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Algebraic Relations Via a Monte Carlo Simulation

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.