On types and classes of commuting matrices over finite fields

TL;DR

This paper explores the structure of commuting matrix classes over finite fields, reducing the problem to nilpotent classes, and classifies when such classes commute, providing new insights into their algebraic properties.

Contribution

It introduces a reduction of the commuting matrix problem to nilpotent classes using class types and classifies commuting pairs over finite fields, extending previous work.

Findings

Complete description of determinants in centralizer algebras.

Classification of nilpotent classes commuting over finite fields.

Identification of classes commuting universally across all nilpotent matrices.

Abstract

This paper addresses various questions about pairs of similarity classes of matrices which contain commuting elements. In the case of matrices over finite fields, we show that the problem of determining such pairs reduces to a question about nilpotent classes; this reduction makes use of class types in the sense of Steinberg and Green. We investigate the set of scalars that arise as determinants of elements of the centralizer algebra of a matrix, providing a complete description of this set in terms of the class type of the matrix. Several results are established concerning the commuting of nilpotent classes. Classes which are represented in the centralizer of every nilpotent matrix are classified--this result holds over any field. Nilpotent classes are parametrized by partitions; we find pairs of partitions whose corresponding nilpotent classes commute over some finite fields, but not over others. We conclude by classifying all pairs of classes, parametrized by two-part partitions, that commute. Our results on nilpotent classes complement work of Ko\v{s}ir and Oblak.