Orthogonal invariants of skew-symmetric matrices

TL;DR

This paper investigates the algebraic structure of invariants of multiple skew-symmetric matrices under orthogonal transformations, providing minimal generators and parameters for specific matrix sizes and counts over fields with characteristic not two.

Contribution

It establishes minimal generating sets and homogeneous systems of parameters for invariants of skew-symmetric matrices under orthogonal group actions for certain dimensions and numbers of matrices.

Findings

Minimal generators for n=3, d>0

Homogeneous systems of parameters for n=3,4,5 with specified d

Results over fields with characteristic not two

Abstract

The algebra of invariants of d-tuples of n x n skew-symmetric matrices under the action of the orthogonal group by simultaneous conjugation is considered over an infinite field of characteristic different from two. For n=3 and d>0 a minimal set of generators is established. A homogeneous system of parameters (i.e., an algebraically independent set such that the algebra of invariants is a finitely generated free module over subalgebra generated by this set) is described for n=3 and d>0, for n=4 and d=2,3, for n=5 and d=2.