On types and classes of commuting matrices over finite fields

JR Britnell, MJ Wildon

Research output: Contribution to journalArticle (Academic Journal)peer-review

8 Citations (Scopus)


This paper addresses various questions about pairs of similarity classes of matrices that 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 that 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 Oblak and of Košir and Oblak.
Translated title of the contributionOn types and classes of commuting matrices over finite fields
Original languageEnglish
Pages (from-to)470 - 492
Number of pages23
JournalJournal of the London Mathematical Society
Volume83, no 2
Publication statusPublished - Apr 2011

Bibliographical note

Publisher: Oxford University Press

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