Algebraic Relations Via a Monte Carlo Simulation

TL;DR

This paper investigates the algebraic relations among invariants of complex orthogonal group actions on matrices, introducing Monte Carlo methods to efficiently compute relations and analyze their growth with matrix size.

Contribution

It presents two methods for finding relations among invariants: a classical Young symmetrizer approach and a novel Monte Carlo simulation technique for larger matrices.

Findings

Relation space dimension grows linearly with n for degree n+1 invariants.

Monte Carlo method provides a computationally efficient way to find relations.

The classical approach becomes infeasible as n increases.

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 invariant polynomials. 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. First, we construct a basis using an idempotent of the group algebra referred to as Young symmetrizers, but this quickly becomes computationally expensive as $n$ increases. Thus, we propose a more computationally efficient method for this problem by repeatedly generating random matrices using a Monte Carlo algorithm.