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Algebraic Relations Via a Monte Carlo Simulation

Ms. Alison Becker
University of Wisconsin-Milwaukee
PhD Graduate Student – Fellow

“The conjugation action of the complex orthogonal group on the polynomial functions on n×n matrices gives rise to a graded algebra of invariants, P(Mn)^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 C[Sn] referred to as Young symmetrizers, and we propose a more computationally efficient method for this problem using a Monte Carlo algorithm.”

Committee Members:
Profs. Jeb Willenbring (Advisor); Allen Bell, Craig Guilbault, Gabriella Pinter, & Yi Ming Zou