On types and classes of commuting matrices over finite fields

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 Košir and Oblak. Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 On types and classes of commuting matrices over finite fields John R. Britnell and Mark Wildon 1. General introduction Let Fq be a finite field, and let C and D be classes of similar matrices in Matn(Fq). We say that C and D commute if there exist commuting matrices X and Y such that X ∈ C and Y ∈ D. In this paper we are concerned with the problem of deciding which similarity classes commute. A matrix is determined up to similarity by its rational canonical form. This however is usually too sharp a tool for our purposes, and many of our results are instead stated in terms of the class type of a matrix. This notion, which seems first to have appeared in the work of Steinberg [14], is important in Green’s influential paper [8] on the characters of finite general linear groups. Lemma 2.1 of that paper implies that the type of a matrix determines its centralizer up to isomorphism; this fact is also implied by our Theorem 2.7, which says that two matrices with the same class type have conjugate centralizers. The main body of this paper is divided into three sections. In §2 we develop a theory of commuting class types; the results of this section reduce the general problem of determining commuting classes to the case of nilpotent classes. A key step in this reduction is Theorem 2.8, which states that if similarity classes C and D commute, then any class of the type of C commutes with any class of the type of D. Relationships between class types and determinants are discussed in §3. We provide a complete account of those scalars which appear as determinants in the centralizer of a matrix of a given type; this result, stated as Theorem 3.1, has appeared without proof in [2, §3.4], and as we promised there, we present the proof here. We also discuss the problem of determining which scalars appear as the determinant of a matrix of a given type. This problem appears intractable in general, and we provide only a very partial answer. But we identify a special case of the problem which leads to a difficult but highly interesting combinatorial problem, to which we formulate Conjecture 3.14 as a plausible solution. In §4 we make several observations concerning the problem of commuting nilpotent classes; this is a problem which has attracted attention in several different contexts over the years, and there is every reason to suppose that it is hard. Among other results, we determine 2000 Mathematics Subject Classification 15A27 (primary), 15A15, 15A21 (secondary). The research was supported in part by the Heilbronn Institute for Mathematical Research. TYPES AND CLASSES OF COMMUTING MATRICES Page 3 of 30 in Theorem 4.6 the nilpotent classes which commute with every other nilpotent class of the same dimension, and in Theorem 4.10 we classify all pairs of commuting nilpotent classes of matrices whose nullities are at most 2. We describe a construction on matrices which produces interesting and non-obvious examples of commuting nilpotent classes. This construction motivates Theorem 4.8, which says that for every prime p and positive integer r, there exists a pair of classes of nilpotent matrices which commute over the field Fpa if and only if a > r. As far as the authors are aware, it has not previously been observed that the commuting of nilpotent classes, as parameterized by partitions, is dependent on the field of definition. More detailed outlines of the results of §2, §3 and §4 are to be found at the beginnings of those sections. 1.1. Background definitions We collect here the main prerequisite definitions concerning partitions, classes and class types that we require. Partitions. We define a partition to be a weakly decreasing sequence of finite length whose terms are positive integers; these terms are called the parts of the partition. We shall denote the j-th part of a partition λ by λ(j). The sum of the parts of λ is written as |λ|. Given partitions λ and μ, we write λ+ μ for the partition of |λ|+ |μ| whose multiset of parts is the union of the multisets of parts of λ and of μ. We shall write 2λ for λ+ λ, and similarly we shall define tλ for all integers t ∈ N0. A partition μ will be said to be t-divisible if it is expressible as tλ for some partition λ; if sλ = tμ then we may write μ = stλ. We shall require the dominance order D on partitions. For two partitions λ and μ we say that λ dominates μ, and write λD μ (or μE λ) if