TYPES AND CLASSES OF COMMUTING MATRICES Page 1 of 30

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 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 F_{q} be a finite field, and let C and D be classes of similar matrices in Mat_{n}(F_{q}). 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.

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 F_{p^{a}} 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 ∈ N_{0}. 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 = \frac{s}{t}λ.

We shall require the dominance order ⊵ on partitions. For two partitions λ and μ we say that λ dominates μ, and write λ ⊵ μ (or μ ⊴ λ) if

∑_{i=1}^{j} λ(i) ≥ ∑_{i=1}^{j} μ(i)

for all j ∈ N. (If i exceeds the number of parts in a partition, then the corresponding part is taken to be 0.)

Let λ be a partition with largest part λ(1) = a. The conjugate partition \overline{λ} is defined to be (\overline{λ}(1), …, \overline{λ}(a)), where \overline{λ}(j) is the number of parts of λ of size at least j. It is a well-known fact (see for instance [11, 1.11]) that the conjugation operation on partitions reverses the dominance order; that is, λ ⊵ μ if and only if \overline{μ} ⊵ \overline{λ}.

A geometric interpretation of the dominance order is developed by Gerstenhaber in [5] and [6]; the issues with which the latter paper is concerned are similar in many respects to those considered in §4 of the present paper, although Gerstenhaber’s approach using algebraic varieties is very different.

Page 4 of 30 JOHN R. BRITNELL AND MARK WILDON

Similarity classes. Let K be a field. A class of similar matrices in Mat_{n}(K) is determined by the following data: a finite set F of irreducible polynomials over K, and for each f ∈ F a partition λ_{f} of a positive integer, such that

n = ∑_{f∈F} |λ_{f}| deg f.

The characteristic polynomial of a matrix M in this class is ∏_{f} f^{|λ_{f} |}. There is a decomposition of V given by

V = ⨁_{f} ⨁_{j} V_{f}(j),

where M acts indecomposably on the subspace V_{f}(j) with characteristic polynomial f^{λ_{f} (j)}. This decomposition is, in general, not unique. By a change of basis, we may express M as ⨁_{f} ⨁_{j} P_{f}(j), where P_{f}(j) is a matrix representing the action of M on V_{f}(j); we say that P_{f}(j) is a cyclic block of M.

If F = {f_{1}, …, f_{t}} and the associated partitions are λ_{1}, …, λ_{t} respectively, then we shall define the cycle type of M to be the formal expression

cyc(M) = f_{1}^{λ_{1}} ⋯ f_{t}^{λ_{t}}.

The order in which the polynomials appear in this expression is, of course, unimportant.

Nilpotent classes. We shall denote by N(λ) the similarity class of nilpotent matrices with cycle type f_{0}^{λ}, where f_{0}(x) = x. We denote by J(λ) the unique matrix in upper-triangular Jordan form in the similarity class N(λ).

If λ = (λ(1), …, λ(k)) we shall omit unnecessary brackets by writing N(λ(1), …, λ(k)) for N(λ) and J(λ(1), …, λ(k)) for J(λ).

Class types. More general than the notion of similarity class is that of class type. If M is a matrix of cycle type f_{1}^{λ_{1}} ⋯ f_{t}^{λ_{t}}, where for each i the polynomial f_{i} has degree d_{i}, then the class type of M is the formal string

ty(M) = d_{1}^{λ_{1}} ⋯ d_{t}^{λ_{t}}.

Here too, the order of the terms is unimportant.

Any string of this form will be called a type. The dimension of the type d_{1}^{λ_{1}} ⋯ d_{t}^{λ_{t}} is defined to be d_{1}|λ_{1}| + ⋯ + d_{t}|λ_{t}|. We shall say that the type T is representable over a field K if there exists a matrix of class type T with entries in K; the dimension of such a matrix is the same as the dimension of the type. Clearly not all types are representable over all fields; for instance the type 1^{λ}1^{μ}1^{ν} is not representable over F_{2} since there are only two distinct linear polynomials over this field; similarly 3^{λ} is not representable over R since there are no irreducible cubics over R.

Similar matrices have the same cycle type and the same class type, and so we may meaningfully attribute types of either kind to similarity classes.

We shall say that a class type T is primary if it is d^{λ} for some d and λ. Otherwise T is compound. If d^{λ} appears as a term in the type T, we say that d^{λ} is a primary component of T. We may also say that a matrix, a similarity class of matrices, or a cycle type is primary or compound, according to its class type, and we may refer to its primary components.

We have already defined what it means for two similarity classes to commute. We generalise this idea to types, as follows.

Definition. Let S and T be class types. We say that S and T commute over a field K if there are matrices X and Y over K such that X has class type S, and Y has class type T, and X and Y commute.

The field K will not always be mentioned explicitly if it is clear from the context.

2. Commuting types of matrices

This section proceeds as follows. In §2.1 we prove several results relating the class type of a polynomial in a matrix M to the class type of M, leading up to Theorem 2.6: that two similarity classes have the same class type if and only if they contain representatives which are polynomial in one another. This result is then used in the proof of Theorem 2.8, which states that two similarity classes commute if and only if their class types commute.

Using Theorem 2.8, we proceed to reduce our original problem of deciding which similarity classes commute, first to the case of primary types in §2.2, and thence to the case of nilpotent classes in §2.3. At the end of §2.3 we give examples illustrating both steps of this reduction.

2.1. Polynomials and commuting types

If M is a matrix of primary class type d^{λ} then it has associated with it a single irreducible polynomial f such that its cycle type is f^{λ}. It is clear that f(M) is nilpotent. The following lemma and proposition describe its associated partition.

Lemma 2.1. Let M be a matrix of cycle type f^{λ}, where deg f = d. For each j, let m_{j} be the number of parts of λ of size j. Then

dm_{j} = (null f(M)^{j} − null f(M)^{j−1}) − (null f(M)^{j+1} − null f(M)^{j}).

Proof. Let P be a cyclic block of M. If the dimension of P is dh then the characteristic polynomial of P is f^{h}. If j ≥ h, then null f(P)^{j} = dh; otherwise null f(P)^{j} = dj.

Since M is a direct sum of cyclic blocks of dimensions dλ(1), dλ(2), …, it follows that

null f(M)^{j} = ∑_{h≤j} dhm_{h} + ∑_{h>j} djm_{h},

and hence

null f(M)^{j+1} − null f(M)^{j} = ∑_{h>j} dm_{h}.

This implies the lemma.

Proposition 2.2. Let M be a matrix of primary type d^{λ}. If the cycle type of M is f^{λ} then f(M) is nilpotent of type 1^{dλ}.

Proof. Since f(M) is nilpotent, it is primary and its associated polynomial, f_{0}(x) = x, is linear. The result is now immediate from Lemma 2.1.

We use the preceding proposition to give some information about the type of F (M), where M is a primary matrix and F is any polynomial. The following lemma will be required.

Lemma 2.3. Let M and N be nilpotent matrices with associated partitions μ and ν respectively. Then μ ⊴ ν if and only if rank M^{j} ≤ rank N^{j} for all j ∈ N.

Proof. The rank of M^{j} is equal to the sum of the j smallest parts of the conjugate partition \overline{μ}. The rank of N^{j} can be calculated similarly in terms of \overline{ν}. It follows easily that rank M^{j} ≤ rank N^{j} for all j if and only if \overline{μ} ⊵ \overline{ν}. The lemma now follows from the fact that the dominance order ⊵ is reversed by conjugation of partitions.

Proposition 2.4. Let X be a primary matrix of class type d^{λ} with entries from a field K, and let F ∈ K[x] be any polynomial. The type of F (X) is e^{μ} for some e dividing d, and some partition μ such that e|μ| = d|λ| and eμ ⊴ dλ.

Proof. Let the cycle type of X be f^{λ} where f is an irreducible polynomial of degree d. If α is a root of f in a splitting field, then the eigenvalues of F (X) are the conjugates over K of F (α). Hence F (X) is of primary type, and if g ∈ K[x] is the irreducible polynomial associated with F (X), then the degree of g divides d. Let e^{μ} be the type of F (X).

Let Y = F (X). We observe that g(Y) = (g ∘ F)(X) is a nilpotent matrix, and hence f divides g ∘ F; let g ∘ F = kf. By Proposition 2.2, f(X) has type 1^{dλ}, while g(Y) has type 1^{eμ}. For each i ∈ N_{0} we have g(Y)^{i} = k(X)^{i}f(X)^{i} and hence im g(Y)^{i} ⊆ im f(X)^{i}. It follows that rank g(Y)^{i} ≤ rank f(X)^{i} for every i ∈ N. Now from Lemma 2.3 we see that eμ ⊴ dλ, as required.

When K is a finite field, Proposition 2.4 has the following partial converse.

