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

Artem Lopatin

TL;DR

This work addresses the $GL_2$-invariant theory of $m$-tuples of $2\times 2$ nilpotent matrices over arbitrary fields, culminating in a complete orbit classification and a minimal separating set for the invariant algebra ${\mathcal O}(\mathcal N_2^m)^{GL_2}$. The author provides an explicit orbit representative theorem (Theorem 3.1), a finite-field orbit count, and constructive separating sets that are proved minimal (Theorems 4.1 and 5.1), along with bounds on separating degrees and related corollaries. The results advance understanding of separating invariants in matrix-invariant theory, especially for nilpotent tuples and finite fields, with implications for orbit-counting and computational invariant theory. The study also proposes broader conjectures regarding the polynomial nature of orbit counts in $q$ and outlines precise formulas for separating-set sizes across settings.

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.

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

TL;DR

This work addresses the -invariant theory of -tuples of nilpotent matrices over arbitrary fields, culminating in a complete orbit classification and a minimal separating set for the invariant algebra . The author provides an explicit orbit representative theorem (Theorem 3.1), a finite-field orbit count, and constructive separating sets that are proved minimal (Theorems 4.1 and 5.1), along with bounds on separating degrees and related corollaries. The results advance understanding of separating invariants in matrix-invariant theory, especially for nilpotent tuples and finite fields, with implications for orbit-counting and computational invariant theory. The study also proposes broader conjectures regarding the polynomial nature of orbit counts in and outlines precise formulas for separating-set sizes across settings.

Abstract

The algebra of -invariants of -tuples of 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 matrices are not known. Working over an arbitrary field we classified all -orbits on -tuples of nilpotent matrices for all . As a consequence, we obtained a minimal separating set for the algebra of -invariant polynomial functions of -tuples of 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.
Paper Structure (12 sections, 5 theorems, 48 equations)

This paper contains 12 sections, 5 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Algebras of invariants
  3. Separating invariants
  4. Matrix invariants
  5. Nilpotent matrices
  6. Results
  7. Notations
  8. Minimal separating invariants for $\mathcal{O}(W)^G$ over a finite field
  9. Classification of ${\rm GL}_2$-orbits on $\mathcal{N}_2^m$
  10. Separating set for $\mathcal{O}(\mathcal{N}_2^m)^{{\rm GL}_2}$
  11. The case of infinite field
  12. Corollaries

Key Result

lemma 1

If ${\underline{A}} \in\mathcal{N}_2^2$ has no zeros, then ${\underline{A}} \sim (E_{12},\alpha E_{12})$ or ${\underline{A}} \sim (E_{12}, \alpha E_{21})$ for some non-zero $\alpha\in{\mathbb{F}}$.

Theorems & Definitions (28)

  • proof
  • lemma 1
  • proof
  • remark 1
  • proof
  • lemma 2
  • proof
  • proof
  • remark 2
  • remark 3
  • ...and 18 more