Commuting Matrices Over Finite Fields

• Commuting matrices over finite fields are defined by conditions on partition invariants that determine when two matrices commute.
• The analysis leverages centralizer algebras where determinant and module theoretic invariants yield explicit arithmetic constraints and asymptotic formulas.
• Methodologies include classification via nilpotent Jordan forms, generating functions for enumeration, and graph-theoretic insights linking algebra and geometry.

Commuting matrices over finite fields represent a deep intersection of linear algebra, representation theory, and finite field arithmetic. The structure and classification of such matrices is governed by subtle invariants, intricate reduction techniques, and striking links with module theory and algebraic geometry. Central developments over the past century—culminating in detailed studies of similarity classes, branching rules, generating functions, and the geometry of commuting varieties—have resulted in both enumerative and structural theorems, many with explicit asymptotics and fine arithmetic sensitivity.

1. Structural Reduction to Nilpotent Classes

A core principle is that questions about commuting similarity classes and class types (in the sense of Steinberg and Green) over finite fields reduce to questions about nilpotent classes, which are naturally parametrized by partitions (Britnell et al., 2010). For two matrices $X$, $Y$ of class types $S$ and $T$, polynomial relations enable a reduction to nilpotent representatives if the types are primary. More precisely, for $S = c^{(\lambda)}$ and $T = d^{(\mu)}$, with $h = \gcd(c,d)$ and $\ell = \operatorname{lcm}(c,d)$, the types commute over $\mathbb{F}_q$ provided the partitions $\lambda$ and $\mu$ satisfy divisibility conditions derived from $h$ and $\ell$ and the associated nilpotent classes $N(h\lambda)$ and $N(h\mu)$ commute.

This reduction is significant because partition data alone become sufficient to determine whether representatives with prescribed types can commute, bypassing the need for full rational canonical forms.

2. Centralizer Algebras and the Part-Size Determinant Invariant

A foundational result describes the determinants realized by the centralizer algebra $\operatorname{Cent}(M)$ of a matrix $M \in \operatorname{Mat}_n(\mathbb{F}_q)$ (Britnell et al., 2010). If $M$ has part-size invariant $k$—the highest common factor of all Jordan block sizes—then every determinant in $\operatorname{Cent}(M)$ is a $k$-th power in $\mathbb{F}_q$: specifically,

$$\{\det(Y) \mid Y \in \operatorname{Cent}(M)\} = \{z^k : z \in \mathbb{F}_q\}.$$

Part-size invariants thus control the available determinants in centralizers, and every $k$-th power arises as a determinant. The module-theoretic analysis, particularly via composition factors aligned with block heights, ensures the divisibility of determinants and aligns group-theoretic centralizing subgroups with arithmetic invariants.

3. Analysis and Classification of Commuting Nilpotent Classes

Nilpotent matrices in $\operatorname{Mat}_n(\mathbb{F}_q)$ are classified by partitions of $n$ and Jordan type. Several criteria determine when nilpotent classes commute:

• Proposition 4.1 (Britnell et al., 2010): A nilpotent Jordan block $J(n)$ commutes with a conjugate of any nilpotent matrix with parts differing by at most 1 (i.e., “almost rectangular” partitions).
• Proposition 4.2: If two partitions $\lambda, \mu$ admit a common refinement into almost rectangular pieces, then $N(\lambda)$ and $N(\mu)$ commute.
• Field-dependent phenomena emerge: For pairs like $(n,n)$ and $(n+1,n-1)$, commute properties depend on arithmetic in $\mathbb{F}_{p^a}$, specifically on divisibility by factors proportional to $p(p^{2a} - 1)$ [(Britnell et al., 2010), Prop 4.7]. For every prime $p$ and integer $r$, corresponding pairs commute over $\mathbb{F}_{p^a}$ if and only if $a > r$.

Such results show that commutativity in nilpotent classes, while predominately partition-governed, can sharply exhibit field dependence in special cases.

4. Classification Frameworks and Universally Commuting Classes

• Theorem 4.6 (Britnell et al., 2010): “Universally commuting” nilpotent classes—those that commute with all nilpotent classes of the same dimension—are exactly those partitions with no parts larger than 2, plus the single case $(3)$.
• Theorem 4.10 provides a complete classification for commuting pairs of nilpotent classes labeled by two-part partitions. For $n = 2m$, pairs like $(m+1,m-1)$ and $(m,m)$ may commute under further field restraints; for $n = 2m+1$, variations such as $(m+1, m)$ arise.

