On m-tuples of nilpotent 2x2 matrices over an arbitrary field

TL;DR

This paper classifies all orbits of m-tuples of 2x2 nilpotent matrices under GL2 action over any field, providing minimal separating sets for the invariant algebra, especially over finite fields.

Contribution

It offers a complete classification of GL2-orbits on m-tuples of 2x2 nilpotent matrices over arbitrary fields and constructs minimal separating sets for their invariants.

Findings

Classified all GL2-orbits on m-tuples of 2x2 nilpotent matrices

Constructed minimal separating sets for the invariant algebra

Determined the minimal size of separating sets over finite fields

Abstract

The algebra of ${\rm GL}_n$-invariants of $m$-tuples of $n\times n$ matrices with respect to the action by simultaneous conjugation is a classical topic in case of infinite base field. On the other hand, in case of a finite field generators of polynomial invariants even in case of a pair of $2\times 2$ matrices are not known. Working over an arbitrary field we classified all ${\rm GL}_2$-orbits on $m$-tuples of $2\times 2$ nilpotent matrices for all $m>0$. As a consequence, we obtained a minimal separating set for the algebra of ${\rm GL}_2$-invariant polynomial functions of $m$-tuples of $2\times 2$ nilpotent matrices. We also described the least possible number of elements of a separating set for an algebra of invariant polynomial functions over a finite field.