Complete invariants for simultaneous similarity

TL;DR

This paper develops complete invariants for classifying tuples of matrices under simultaneous similarity over any field, reducing complex actions to simpler forms and providing explicit orbit separation methods.

Contribution

It introduces discrete and continuous invariants that fully determine orbits under simultaneous similarity, generalizing to module varieties and simplifying the group action analysis.

Findings

Decomposition of the variety into finitely many stable subsets.

Construction of invariant morphisms separating orbits.

Reduction of complex group actions to simpler forms.

Abstract

Always dealing with an arbitrary field we consider the variety $(k^{n\times n})^{p}$ under the action of $GL_{n}$ by simultaneous similarity. We define discrete and continuous invariants which completely determine the orbits. The discrete invariants induce a disjoint decomposition of the variety into finitely many locally closed $GL_{n}$-stable subsets and for each of these we construct finitely many invariant morphisms to $k$ separating the orbits. The complicated action of $GL_{n}$ by similarity is reduced to left multiplication of a product of $GL_{l_{i}}$'s on a product of $k^{l_{i}\times m_{i}}$'s. An analogous result holds for the left-right action of $GL_{m}\times GL_{n}$ on $(k^{m\times n })^{p}$ and more generally for all varieties of finite dimensional modules over some finitely generated algebra.