Proposition 2.5. If X is a primary matrix of class type d^{λ} with entries from F_{q}, and D is a similarity class of matrices also of this class type, then there is a polynomial F ∈ F_{q}[x] such that F (X) ∈ D.

Proof. Let f ∈ F_{q}[x] be the irreducible polynomial associated with X. Suppose that the additive Jordan–Chevalley decomposition of X is \overline{X} + N, where \overline{X} is semisimple and N is nilpotent; recall that \overline{X} and N can be expressed as polynomials in X. Without loss of generality, we may suppose that

\overline{X} = diag(P, …, P),

where the cyclic block P has minimum polynomial f.

Let g be the irreducible polynomial associated with the similarity class D, and let α and β be roots of f and g respectively in F_{q^{d}}. There exists a polynomial G ∈ F_{q}[x], coprime with f, such that G(α) = β. If we define

Q = G(P),

then Q has minimum polynomial g. Let

\overline{Y} = diag(Q, …, Q).

Then \overline{Y} = G(\overline{X}), and since \overline{X} is polynomial in X, it follows that \overline{Y} is too. Moreover, if we set Y = \overline{Y} + N, then Y is polynomial in X, and it is clear that Y lies in the similarity class D.

Let C and D be similarity classes of Mat_{n}(F_{q}). We say that D is polynomial in C if there exists a polynomial F with coefficients in F_{q} such that F (X) ∈ D for all X ∈ C.

Theorem 2.6. Let C and D be similarity classes of Mat_{n}(F_{q}). The classes C and D have the same type if and only if C and D are polynomial in one another.

Proof. We observe that applying a polynomial to a matrix cannot increase its number of primary components. So if C and D are polynomial in one another, then they have the same number of components. Moreover there is a pairing between the primary components C_{1}, …, C_{t} of C and D_{1}, …, D_{t} of D such that C_{i} and D_{i} are polynomial in one another for all i. It will therefore be sufficient to prove the result in the case that both C and D are primary. Suppose that ty(C) = d^{λ} for some d ∈ N and some partition λ. It follows from Proposition 2.4 that D has class type e^{μ} where e divides d and eμ ⊴ dλ. By symmetry we see that e = d and λ = μ, as required.

For the converse, suppose that ty(C) = ty(D). Let T_{1}, T_{2}, …, T_{t} be the primary components of ty(C), and let X = diag(X_{1}, …, X_{t}) be an element of C such that ty(X_{i}) = T_{i} for all i. Let the minimum polynomial of the block X_{i} be f_{i}^{a_{i}}, where f_{i} is irreducible. By Proposition 2.5, there exist polynomials F_{1}, …, F_{t} ∈ F_{q}[x] such that diag(F_{1}(X_{1}), …, F_{t}(X_{t})) ∈ D. By the Chinese Remainder Theorem, there exists a polynomial F ∈ F_{q}[x] such that

F (x) ≡ F_{i}(x) mod f_{i}^{a_{i}}(x) for all i.

And now we see that F (X) ∈ D, as required.

It was proved by Green [8, Lemma 2.1] that the type of a matrix determines its centralizer up to isomorphism. Using Theorem 2.6 we may prove the following stronger result.

Theorem 2.7. Let X and Y be matrices in Mat_{n}(F_{q}) with the same class type. Let Cent X and Cent Y be the centralizers in Mat_{n}(F_{q}) of X and Y respectively. Then Cent X and Cent Y are conjugate by an element of GL_{n}(F_{q}).

Proof. By Theorem 2.6 there exist polynomials F and G such that F (X) is conjugate to Y and G(Y) is conjugate to X. Now the centralizer Cent F (X) is a subalgebra of Cent X which is conjugate to Cent Y; similarly the centralizer Cent G(Y) is a subalgebra of Cent Y which is conjugate to Cent X. Since Cent X and Cent Y are finite, it is clear that Cent X = Cent F (X) and that Cent Y = Cent G(Y), which suffices to prove the theorem.

An obvious corollary of Theorem 2.6, which has been stated in [3, §3.2], is that classes of the same type commute. We are now in a position to establish a stronger result. Recall that types S and T are said to commute if there exist commuting matrices X and Y with types S and T respectively.

Theorem 2.8. Let C and D be similarity classes of matrices over F_{q}. Then C and D commute if and only if ty(C) and ty(D) commute.

Proof. One half of the double implication is trivial, since if the similarity classes commute then by definition the class types do. For the other half, notice that if ty(C) and ty(D) commute then there exist commuting similarity classes C′ and D′ such that ty(C′) = ty(C) and ty(D′) = ty(D). Let X′ and Y′ be commuting matrices from C′ and D′ respectively. Then there exist polynomials F and G such that F (X′) ∈ C and G(Y′) ∈ D, and clearly F (X′) and G(Y′) commute.

We remark that Theorems 2.6, 2.7 and 2.8 do not hold for matrices over an arbitrary field. There are counterexamples in Mat_{2}(Q), for instance. Let Q(α) and Q(β) be distinct quadratic extensions of Q. Let C and D be the similarity classes of rational matrices with characteristic polynomials x^{2} − α and x^{2} − β respectively; then ty(C) = ty(D) = 2^{(1)}. Since the eigenvalues α and β are not polynomial in one another, it is clear that neither are C and D. Moreover, the classes C and D do not commute. It is for this reason that our consideration of commuting types is for the most part restricted to matrices with entries from a finite field.

2.2. Reduction to primary types

The next step in our strategy is to reduce the question of which class types commute to the corresponding question about primary types. This is accomplished in Proposition 2.9 below.

We shall need the following two definitions.

Definition. A separation operation on a type T is the replacement of a primary component d^{λ} of T by d^{μ}d^{ν}, where λ = μ + ν. A separation of T is a type obtained from T by repeated applications of separation operations.

Definition. Let S and T be types. We shall say that S and T commute componentwise over a field K if the primary components of S and T can be ordered so that S = c_{1}^{λ_{1}} ⋯ c_{t}^{λ_{t}} and T = d_{1}^{μ_{1}} ⋯ d_{t}^{μ_{k}}, where c_{i}^{λ_{i}} commutes with d_{i}^{μ_{i}} over K for each i.

This definition, it should be noted, does not preclude the possibility that types S and T commute componentwise, even if one or both of them cannot be represented over the field K. For example, 1^{(1,1,1)} commutes componentwise with 1^{(1)}1^{(1)}1^{(1)} over F_{2} according to the definition, even though the latter type is not representable. The examples at the end of §2.3 illustrate why this freedom is desirable.

Proposition 2.9. Let S and T be types which are representable over a finite field F_{q}. Then S and T commute over F_{q} if and only if there exist separations S^{⋆} of S and T^{⋆} of T such that S^{⋆} and T^{⋆} commute componentwise.

Proof. Let X and Y be commuting matrices with entries from F_{q}, whose types are S and T respectively. It is well known and easy to show that there exists a decomposition V = V_{1} ⊕ ⋯ ⊕ V_{t} such that both X and Y act as transformations of primary type on each of the summands V_{i}. Suppose that the action of X on V_{i} has type c_{i}^{λ_{i}}, and the action of Y has type d_{i}^{μ_{i}}. Then it is clear that the primary types c_{i}^{λ_{i}} and d_{i}^{μ_{i}} commute, that c_{1}^{λ_{1}} ⋯ c_{t}^{λ_{t}} is a separation of S and that d_{1}^{μ_{1}} ⋯ d_{t}^{μ_{t}} is a separation of T.

For the converse, suppose that the primary types c_{i}^{λ_{i}} and d_{i}^{μ_{i}} commute, that c_{1}^{λ_{1}} ⋯ c_{t}^{λ_{t}} is a separation of S and that d_{1}^{μ_{1}} ⋯ d_{t}^{μ_{t}} is a separation of T. Then, from Theorem 2.8, it follows that for any choice of irreducible polynomials f_{i} of degree c_{i} and g_{i} of degree d_{i}, the classes f_{i}^{λ_{i}} and g_{i}^{μ_{i}} commute. If X_{i} and Y_{i} are commuting representatives of these respective classes, then the matrices X = diag(X_{1}, …, X_{t}) and Y = diag(Y_{1}, …, Y_{t}) commute. Now each primary type c_{i}^{λ_{i}} derives (under separation operations) from a particular component of S. If we select our polynomials f_{i} in such a way that blocks deriving from the same component of S have the same polynomial, then we find that ty(X) = S. Similarly we can choose the polynomials g_{i} so that ty(Y) = T, and it follows that S and T commute.

2.3. Reduction to nilpotent classes

We now complete the reduction of our general problem of commuting classes to the case of nilpotent classes. Recall that we denote by N(λ) the similarity class of nilpotent matrices with cycle type f_{0}^{λ}, where f_{0}(x) = x. Recall also that a partition is said to be t-divisible if it is tν for some partition ν.

