Date of Award
August 2020
Degree Type
Dissertation
Degree Name
Doctor of Philosophy
Department
Mathematics
First Advisor
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.
Recommended Citation
Becker, Alison Elaine, "Algebraic Relations Via a Monte Carlo Simulation" (2020). Theses and Dissertations. 2455.
https://dc.uwm.edu/etd/2455