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 $(A{B}_1{C}^{-1},\dots ,A{B}_m{C}^{-1})$. 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 $\le D$ 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 $\beta (R(2,m))\le 4$.

16 pages

Cited by in corpus (2)

• Non-commutative Edmonds' problem and matrix semi-invariants
• Tripartite-to-bipartite Entanglement Transformation by Stochastic Local Operations and Classical Communication and the Structure of Matrix Spaces