On automorphisms of matrix invariants

Abstract:

Let ${Q_{m,n}}$ be the space of $m$-tuples of $n \times n$-matrices modulo the simultaneous conjugation action of $PG{L_n}$. Let ${Q_{m,n}}(\tau )$ be the set of points of ${Q_{m,n}}$ of representation type $\tau$. We show that for $m \geq n + 1$ the group $\operatorname {Aut}({Q_{m,n}})$ of representation type preserving algebraic automorphisms of ${Q_{m,n}}$ acts transitively on each ${Q_{m,n}}(\tau )$. Moreover, the action of $\operatorname {Aut}({Q_{m,n}})$ on the Zariski open subset ${Q_{m,n}}(1,n)$ of ${Q_{m,n}}$ is $s$-transitive for every positive integer $s$. We also prove slightly weaker analogues of these results for all $m \geq 3$.