On Numbers of Tuples of Nilpotent Matrices over Finite Fields under Simultaneous Conjugation

TL;DR

This paper derives a closed-form polynomial formula for counting absolutely indecomposable orbits of nilpotent matrix tuples over finite fields under simultaneous conjugation, revealing their algebraic structure and conjecturing non-negative coefficients.

Contribution

It provides the first explicit polynomial formula for counting such orbits, extending Hua's methodology to nilpotent matrices over finite fields.

Findings

Number of absolutely indecomposable orbits is finite for fixed matrix size.

The orbit counts are polynomials in the size of the finite field.

Coefficients of these polynomials are conjectured to be non-negative.

Abstract

The problem of classifying tuples of nilpotent matrices over a field under simultaneous conjugation is considered "hopeless". However, for any given matrix order over a finite field, the number of concerned orbits is always finite. This paper gives a closed formula for the number of absolutely indecomposable orbits using the same methodology as Hua [5]; those orbits are non-splittable over field extensions. As a consequence, those numbers are always polynomials in the cardinality of the base field with integral coefficients. It is conjectured that those coefficients are always non-negative.