## Theses and Dissertations

August 2020

Dissertation

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

Jeb F Willenbring

#### Committee Members

Allen Bell, Yi Ming Zou, Craig Guilbault, Gabriella Pinter

#### Keywords

Invariant theory, Monte Carlo, Representation theory

#### Abstract

The conjugation action of the complex orthogonal group on the polynomial functions on $n \times n$ matrices gives rise to a graded algebra of invariants, $\mathcal{P}(M_n)^{O_n}$. A spanning set of this algebra is in bijective correspondence to a set of unlabeled, cyclic graphs with directed edges equivalent under dihedral symmetries. When the degree of the invariants is $n+1$, we show that the dimension of the space of relations between the invariants grows linearly in $n$. Furthermore, we present two methods to obtain a basis of the space of relations; we construct a basis using an idempotent of the group algebra $\mathbb{C}[S_n]$ referred to as Young symmetrizers, and we propose a more computationally efficient method for this problem using a Monte Carlo algorithm.

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