Date of Award

Fall 2024

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.

Document Type

Dissertation

First Advisor

Edward Hinson

Second Advisor

Maria Basterra

Third Advisor

Junaho Shen

Department or Program

Mathematics

Degree Name

Doctor of Philosophy

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