Minimal generating and separating sets for O(3)-invariants of several matrices

TL;DR

This paper identifies minimal separating and generating sets for invariants of matrices under orthogonal group actions, covering various matrix types and sizes over fields with characteristic not two.

Contribution

It provides explicit minimal separating sets for several matrix invariant algebras under O(3) and related groups, and a minimal generating set for three symmetric 3x3 matrices.

Findings

Minimal separating sets for O(3)-invariants of matrices

Minimal generating set for three symmetric 3x3 matrices

Results valid over fields with characteristic not two

Abstract

Given an algebra $F[H]^G$ of polynomial invariants of an action of the group $G$ over the vector space $H$, a subset $S$ of $F[H]^G$ is called separating if $S$ separates all orbits that can be separated by $F[H]^G$. A minimal separating set is found for some algebras of matrix invariants of several matrices over an infinite field of arbitrary characteristic different from two in case of the orthogonal group. Namely, we consider the following cases: 1) $GL(3)$-invariants of two matrices; 2) $O(3)$-invariants of $d>0$ skew-symmetric matrices; 3) $O(4)$-invariants of two skew-symmetric matrices; 4) $O(3)$-invariants of two symmetric matrices. A minimal generating set is also given for the algebra of orthogonal invariants of three $3\times 3$ symmetric matrices.