Theorem 2.10. Let S = c^{λ} and T = d^{μ} be primary types of the same dimension. Let h = hcf(c, d) and ℓ = lcm(c, d). Then S and T commute over F_{q} if and only if λ is \frac{d}{h}-divisible, μ is \frac{c}{h}-divisible, and the nilpotent classes N(\frac{h}{d}λ) and N(\frac{h}{c}μ) commute over F_{q^{ℓ}}.

Proof. Suppose that S and T commute over F_{q}. Let X and Y be commuting elements of Mat_{n}(F_{q}) with cycle types f^{λ} and g^{μ} respectively, where deg f = c and deg g = d. Let α_{1}, …, α_{c} be the roots of f and β_{1}, …, β_{d} the roots of g in the extension field F_{q^{ℓ}}. Over this extension field, it is easy to see that the cycle types of X and Y are given by

cyc(X) = (x − α_{1})^{λ} ⋯ (x − α_{c})^{λ},
cyc(Y) = (x − β_{1})^{μ} ⋯ (x − β_{d})^{μ}.

Let W = F_{q^{ℓ}}^{n}, and let W_{ij} denote the maximal subspace of W on which X − α_{i}I and Y − β_{j}I are both nilpotent. (So W = ⨁_{ij} W_{ij}.) Let λ_{ij} and μ_{ij} be the partitions such that the type of X on W_{ij} is 1^{λ_{ij}} and the type of Y on W_{ij} is 1^{μ_{ij}}. Then clearly ∑_{j=1}^{d} λ_{ij} = λ for all i, while ∑_{i=1}^{c} μ_{ij} = μ for all j.

Since X and Y have entries in F_{q}, it follows that the Frobenius automorphism ξ ↦ ξ^{q} of F_{q^{ℓ}} induces an isomorphism between the F_{q^{ℓ}}⟨X, Y ⟩-modules V_{ij} and V_{i′j′} whenever i − j ≡ i′ − j′ mod h. Hence

λ_{ij} = λ_{i′j′} and μ_{ij} = μ_{i′j′} whenever i − j ≡ i′ − j′ mod h.

Therefore the partitions λ_{ij} for i ∈ {1, …, c} and j ∈ {1, …, d} are determined by the partitions λ_{1k} for k ∈ {1, …, h}, and since

λ = \frac{d}{h} ∑_{k=1}^{h} λ_{1k},

it follows that λ is \frac{d}{h}-divisible. Similarly, μ is \frac{c}{h}-divisible.

Now clearly the actions of X and Y on the subspace ⨁_{k=1}^{h} V_{1k} commute. The type of X on this submodule (defined over F_{q^{ℓ}}) is 1^{λ_{11}} ⋯ 1^{λ_{1h}}, which is a separation of 1^{\frac{h}{d} λ}. Similarly the type of Y on the submodule is a separation of 1^{\frac{h}{c} μ}. Hence, by the ‘if’ direction of Proposition 2.9, the types 1^{\frac{h}{d} λ} and 1^{\frac{h}{c} μ} commute over F_{q^{ℓ}}. In particular, it follows from Theorem 2.8 that the nilpotent classes N(\frac{h}{d}λ) and N(\frac{h}{c}μ) commute over this field.

For the converse, let λ′ = \frac{h}{d}λ and μ′ = \frac{h}{c}μ, and suppose that the nilpotent classes N(λ′) and N(μ′) commute over F_{q^{ℓ}}. We shall denote by m the integer |λ′|, which of course is equal to |μ′|. Let α and β be elements of F_{q^{ℓ}} whose degrees over F_{q} are c and d respectively. Since N(λ′) and N(μ′) commute over F_{q^{ℓ}}, so do the classes with cycle types (x − α)^{λ′} and (x − β)^{μ′}. Let X and Y be commuting elements of these respective classes. Let ϕ be an embedding of the matrix algebra Mat_{m}(F_{q^{ℓ}}) into Mat_{ℓm}(F_{q}); then it is not hard to see that ϕ(X) has class type c^{λ} and ϕ(Y) has class type d^{μ}. It follows that these types commute over F_{q}.

It is worth noting that Theorem 4.8 below implies that the references to particular fields in the statement of Theorem 2.10 are essential. The following special case of the theorem, however, does not depend on the field of definition.

Proposition 2.11. Let d, k ∈ N. The types d^{(k)} and 1^{(k,...,k)} commute over any field.

Proof. If the field in question is finite, then the proposition follows from Theorem 2.10. For it suffices to show that the type \frac{1}{d}(k, …, k) = (k) commutes with itself over F_{q^{d}}, and certainly this is the case.

A straightforward modification of the last paragraph of the proof of Theorem 2.10 would allow us to deal with arbitrary fields; however we prefer the following short argument involving tensor products. Let f be an irreducible polynomial of degree d and let P be the companion matrix of f. The type d^{(k)} is represented by the dk × dk matrix

P^{(k)} = (\begin{matrix}P & I \\ & P & I \\ & & ... & ... \\ & & & P \end{matrix}) .

Let J = J(k) be the k-dimensional Jordan block with eigenvalue 1. It is clear that P^{(k)} commutes with the tensor product I ⊗ J (which is obtained from the matrix above by substituting I for each occurrence of P). And I ⊗ J is conjugate to J ⊗ I = diag(J, …, J), which has type 1^{(k,...,k)}. Hence the types d^{(k)} and 1^{(k,...,k)} commute.

We end this section with two examples of how the steps in our reduction can be carried out, which illustrate the various results of this section.

Example. Let p, q, r, s and t be the following irreducible polynomials over F_{2}:

linear:  p(x) = x, q(x) = x + 1;
quadratic: r(x) = x^{2} + x + 1;
cubic:  s(x) = x^{3} + x + 1, t(x) = x^{3} + x^{2} + 1.

Let C be the similarity class of matrices over F_{2} with cycle type p^{(12,12)}q^{(2,2,2)}r^{(3)}s^{(1)} and let D be the similarity class with cycle type r^{(7,5)}t^{(2,2,1)}. We shall prove that C commutes with D.

By Theorem 2.8, this is equivalent to showing that the types

S = 1^{(12,12)}1^{(2,2,2)}2^{(3)}3^{(1)},
T = 2^{(7,5)}3^{(2,2,1)}

commute. This, in turn, will follow from Proposition 2.9, if we can show that S commutes componentwise with the separation T^{⋆} = 2^{(7,5)}3^{(2)}3^{(2)}3^{(1)} of T. (This example was chosen to make the point that it is not necessary that the separated types can be represented over F_{2}.) By Theorem 2.10 we see that 1^{(12,12)} commutes with 2^{(7,5)} over F_{2} if and only if 1^{(6,6)} commutes with 1^{(7,5)} over F_{4}; that this is the case follows from Proposition 4.7 below, which implies that the nilpotent classes N(6, 6) and N(7, 5) commute over F_{4}. It is immediate from Theorem 2.10 that 1^{(2,2,2)} commutes with 3^{(2)}, and that 2^{(3)} commutes with 3^{(2)}. Hence S and T^{⋆} commute componentwise, and so C and D commute.

The converse directions of Proposition 2.9 and Theorem 2.10 can in principle be used as part of a argument that two similarity classes do not commute; again, results about commuting of nilpotent classes will generally be needed to complete such an argument. The following example is illustrative.

Example. Let the polynomials p, q, r, s and t, the class C, and the type S be as in the previous example. Let D be the similarity class over F_{2} with cycle type r^{(8,4)}t^{(2,2,1)}. The class type of D is

T = 2^{(8,4)}3^{(2,2,1)}.

Suppose that a separation T^{∗} of T commutes componentwise with a separation S^{∗} of S; then one of 2^{(8)} or 2^{(8,4)} is a component of T^{∗}. The first possibility is ruled out since S^{∗} can have no component of dimension 16. The only possible component of S^{∗} of dimension 24 is 1^{(12,12)}, and so if our supposition is correct, then the primary types 2^{(8,4)} and 1^{(12,12)} must commute over F_{2}. By Theorem 2.10, this is the case only if 1^{(8,4)} and 1^{(6,6)} commute. But by Proposition 4.9 below, the nilpotent classes N(8, 4) and N(6, 6) do not commute over any field. It follows that C and D do not commute.

3. Types and determinants

The main object of this section is to establish Theorem 3.1, concerning determinants of elements of centralizer algebras. The following definition is key.

Definition. Let M be a matrix with class type d_{1}^{λ_{1}} ⋯ d_{t}^{λ_{t}}. The part-size invariant of M is defined to be the highest common factor of all of the parts of the partitions λ_{1}, …, λ_{t}.

Theorem 3.1. Let M ∈ Mat_{n}(F_{q}) have part-size invariant k. The determinants which occur in the centralizer of M in Mat_{n}(F_{q}) are precisely the k-th powers in F_{q}.

