Date of Award

August 2020

Degree Type


Degree Name

Doctor of Philosophy



First Advisor

Jeb F Willenbring

Committee Members

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


Invariant theory, Monte Carlo, Representation theory


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.

Included in

Mathematics Commons