On generating the ring of matrix semi-invariants

Abstract

For a field $\mathbb{F}$, let $R(n, m)$ be the ring of invariant polynomials for the action of $\mathrm{SL}(n, \mathbb{F}) \times \mathrm{SL}(n, \mathbb{F})$ on tuples of matrices -- $(A, C)\in\mathrm{SL}(n, \mathbb{F}) \times \mathrm{SL}(n, \mathbb{F})$ sends $(B_1, \dots, B_m)\in M(n, \mathbb{F})^{\oplus m}$ to $(AB_1C^{-1}, \dots, AB_mC^{-1})$. In this paper we call $R(n, m)$ the \emph{ring of matrix semi-invariants}. Let $\beta(R(n, m))$ be the smallest $D$ s.t. matrix semi-invariants of degree $\leq D$ generate $R(n, m)$. Guided by the Procesi-Razmyslov-Formanek approach of proving a strong degree bound for generating matrix invariants, we exhibit several interesting structural results for the ring of matrix semi-invariants $R(n, m)$ over fields of characteristic $0$. Using these results, we prove that $\beta(R(n, m))=\Omega(n^{3/2})$, and $\beta(R(2, m))\leq 4$.