Part of the motivation for this investigation comes from the authors’ paper [2] on the distribution of conjugacy classes of a group G across the cosets of a normal subgroup H, where G/H is abelian. The centralizing subgroup of a class C with respect to H was defined to be the subgroup Cent_{G}(g) · H, where g ∈ C may be chosen arbitrarily. It was proved that if G is finite and G/H is cyclic, then the classes with centralizing subgroup K are uniformly distributed across the cosets of H in K.

Theorem 3.1 treats the case where G = GL_{n}(F_{q}) and H = SL_{n}(F_{q}). It is clear that the subgroups K lying in the range H ≤ K ≤ G may be defined in terms of the determinants of their elements; specifically, the index |K : H| is equal to the order of the subgroup of F_{q}^{×} generated by the determinants of the matrices in K. Hence, in order to calculate the centralizing subgroup of a matrix, we must decide which determinants occur in its centralizer. The following corollary of Theorem 3.1 shows that the answer to this question depends only on the class type of the matrix concerned.

Corollary 3.2. Let M ∈ GL_{n}(F_{q}) have part-size invariant k, and let c = hcf(q − 1, k). The centralizing subgroup of the conjugacy class of M is the unique index c subgroup of GL_{n}(F_{q}) containing SL_{n}(F_{q}).

In §3.1 below we prove a special case of Theorem 3.1, namely that the determinants in the centralizer of a nilpotent matrix are k-th powers, where k is the part-size invariant. The proof of Theorem 3.1 is completed in §3.2. We end in §3.3 by discussing the natural—but surprisingly hard—question of which scalars can appear as the determinant of a matrix of a given type.

3.1. Determinants in the centralizer of a nilpotent matrix

In this section we let M ∈ Mat_{n}(F_{q}) be a nilpotent matrix lying in the similarity class N(λ). Let A = Cent M be the subalgebra of Mat_{n}(F_{q}) consisting of the matrices that centralize M. We shall find the composition factors of V = F_{q}^{n} as a right A-module; using this result we describe the determinants of the matrices of A. For some related results on the lattice of A-submodules of V, the reader is referred to [7, Chapter 14].

Definition. For v ∈ V we define the height of v, written ht(v), to be the least integer h such that v ∈ ker M^{h}.

Definition. We shall say that a vector u ∈ V is a cyclic vector for M if u is not in the image of M.

The proof of the following well-known lemma is straightforward, and is omitted.

Lemma 3.3. An element Y ∈ A is uniquely determined by its effect on the cyclic vectors of M. If u_{1}, …, u_{t} are linearly independent cyclic vectors and v_{1}, …, v_{t} are any vectors such that ht(v_{i}) ≤ ht(u_{i}) for every i, then there is an element Y ∈ A such that u_{i}Y = v_{i} for each i.

As in Lemma 2.1, we let m_{h} be the number of parts of λ of size h. For h ∈ N_{0}, we shall write V_{h} for ker M^{h}.

Proposition 3.4. For each h ∈ N, the subspace V_{h} is an A-submodule of V containing V_{h+1}M + V_{h−1} as an A-submodule. Moreover if m_{h} ≠ 0 then

V_{h} / (V_{h+1}M + V_{h−1})

is a simple A-module of dimension m_{h}.

Proof. The proof of the first statement is straightforward, and we omit it; we shall outline a proof of the second statement.

Let u_{1}, …, u_{m_{h}} be a maximal set of linearly independent cyclic vectors each of height h. It is not hard to see that u_{1}, …, u_{m_{h}} span a complement in V_{h} to V_{h+1}M + V_{h−1}. By the previous lemma, for any vectors v_{1}, …, v_{m_{h}} in V_{h}, there exists Y ∈ A such that u_{i}Y = v_{i} for each i. This implies that A acts as a full matrix algebra in its action on the quotient module V_{h}/(V_{h+1}M + V_{h−1}). Hence the quotient module is simple.

For h such that m_{h} ≠ 0, let S_{h} = V_{h}/(V_{h+1}M + V_{h−1}) be the simple A-module constructed in Proposition 3.4. If h ≠ h′ and both S_{h} and S_{h′} are defined, then by Lemma 3.3, it is possible to define a matrix Y ∈ A such that Y acts as the identity on the cyclic vectors spanning S_{h}, and as the zero map on the cyclic vectors spanning S_{h′}. The simple modules S_{h} and S_{h′} are therefore non-isomorphic as A-modules.

Proposition 3.5. The A-module V has a composition series in which the simple A-module S_{h} appears with multiplicity h.

Proof. The action of the nilpotent matrix M on V_{h} induces a non-zero homomorphism of simple A-modules

\frac{V_{h}M^{i−1}}{V_{h+1}M^{i} + V_{h−1}M^{i−1}} ⟶ \frac{V_{h}M^{i}}{V_{h+1}M^{i+1} + V_{h−1}M^{i}}

for each i such that 1 ≤ i ≤ h − 1. This gives us h distinct composition factors of V_{h}, each isomorphic to S_{h}. It now follows from the Jordan–Hölder theorem that in any composition series of V, the simple module S_{h} appears at least with multiplicity h. Finally, by comparing dimensions using the equation

dim V = n = ∑_{h} hm_{h} = ∑_{h} h dim S_{h},

we see that equality holds for each h, and that the A-module V has no other composition factors.

Proposition 3.6. If M is nilpotent, and has part-size invariant k, then the determinants that appear in Cent M are k-th powers in F_{q}.

Proof. Given Y ∈ Cent M let Y_{h} denote the matrix in Mat_{m_{h}}(F_{q}) which gives the action of Y on the simple A-module S_{h}. Using the composition series given by the previous theorem to compute det Y we get

det Y = ∏_{{h \atop m_{h}≠0}}(det Y_{h})^{h}.

Since the part-size invariant of m is the highest common factor of the set {h | m_{h} ≠ 0}, we see that det Y is a k-th power.

It is worth remarking that it is also possible to prove Proposition 3.5 in a way that gives the required composition series in an explicit form. We have avoided this approach in order to keep the notation as simple as possible. The following example indicates how to construct a suitable basis of V in a small case.

Example. Let M ∈ Mat_{5}(F_{q}) be a nilpotent matrix in the similarity class N(2, 2, 1). Let u_{1}, u_{2} be cyclic vectors of M of height 2, and let v be a cyclic vector of M of height 1. Then with respect to the basis u_{1}, u_{2}, v, u_{1}M, u_{2}M of F_{q}^{5}, the centralizer of M consists of all matrices of the form

(\begin{matrix}α & β & ⋆ & ⋆ & ⋆ \\ γ & δ & ⋆ & ⋆ & ⋆ \\ & & ζ & ⋆ & ⋆ \\ & & & α & β \\ & & & γ & δ \end{matrix})

where gaps denote zero entries, and ⋆ is used to denote an entry we have no need to specify explicitly. The key to obtaining this matrix in the required form is to order the elements of the basis correctly. The following principles determine a suitable ordering on the basis: elements come in decreasing order of height; cyclic vectors come first among elements of the same height; if b_{i} comes before b_{j} then b_{i}M comes before b_{j}M.

3.2. Proof of Theorem 3.1

The proof has two steps. We first show that if M is a matrix with entries in F_{q} and part-size invariant k, then every k-th power in F_{q} appears as the determinant of a matrix in Cent M. In the second, we use Proposition 3.6 to show that no other powers can appear.

We begin with the following lemma.

Lemma 3.7. Let S(k) be the set of k-th powers in F_{q}. Let d ∈ N and ϑ ∈ F_{q}^{×}. Then the number of irreducible polynomials of degree d over F_{q} with constant term ϑ is

\frac{1}{d(q − 1)} ∑_{{k|d \atop S(k)∋ϑ}} μ(k) hcf(q − 1, k)(q^{d/k} − 1).

This number is non-zero for all choices of d and ϑ and for all q.

Proof. We give an elementary proof of the existence of a polynomial with degree d and constant term ϑ. For the number of polynomials, see for instance [1, §5.2].

Let α be a generator of the multiplicative group F_{q^{d}}^{×}, and let β = 𝔫(α) where 𝔫 : F_{q^{d}}^{×} → F_{q}^{×} is the norm homomorphism. It is clear that β generates F_{q}^{×}. Let c be such that 0 < c < q and (−1)^{d}ϑ = β^{c}. Since F_{q^{d}} has no proper subfield of index less than q, and since the multiplicative order of α^{c} is at least (q^{d} − 1)/c, it is easy to see that α^{c} cannot lie in a proper subfield of F_{q^{d}}. It follows that the minimum polynomial of α^{c} over F_{q} has degree d and constant term ϑ, as required.

