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

Understand

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.