Detailed cyclic basis analysis and partition refinement underpin these theorems, which delineate explicit commutative cases among all possible nilpotent types.

5. Simultaneous Similarity, Enumeration, and Generating Functions

The classification of commuting tuples $(A_1,\ldots,A_k)$ up to simultaneous similarity is equivalent to classification of $\mathbb{F}_q[x_1,\ldots,x_k]$-modules of dimension $n$ (Sharma, 2014). The number $c_{n,k}(q)$ of isomorphism classes is captured by rational generating functions whose coefficients are polynomials in $q$ with non-negative integer coefficients, verified for $n \leq 4$. For example,

$$h_3(t) = \frac{1 + q^2 t^2}{(1 - qt)(1 - q^2 t)(1 - q^3 t)}.$$

Branching matrices are utilized for recursive enumeration, and explicit count formulas arise from rational canonical form analysis combined with branching rules—illuminated for matrices up to size $n = 4$, with conjectured generalization.

6. Graph-Theoretic and Functorial Perspectives

Studies of the commuting graph $\Gamma(\operatorname{Mat}_n(F))$ for fields $F$ with finite algebraic closure show that for $n \geq 3$ the graph is connected with diameter four (Miguel, 2015). Vertices correspond to noncentral matrices, and edges represent commutation. Key structural matrices (nontrivial idempotents, nonzero nilpotents) suffice to connect any two noncentral matrices through paths of length at most four.

Compressed commuting graphs further refine this viewpoint by identifying vertices generating the same subring, yielding a functor from rings to graphs (Boroja et al., 2023). In finite fields, the compressed graph is a complete graph indexed by divisors of the field degree, and, for $2\times2$ matrix rings, is a union of complete graphs determined by minimal polynomial types and centralizer structure.

7. Enumeration, Asymptotics, and Product Expansions

Feit and Fine's generating function for commuting pairs is a central result (Fulman et al., 2016), further refined by detailed product-expansion-based asymptotics (Bringmann et al., 19 Sep 2025):

$$\sum_{n=0}^\infty \frac{Q_{\mathbb{F}_{p^r}}(n)}{p^{rn^2} f_{\mathbb{F}_{p^r}}(n)} w^n = \prod_{l \geq 1, j \geq 0} \frac{1}{1 - p^{r(1-j)} w^l},$$

with $$f_{\mathbb{F}_{p^r}}(n) = \prod_{j=1}^n (1 - p^{-rj})$$.

The main term in the asymptotic expansion is

$$Q_{\mathbb{F}_{p^r}}(n) = p^{r(n^2+n)} \prod_{j \geq 1} (1-p^{-rj})^{-j} + O(p^{rn^2 + (r/2)n}),$$

thus confirming the dimension ($n^2 + n$) heuristic of commuting matrix varieties. Laurent expansions, removable singularity methods, and modular q-series identities are key elements, with possible parallels to the theory of partitions and Cohen–Lenstra heuristics.

8. Comparative Insights and Relation to Broader Theory

The classification and enumeration of commuting matrices over finite fields complement and extend work on commuting nilpotent matrices by Košir and Oblak, especially via expanded analysis of class types, determinant invariants, and field dependence (Britnell et al., 2010). Module-theoretic approaches (e.g., Gerstenhaber and Wadsworth's results on algebra size vs. module dimension (Bergman, 2013)) offer algebraic context, with explicit confirmation for the two-generator case and open questions for higher generator sets.

Compression-based functorial techniques (Boroja et al., 2023) and nearly commuting varieties (Kadyrsizova, 2017) further illuminate singularity and geometry, while enumeration in solvable groups—upper triangular and unitriangular—provides explicit formulas for simultaneous conjugacy classes and commuting probabilities, central for probabilistic and combinatorial group theory (Kaur et al., 2020).

---

This synthesis illustrates that commuting matrices over finite fields exhibit rich combinatorial, algebraic, and geometric structures governed by partition data, centralizer properties, and generating functions. Sharp field-dependence, precise asymptotics, and extensive classification theorems await further research, particularly regarding multidimensional enumeration, recursive formulas, and geometric invariants arising from commuting varieties.