Proposition 3.8. Let P be a matrix with class type d^{(j)}. Then for any ϑ ∈ F_{q}, there exists a matrix in Cent P with determinant ϑ^{j}.

Proof. We may assume that ϑ is non-zero. By Lemma 3.7 there exists an irreducible polynomial f over F_{q} with degree d and constant term (−1)^{d}ϑ. Let C be the similarity class containing P, and let D be the class of matrices with cycle type f^{(j)}. Since C and D have the same class type, it follows from Theorem 2.6 that they commute. Therefore P commutes with an element of D. It is clear from the construction of D that its elements have determinant ϑ^{j}, as required.

We now extend Proposition 3.8 to a general matrix.

Proposition 3.9. If M is a matrix with part-size invariant k, then for any ζ ∈ F_{q}, there exists a matrix in Cent M with determinant ζ^{k}.

Proof. Let P_{1}, …, P_{s} be the distinct cyclic blocks of M; so M is conjugate to ⨁_{i} P_{i}. For each i let the class type of the block B_{i} be d_{i}^{h_{i}}. By Proposition 3.8, for any scalars ϑ_{i} that we choose, there exist matrices X_{1}, …, X_{s} such that X_{i} ∈ Cent B_{i} for all i, and det X_{i} = ϑ_{i}^{h_{i}}. Thus M commutes with a conjugate of the matrix diag(X_{1}, …, X_{s}), which has determinant ∏_{i} ϑ_{i}^{h_{i}}.

It will therefore be enough to show that there exist non-zero scalars ϑ_{1}, …, ϑ_{s} such that ∏_{i} ϑ_{i}^{h_{i}} = ζ^{k}. But we know that k = hcf(h_{1}, …, h_{s}), and so there exist integers a_{i} such that k = ∑_{i} a_{i}h_{i}; it follows that we can simply take ϑ_{i} = ζ^{a_{i}} for all i.

We now turn to the second step in the proof of Theorem 3.1.

Proposition 3.10. Let M be a matrix with part-size invariant k. The determinant of an element of Cent M is a k-th power in F_{q}.

Proof. Let M act on V = F_{q}^{n}. For each irreducible polynomial f over F_{q} which divides the minimal polynomial of M, let V_{f} be the largest subspace of V on which f(M) acts nilpotently. Then V = ⨁ V_{f}, and each summand V_{f} is invariant under Cent M. It follows that if Y ∈ Cent M then det Y = ∏ det Y_{f}, where Y_{f} is the restriction of Y to V_{f}. Therefore, it will be sufficient to show that det Y is a k-th power for each f.

Let λ = (h_{1}, …, h_{s}) be the partition associated with a given f in the rational canonical form of M. From the definition of the part-size invariant, each of the parts h_{i} is divisible by k. Let M_{f} be the restriction of M to V_{f}, and let Y_{f} ∈ Cent M_{f}.

By Proposition 2.2, f(M_{f}) is nilpotent with associated partition dλ, where d is the degree of f. It is clear, then, that the part-size invariant of f(M_{f}) is k. Since Y_{f} is in the centralizer of f(M_{f}), it follows from Proposition 3.6 that det Y_{f} is a k-th power in F_{q}, as required.

Combining the results of Propositions 3.9 and 3.10 gives Theorem 3.1.

3.3. Determinants in classes of a given type

It is natural to ask which determinants are represented among matrices of a given type. This question leads to a hard problem in arithmetic combinatorics, to which we have been able to find only a partial solution.

It is clear that if T is a type representable over the field F_{q}, then there is a matrix of type T with zero determinant if and only if T has a primary component 1^{λ} for some λ. This leaves us to decide which non-zero determinants can arise. For primary types this question is easily answered.

Lemma 3.11. Let λ be a partition of k ∈ N, let d ∈ N, and let ϑ ∈ F_{q}^{×}. There is an invertible matrix over F_{q} with type d^{λ} and determinant ϑ if and only if ϑ is a k-th power in F_{q}^{×}.

Proof. If M is a matrix of type d^{λ} then M has characteristic polynomial f^{k}. The determinant of M is therefore a k-th power. That every k-th power in F_{q}^{×} is obtained in this way follows easily from Lemma 3.7.

The following pair of propositions establish a sufficient condition on a type for it to represent all non-zero determinants.

Proposition 3.12. Let d ∈ N be coprime with q − 1, and let T = d^{λ_{1}} ⋯ d^{λ_{t}} be a type representable over F_{q}. If L = |λ_{1}| + ⋯ + |λ_{t}| is also coprime to q − 1, then every element of F_{q}^{×} is the determinant of a matrix of type T.

Proof. It is an easy consequence of Lemma 3.7 that if d is coprime with q − 1, then there are the same number of irreducible polynomials of degree d with any non-zero constant term. It follows that, for a generator ϑ of the cyclic group F_{q}^{×}, there exists a permutation σ of the set of irreducible polynomials of degree d, such that f^{σ}(0) = ϑf(0) for all f.

Let C be a similarity class of type T, whose members have determinant α. Consider the class C′ obtained from C by applying the permutation σ to the irreducible polynomials which appear in its cycle type. It is easy to see that C′ has the same type as C, and that the members of C′ have determinant αϑ^{L}, where L is as in the statement of the proposition. Now ϑ^{L} is a generator of F_{q}^{×} since L is coprime with q − 1, and so it is clear that by repeated applications of the permutation σ we can obtain any non-zero determinant of our choice.

Proposition 3.13. Let T be a type representable over a finite field F_{q}. For each d let L_{d} be the sum of the sizes of the partitions associated with the components of degree d in T. If dL_{d} is coprime with q − 1 for any d, then every element of F_{q}^{×} is a determinant of a matrix of type T.

Proof. This follows immediately from Proposition 3.12.

It should be noted that Proposition 3.13 does not come close to giving a necessary condition for a type to contain all non-zero determinants. Finding conditions which are both necessary and sufficient appears to be a highly intractable problem.

A special case of considerable interest is that of linear types, of the form 1^{λ_{1}} ⋯ 1^{λ_{t}}. (These are precisely the types of triangular matrices over F_{q}.) We make use of the following definition.

Definition. Let A be an abelian group of order m (written multiplicatively) and let π = (π_{1}, …, π_{m}) ∈ Z^{m}. We say that an element x ∈ A is π-expressible if there exists an ordering g_{1}, …, g_{m} of the elements of G such that x = g_{1}^{π_{1}} ⋯ g_{m}^{π_{m}}.

The relevance of this definition to our problem is easily explained. Let T be the linear type 1^{λ_{1}} ⋯ 1^{λ_{t}} where t ≤ q − 1. Let π ∈ Z^{q−1} be defined by

π = (|λ_{1}|, …, |λ_{t}|, 0, …, 0).

Then we observe that the non-zero determinants represented in T are precisely the π-expressible elements of F_{q}^{×}.

If A is an abelian group of exponent n then we observe that adding multiples of n to the entries of π does not affect π-expressibility in A; we may therefore assume that all of the entries of π satisfy 0 ≤ π_{i} ≤ n − 1. Similarly, reordering the entries of π cannot affect π-expressibility, and so we may suppose that they appear in decreasing order.

Numerical evidence obtained by the authors supports the following conjecture.

Conjecture 3.14. Let A be a cyclic group of order m. Let π = (π_{1}, …, π_{m}) ∈ (Z/mZ)^{m}, where π_{1} ≥ ⋯ ≥ π_{m}. Let π′ be the partition obtained from π by subtracting π_{m} from each part (thereby ensuring that the last part is 0). Then every element of A is π-expressible unless one of the following holds:

(i) π′ = (m − r, r, 0, …, 0) for some r, or (ii) There exists an integer p > 1 which divides each part of π′, and which also divides m.

This conjecture is known to be true in the case that m is a prime (see [4, Theorem 1.2]). For our purposes, we would like it to be true for A = F_{q}^{×} for all q; that is, whenever m + 1 is a power of a prime. This would provide a complete classification of the determinants occurring in linear types. In the very special case when q = 2^{r} and |F_{q}^{×}| = 2^{r} − 1 is a Mersenne prime, the result of [4] already gives such a classification.

4. Commuting nilpotent classes

In §2 the question of which similarity classes of matrices over a finite field commute was reduced to the analogous problem for nilpotent classes. The question of which nilpotent classes commute with a given nilpotent class N(λ) appears to be a very hard problem, and we shall not attempt to answer it in any generality. We shall, however, treat a variety of special cases, and make a number of observations which, so far as we have been able to determine, do not appear in the existing literature. Our approach is elementary, and leads to results which, for the most part, apply to matrices defined over an arbitrary field. (For some other recent results on the problem of commuting nilpotent classes over algebraically closed fields, obtained by the methods of Lie theory, the reader is referred to [10] and [13].)

