Algorithms for orbit closure separation for invariants and semi-invariants of matrices

TL;DR

This paper develops efficient algorithms to determine orbit closure intersection for matrix invariants under two group actions and improves bounds on separating invariants, advancing computational invariant theory.

Contribution

It introduces new algorithms for orbit closure separation under simultaneous conjugation and left-right actions, and refines bounds on separating invariants.

Findings

Algorithms decide orbit closure intersection efficiently.

Bounds on degrees of separating invariants are improved.

Enhanced understanding of invariant theory for matrix group actions.

Abstract

We consider two group actions on $m$-tuples of $n \times n$ matrices. The first is simultaneous conjugation by $\operatorname{GL}_n$ and the second is the left-right action of $\operatorname{SL}_n \times \operatorname{SL}_n$. We give efficient algorithms to decide if the orbit closures of two points intersect. We also improve the known bounds for the degree of separating invariants in these cases.