)$, the set of unimodular vectors over a commutative ring with identity. Finally, we investigate $\EchiGL$ and $\EstarGL$-orbits in $Um(n,R)$, and pose some questions for potential work going forward.

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