Our results may be summarized as follows. Proposition 4.1 describes the nilpotent classes that commute with N(λ) when λ has a single part. This result has appeared previously in [12]; our Proposition 4.2 is similar to, but slightly stronger than, the result which appears there as Proposition 2.

Similarly, we deal in Proposition 4.4 with the case that λ = (n − 1, 1) for some n, and in Proposition 4.5 with the case that λ = (2, …, 2). Using these results we are able to classify those nilpotent classes that commute with every nilpotent class of the same dimension; this is Theorem 4.6.

We next establish a condition for the nilpotent classes N(n, n) and N(n + 1, n − 1) to commute; these classes are found to commute over any infinite field, and over the finite field F_{p^{r}} provided that p(p^{2r} − 1)/e does not divide n, where e = 1 if p = 2 and e = 2 otherwise. As well as augmenting our list of commuting classes, this result is particularly significant, since it demonstrates that commuting of classes is in some cases dependent on the field of definition. Finally, we use the results just mentioned to classify those commuting nilpotent classes whose associated partitions have no more than two parts; this result, stated as Theorem 4.10, is valid over any field.

The following definition will be useful in what follows.

Definition. Let M be a nilpotent transformation of a space V. A cyclic basis for M is a basis B of V with the property that for each v ∈ B, either vM = 0, or else vM ∈ B.

Earlier in §3.1 we defined a cyclic vector for M to be a vector which is not in the image of M. Let M ∈ N(h_{1}, …, h_{k}), and let B be a cyclic basis for M. Then B contains cyclic vectors v_{1}, …, v_{k}, where ht v_{i} = h_{i} for all i; in fact

B = {v_{i}M^{j} | 1 ≤ i ≤ k, 0 ≤ j < h_{i}}.

By Lemma 3.3, an element of Cent M is determined by its action on v_{1}, …, v_{k}.

4.1. Cyclic nilpotent classes and partition refinements

Recall that J(λ) is the unique upper-triangular matrix in Jordan form in the similarity class N(λ). The next proposition describes which classes commute with J(n). We shall use the well-known fact that the elements of the centralizer algebra Cent J(n) are the polynomials in J(n)—see for example [9, Ch. III, Corollary to Theorem 17].

Proposition 4.1. Let λ = (h_{1}, …, h_{k}) be a partition of n. Then J(n) commutes with a conjugate of J(λ) if and only if h_{1} − h_{k} ≤ 1.

Proof. Write E_{i} for the matrix whose (x, y)-th entry is 1 if k = y − x, and 0 otherwise. The matrices E_{0}, E_{1}, …, E_{n−1} form a basis for the centralizer algebra of J(n). Let M be a non-zero nilpotent element of this algebra; then for some d in the range 0 < d ≤ n − 1 we can write

M = ∑_{i≥d} α_{i}E_{i},

for scalars α_{i}, with α_{d} ≠ 0.

It is easy to check that null M^{s} = min(sd, n) for all integers s. Let h be the least integer such that hd ≥ n. Then it follows from Lemma 2.1 that M is conjugate to J(λ), where

λ = (h, …, h, h − 1, …, h − 1)

is the partition with n − hd parts of size h − 1 and (h + 1)d − n parts of size h. This establishes the proposition.

The terminology in the first of the following definitions is borrowed from [10, §3].

Definition. A partition is almost rectangular if its largest part differs from its smallest part by at most 1.

Definition. Let λ and μ be partitions. We say that μ is a refinement of λ if μ is the disjoint union of subpartitions whose sizes are the parts of λ. We say that a refinement of λ is almost rectangular if all of the subpartitions involved are almost rectangular.

For instance, (5, 3, 1) = (3 + 2, 2 + 1, 1) has (3, 2, 2, 1, 1) as an almost-rectangular refinement. It is worth noting that while the relation given by “μ is a refinement of λ” is clearly transitive, the relation given by “μ is an almost rectangular refinement of λ” is not.

Proposition 4.2. Let μ_{1} and μ_{2} be partitions of n. If there exists a partition λ which has both μ_{1} and μ_{2} as almost rectangular refinements, then the conjugacy classes represented by the Jordan blocks J(μ_{1}) and J(μ_{2}) commute.

Proof. Consider the subpartitions ν_{1} of μ_{1} and ν_{2} of μ_{2} whose parts combine to create a single part of λ of size h. Since ν_{1} and ν_{2} are almost rectangular, they yield Jordan blocks whose classes commute with that of J(h). But the centralizer of J(h) consists of polynomials in J(h), and it follows that the classes of J(ν_{1}) and J(ν_{2}) have representatives which are polynomials in J(h). So these representatives commute, and hence J(μ_{1}) and J(μ_{2}) have conjugates which commute.

The preceding proposition is slightly more general than [12, Proposition 2], which states that the nilpotent classes N(λ) and N(μ) commute if μ is an almost rectangular refinement of λ. It is noted in [12] that there exist examples of classes commuting that cannot be explained in this way. We remark that our Proposition 4.2 does not account for all commuting between classes, either. We illustrate this fact with the example and the proposition below; other examples will be seen in subsequent sections.

Example. There is no partition which has both (2, 2) and (3, 1) as an almost rectangular refinement, but the classes N(2, 2) and N(3, 1) commute over any field. We leave the proof of this to the reader, while remarking that it is a special case of any one of Propositions 4.4, 4.5 and 4.7 below.

Proposition 4.3. Let λ be a partition, and let \overline{λ} be its conjugate partition. Then the nilpotent classes with partitions λ and \overline{λ} commute.

Proof. Let λ = (h_{1}, …, h_{k}), where h_{1} ≥ ⋯ ≥ h_{k}. Let N be nilpotent of type λ, and let u_{1}, …, u_{k} be cyclic vectors for N, such that u_{i} has height h_{i} for all i. By Lemma 3.3 there is a unique matrix M ∈ Cent N such that u_{i}M = u_{i+1} for all i, with u_{k}M = 0.

If λ = (5, 5, 3, 2), for instance, then the actions of N and M on the cyclic basis can be represented as follows:

N M
\xymatrix{u_{1} • \ar[r] & • \ar[r] & • \ar[r] & • \ar[r] & • & u_{1} • \ar[d] & • \ar[d] & • \ar[d] & • \ar[d] & • \ar[d] \\ u_{2} • \ar[r] & • \ar[r] & • \ar[r] & • \ar[r] & • & u_{2} • \ar[d] & • \ar[d] & • \ar[d] & • & • \\ u_{3} • \ar[r] & • \ar[r] & • &  &  & u_{3} • \ar[d] & • \ar[d] & • \\ u_{4} • \ar[r] & • &  &  &  & u_{4} • & •}

It is easy to check that M is nilpotent, with associated partition \overline{λ}.

In general there does not exist a partition which has both λ and \overline{λ} as almost rectangular refinements, as is shown by the example illustrating the proof above, or by the case λ = (4, 1, 1).

4.2. Universally commuting classes

The object of this section is to classify, in Theorem 4.6, the partitions to which the following definition refers.

Definition. A partition λ of n is universal with respect to a field K if N(λ) commutes with N(μ) over K for every partition μ of n.

The reference to the field in this definition is in fact redundant; it is a consequence of Theorem 4.6 that a partition which is universal with respect to one field is universal with respect to any field. To prove the theorem, we shall require the following two propositions.

Proposition 4.4. Let λ be a partition of n. The matrix J(n − 1, 1) commutes with a conjugate of J(λ) if and only if one of the following holds:

(i) λ has a part of size 1, and if λ^{−} is obtained from λ by removing this part, then J(n − 1) commutes with a conjugate of J(λ^{−}); Proposition 4.1 provides a classification in this case.

(ii) n is even, and all of the parts of λ are of size 2.

(iii) λ has a part of size 3, and its other parts are of size 1 or 2, with at least one part of

size 1. (iv) n = 3 and λ = (3).

Proof. The centralizer algebra of J(n − 1, 1) has the basis

{E_{i} | 0 ≤ i ≤ n − 2} ∪ {F, G, H},

where

∑_{i} α_{i}E_{i} + βF + γG + δH = ( \begin{matrix}α_{0} & α_{1} & α_{2} & … & α_{n−2} & β \\ 0 & α_{0} & α_{1} & & α_{n−3} & 0 \\ 0 & 0 & α_{0} & & α_{n−4} & 0 \\ ⋮ & & & ... & & ⋮ \\ 0 & 0 & 0 & & α_{0} & 0 \\ 0 & 0 & 0 & … & γ & δ \end{matrix} ) .

A nilpotent element of this algebra must have α_{0} = δ = 0. We suppose that M is such an element, and that M is non-zero. By Lemma 2.1 the partition λ associated with M is determined by the sequence of ranks of powers of M.

If α_{i} = 0 for all i < n − 2, then α_{n−2}, β and γ are the only entries that are possibly nonzero. It is easy to see that the rank sequence (rank I, rank M, rank M^{2}, rank M^{3}) must be either (n, 2, 1, 0) or (n, 1, 0, 0). In the first case the partition associated with M is (3, 1^{n−3}), which is covered by either part (iii) or part (iv) of the lemma. In the second case the partition is (2, 1^{n−2}), which is covered by part (i) or part (ii).

Now suppose that there exists i < n − 2 such that α_{i} ≠ 0. Let m be the least such i. If m < (n − 2)/2 then it is not hard to see that the rank sequence is

(n, n − m − 1, n − 2m − 1, …, 0).

The partition λ given by this data has one more part of size 1 than the partition λ^{−} given by the data

(n − 1, n − m − 1, n − 2m − 1, …, 0).

But λ^{−} corresponds to the rank sequence for an element of the centralizer algebra of J(n − 1), and so this case is covered by case (i) of the lemma. If m > (n − 2)/2 then the same situation occurs if βγ = 0. But if β and γ are both non-zero then the rank sequence obtained is (n − m − 1, 1, 0). The corresponding partition λ is covered by part (iii) of the lemma.

The final case to analyse occurs when n is even and m = (n − 2)/2. If M^{2} ≠ 0 then the situation of the previous paragraph applies. Otherwise the rank sequence is (n, n/2, 0) and all of the parts of λ have size 2, as in part (ii) of the lemma.

Proposition 4.5. Let λ be the partition of 2s which has s parts of size 2, and let μ be any partition of 2s. Then J(λ) commutes with a conjugate of J(μ).

Proof. By a straightforward inductive argument, we may suppose that μ has no subpartition of even size. If μ has only one part then the result follows from Proposition 4.1; so we may assume that μ has exactly two parts, s + t and s − t.

A cyclic basis for N = J(λ) has the form B = {e_{1}, …, e_{s}, f_{1}, …, f_{s}}, where the vectors f_{i} are in the kernel of N, and e_{i}N = f_{i} for all i. Let M be the matrix whose action is defined by e_{i}M = e_{i+1}, f_{i}M = f_{i+1} for 1 ≤ i < s, and

e_{s}M = \begin{cases}f_{s−t+1} & if t > 0, \\ 0 & otherwise, \end{cases}
f_{s}M = 0.

It is easy to see that M commutes with N, hence it suffices to show that M ∈ N(μ). A basis for ker M is given by {f_{s}, e_{s} − f_{s−t}}, so null M = 2. It follows that the partition associated with M has two parts, and since e_{1} is a cyclic vector of height s + t, this partition must be (s + t, s − t), as required.

Theorem 4.6. The universal partitions are precisely those with no part greater than 2, together with λ = (3).

Proof. Suppose that λ has no part of size greater than 2. If all of the parts of λ have size 2, then J(λ) commutes with all nilpotent classes, by Proposition 4.5. Otherwise λ has a subpartition λ_{m} of m for every m ≤ n. Let μ be a partition of n with largest part m. Then since λ_{m} is an almost rectangular refinement of m, it follows from Proposition 4.1 that J(λ_{m}) commutes with a conjugate of J(m). Now if λ′ denotes the partition obtained by deleting the parts of λ_{m} from λ, and if μ′ is obtained by deleting a part of size m from μ, then we may suppose inductively that J(λ′) commutes with a conjugate of J(μ′). It follows that J(λ) commutes with a conjugate of J(μ).

Conversely, suppose that λ has largest part h > 2. If J(λ) commutes with J(n) then by Proposition 4.1 all of its parts have size h or h − 1. Then we see from Proposition 4.4 that J(λ) does not commute with a conjugate of J(n − 1, 1), except in the single case that λ = (3).

4.3. Commuting of classes N(n, n) and N(n + 1, n − 1)

The main object of this section is to prove Proposition 4.7 below, which gives a necessary and sufficient condition for the classes N(n, n) and N(n + 1, n − 1) to commute. This case is of particular interest because the field enters in an essential way. In Theorem 4.8 we use this proposition to show that for every prime p and positive integer r, there exists a pair of classes of nilpotent matrices which commute over the field F_{p^{r}} if and only if s > r.

Proposition 4.7 is motivated by a natural construction on matrices. Suppose that X and Y are commuting matrices over a field K, and let

D = (\begin{matrix}X & 0 \\ 0 & X \end{matrix}) , E = (\begin{matrix}Y & I \\ 0 & Y \end{matrix} ) .

Clearly the matrices D and E commute. We may assume that X and Y (and hence D and E) are nilpotent; then this construction (and other similar ones) may in principal be used to find new cases of commuting nilpotent classes. The partition labelling the class of D is clearly 2λ, where λ labels the class of X. The partition labelling the class of E is harder to calculate, and depends on the characteristic of K.

We have no occasion to make systematic use of this construction in the present paper, but the following example is illustrative. Let X = Y = J(n). Then D ∈ N(n, n). The partition labelling the class of E is (n + 1, n − 1) except in the case that char K divides n, in which case it is (n, n). It follows that N(n, n) and N(n + 1, n − 1) commute over fields of all but finitely many characteristics, the exceptions being the prime divisors of n. We note, however, that the present method gives no information about whether the classes commute in fields of these exceptional characteristics; this gives an indication that the following proposition is non-trivial.

Proposition 4.7. Let p be a prime, and let

e = \{ \begin{matrix}1 & if p = 2, \\ 2 & otherwise. \end{matrix}

Then the nilpotent types (n, n) and (n + 1, n − 1) commute over F_{p^{r}} if and only if n is not divisible by p(p^{2r} − 1)/e.

Proof. Let M be nilpotent of type (n + 1, n − 1), acting on a space V over F_{p^{r}}. Take a cyclic basis {u_{i}, w_{j} | 0 ≤ i ≤ n, 1 ≤ j ≤ n − 1} for V, with u_{i}M = u_{i−1} and w_{j}M = w_{j−1} for all i and j. Let U_{k} and W_{k} denote the subspaces ⟨u_{j} | 0 ≤ j ≤ k⟩ and ⟨w_{j} | 1 ≤ j ≤ k⟩ respectively—we take W_{0} = {0} and W_{n} = W_{n−1}. Let V_{k} denote U_{k} ⊕ W_{k} for all k. For each pair (x, y) with x ∈ V_{n−1} and y ∈ V_{n−2}, there is an unique nilpotent element Y of Cent M such that u_{n}Y = x and w_{n−1}Y = y; it follows from Lemma 3.3 that all of the nilpotent elements of Cent M can be obtained in this way.

Let Y ∈ Cent M be nilpotent, and define α, β, γ, δ by

u_{n}Y ∈ αu_{n−1} + γw_{n−1} + V_{n−2},
w_{n−1}Y ∈ βu_{n−2} + δw_{n−2} + V_{n−3}.

The reader may find helpful the following diagrammatic representation of Y.

[figure: {u_{n} \atop •}; α; γ; {u_{n−1} \atop •}; α; γ; {u_{n−2} \atop •}; α; γ; {u_{n−3} \atop •}; {u_{3} \atop •}; α; γ; {u_{2} \atop •}; α; γ; {u_{1} \atop •}; α; {u_{0} \atop •}; {• \atop w_{n−1}}; δ; β; {• \atop w_{n−2}}; δ; β; {• \atop w_{n−3}}; {• \atop w_{3}}; δ; β; {• \atop w_{2}}; δ; β; {• \atop w_{1}}; β]

The matrix A = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) describes the maps induced by Y,

\overline{Y}_{k} : \frac{V_{k}}{V_{k−1}} ⟶ \frac{V_{k−1}}{V_{k−2}} ,

where k is in the range 1 < k < n. Outside of this range, the map \overline{Y}_{n} has domain ⟨ u_{n} + V_{n−1} ⟩ of dimension 1, while \overline{Y}_{1} has codomain ⟨u_{0}⟩ of dimension 1. These maps are represented by the first row and the first column of A respectively.

Claim. The kernel of Y has dimension 2 if and only if A is non-singular.

Proof of Claim. If A is invertible, then the maps \overline{Y}_{k} are injective for k > 1. It follows easily that if v ∈ ker Y then v ∈ V_{1}. It is now easy to check that v ∈ ⟨u_{1}, βu_{2} − αw_{1}⟩ and so null Y = 2 in this case.

Conversely, suppose that A is singular. If α = β = 0 then V_{1} ⊆ ker Y, and so null Y ≥ 3. So let us suppose that α and β are not both 0. Then there exists z ∈ V_{1} such that zY = u_{0}. Since A is singular, the map \overline{Y}_{2} has a non-trivial kernel, and it follows that there exists v ∈ V_{2} \ V_{1} such that vY ∈ V_{0} = ⟨u_{0}⟩. Say that vY = σu_{0}; now we have a set of three kernel vectors, {u_{0}, βu_{1} − αw_{1}, v − σz}, which is linearly independent since v − σz ∉ V_{1}. So null Y ≥ 3 in this case as well.

The dimension of ker Y tells us the number of parts in the partition associated with the class of Y. This partition therefore has two parts if and only if the matrix A is non-singular. Note that since Y^{n+1} = 0, no part can be larger than n + 1, and therefore the only possible partitions are (n + 1, n − 1) and (n, n). The former corresponds to the class of M itself, while the latter case occurs when Y^{n} = 0, which is the case if and only if u_{n} ∈ ker Y^{n}.

Now we observe that

u_{n}Y^{n} = \overline{Y}_{1} ∘ \overline{Y}_{2} ∘ ⋯ ∘ \overline{Y}_{n}(v_{n} + V_{n−1})
= u_{0}R_{1}A^{n−2}C_{1} ,

where R_{1} and C_{1} are, respectively, the first row and the first column of A. So the partition of Y is (n, n) precisely when R_{1}A^{n−2}C_{1} = (0), or equivalently, when the matrix A^{n} has a zero for its top left-hand entry.

Claim. Every element of GL_{2}(F_{p^{r}}) is either a scalar matrix, or else is conjugate to a matrix with a zero for its top left-hand entry.

Proof of Claim. Every quadratic polynomial over F_{p^{r}} is the characteristic polynomial of a unique similarity class of non-scalar matrices. Thus if X is a non-scalar matrix with characteristic polynomial x^{2} + σx + τ, then X is conjugate to

( \begin{matrix}0 & 1 \\ −τ & −σ \end{matrix} ),

as required.

Now suppose that GL_{2}(F_{p^{r}}) contains a non-scalar element X which is an n-th power in the group. Then X has a conjugate X′ with a zero for its top left-hand entry. Clearly X′ is also an n-th power; by choosing a, b, c, δ to be the entries of an n-th root of X′, we can construct a matrix Y in Cent M whose type is (n, n).

There exist non-scalar n-th powers in GL_{2}(F_{p^{r}}) provided that n is not divisible by the exponent of PGL_{2}(F_{p^{r}}). This exponent is p(p^{2r} − 1)/e, and the proof of Proposition 4.7 is complete.

Remark. This argument also goes to show that the nilpotent types (n, n) and (n − 1, n + 1) commute over any infinite field K, since the exponent of PGL_{2}(K) is infinite.

Theorem 4.8. Let p be a prime, and r ≥ 1. There exist partitions λ and μ, such that N(λ) commutes with N(μ) over the fields F_{p^{a}} for a > r, but not for a ≤ r.

Proof. We use a famous theorem of Zsigmondy [15] which states that if k ≥ 2, t ≥ 3, and (k, t) ≠ (2, 6), then there is a prime divisor of k^{t} − 1 which does not divide k^{s} − 1 for any s such that 1 ≤ s < t.

Let L = lcm({p^{2s} − 1 | 1 ≤ s ≤ r}), and let n = pL/e. We observe that p(p^{2a} − 1)/e divides n whenever a ≤ r. When a > r we invoke Zsigmondy’s Theorem with (k, t) = (p, 2a), or with (k, t) = (4, 3) if p = 2 and t = 3; this tells us that p^{2a} − 1 has a prime divisor q which does not divide p^{2s} − 1 for s < a. Clearly q does not divide n, and so p(p^{2a} − 1)/e does not divide n. It now follows from Proposition 4.7 that the partitions (n, n) and (n + 1, n − 1) have the property stated in the theorem.

Remark. The authors have found no case where the commuting of nilpotent classes depends on the field of definition in dimension less than 12. This is the dimension of the smallest example given by Proposition 4.7: that of N(6, 6) and N(7, 5), which commute over every field except F_{2}.

4.4. Classes corresponding to two-part partitions

We end by establishing a result which, together with results already presented, will allow us to classify, over any field K, pairs of partitions (λ, μ) with at most two parts, such that N(λ) and N(μ) commute over K. We note that classes with at most 2 parts are precisely those whose elements have nullity at most 2.

Proposition 4.9. Let λ = (a, b) and μ = (c, d), where a + b = c + d and a > c ≥ d > b. If N(λ) and N(μ) commute over a field K then c = d and a − b = 2.

Proof. The case that c = d and a − b = 2 has been dealt with in Proposition 4.7 and the ensuing remark. We may therefore suppose that a − b > 2. Let M ∈ N(λ), and let {v, vM, …, vM^{a−1}, w, wM, …, wM^{b−1}} be a cyclic basis for M. Let W = ker M^{a−2}; so W is the span of all the basis vectors apart from v and vM. Suppose that Y is nilpotent and commutes with M; then it is not hard to see that W ⊆ ker Y^{a−2}. Since Y is nilpotent we have vY ∈ αvM + W for some α ∈ K.

Suppose first that α ≠ 0; then we see that vY^{a−1} = α^{a−1}vM^{a−1}, while vY^{a} = 0. Hence v is a cyclic vector for Y of height a. It follows that if the partition associated with Y has only 2 parts then it must be λ.

Suppose alternatively that α = 0, so vY ∈ W. We shall show that null Y ≥ 3, and so the partition associated with Y has more than 2 parts. First observe that vM^{a−2} and vM^{a−1} are in ker Y, since vM^{a−2}Y = vY M^{a−2} ∈ W M^{a−2} = {0}. Furthermore it is easy to show that vM^{a−3}Y and wM^{b−1}Y both lie in ⟨vM^{a−1}⟩, and hence a non-zero linear combination of these two vectors lies in ker Y. We have therefore found three linearly independent vectors in ker Y, as required.

The following theorem simply collects together elements of Propositions 4.1, 4.7 and 4.9; it requires no further proof.

Theorem 4.10. Suppose that λ and μ are partitions of n with at most two parts, and that N(λ) and N(μ) commute over a field K. Assume without loss of generality that the largest part of λ is at least as large as the largest part of μ. Then one of the following holds.

(i) λ = μ. (ii) n = 2m, λ = (n) and μ = (m, m). (iii) n = 2m, λ = (m + 1, m − 1), μ = (m, m) and, if K is finite then the exponent of

PGL_{2}(K) does not divide m. (iv) n = 2m + 1, λ = (n) and μ = (m + 1, m).

References

1. John R. Britnell, ‘Cyclic, separable and semisimple matrices in the special linear groups over a finite field’, J. London Math. Soc. (2) 66 (2002) 605–622.

2. John R. Britnell and Mark Wildon, ‘On the distribution of conjugacy classes between the cosets of a finite group in a cyclic extension’, Bull. London Math. Soc. 40 (5) (2008) 897–906.

3. John R. Britnell and Mark Wildon, ‘Commuting elements in conjugacy classes: An application of Hall’s Marriage Theorem’, J. Group Theory, to appear.

4. András Gács, Tamás Héger, Zoltán Lóránt Nagy and Dömötör Pálvölgyi, ‘Permutations, hyperplanes and polynomials over finite fields’, preprint.

5. Murray Gerstenhaber, ‘On nilalgebras and linear varieties of nilpotent matrices III’, Ann. of Math. 70 (1) (1959) 167–205.

6. Murray Gerstenhaber, ‘On dominance and varieties of commuting matrices’, Ann. of Math. 73 (2) (1961) 324–348.

7. Israel Gohberg, Peter Lancaster and Leiba Rodman, Invariant subspaces of matrices and applications (Wiley, New York, 1986).

8. J. A. Green, ‘The characters of the finite general linear groups’, Trans. Amer. Math. Soc. 80 (2) 1955 402–447.

9. Nathan Jacobson, Lectures in abstract algebra II: Linear Algebra (Van Nostrand, 1953)

10. Tomaž Košir and Polona Oblak, ‘On pairs of commuting nilpotent matrices’, Transformation Groups 14 (1) (2009) 175–182.

11. I. G. Macdonald, Symmetric functions and Hall polynomials (Second Edition, Oxford University Press, Oxford, 1995).

12. Polona Oblak, ‘The upper bound for the index of nilpotency for a matrix commuting with a given nilpotent matrix’, Linear and Multilinear Algebra 56 (6) (2008) 701–711.

13. Dmitri I. Panyushev, ‘Two results on centralisers of nilpotent elements’, J. Pure Appl. Algebra 212 (2008) 774–779.

14. R. Steinberg, ‘A geometric approach to the representations of the full linear group over a Galois field’, Trans. Amer. Math. Soc. 71 (1951) 274–282.

15. K. Zsigmondy, ‘Zur Theorie der Potenzreste’, Monatsh. für Math. u. Phys. 3 (1892) 265–284.

School of Mathematics University of Bristol University Walk Bristol BS8 1TW United Kingdom

j.r.britnell@bristol.ac.uk mark.wildon@bristol.ac.uk