On matrices commuting with their Frobenius

The Frobenius of a matrix $M$ with coefficients in $\bar{\mathbb F}_p$ is the matrix $σ(M)$ obtained by raising each coefficient to the $p$-th power. We consider the question of counting matrices with coefficients in $\mathbb F_q$ which commute with their Frobenius, asymptotically when $q$ is a large power of $p$. We give answers for matrices of size $2$, for diagonalizable matrices, and for matrices whose eigenspaces are defined over $\mathbb F_p$. Moreover, we explain what is needed to solve the case of general matrices. We also solve (for both diagonalizable and general matrices) the corresponding problem when one counts matrices $M$ commuting with all the matrices $σ(M)$, $σ^2(M)$, $\ldots$ in their Frobenius orbit.

💡 Research Summary

The paper investigates the enumeration of n × n matrices over a finite field 𝔽_q (where q is a power of a fixed prime p) that commute with their Frobenius twist σ(M) (the matrix obtained by raising each entry to the p‑th power). Four families of matrices are considered:

1. X(𝔽_q) – all matrices M with M σ(M)=σ(M) M;
2. X_diag(𝔽_q) – the subset of X consisting of diagonalizable matrices (i.e. semisimple over the algebraic closure);
3. X_∞(𝔽_q) – matrices whose entire Frobenius orbit {M, σ(M), σ²(M), …} consists of pairwise commuting elements;
4. X_diag_∞(𝔽_q) – the intersection of X_∞ with the diagonalizable matrices.

The main goal is to determine the asymptotic size of each set as q→∞ while p and n remain fixed. The authors employ the Lang–Weil estimate, which reduces the problem to computing the dimension and the number of top‑dimensional irreducible components of suitable constructible subsets of the affine space M_n(𝔽_p).

Diagonalizable matrices (X_diag).
For a diagonalizable matrix M over 𝔽_p, let E_λ be the eigenspace for eigenvalue λ. The dimensions of the intersections E_λ∩σ(E_μ) are recorded as the number of directed edges λ→μ in a quiver Q_M. The condition Mσ(M)=σ(M)M is equivalent to Q_M having exactly n edges and being balanced (in‑degree equals out‑degree at each vertex). The set of isomorphism classes of such balanced quivers with n edges is denoted Bal_n; it is finite. For each Q∈Bal_n the authors define a constructible set X_diag^Q consisting of matrices whose associated quiver is isomorphic to Q. By describing X_diag^Q as the image of a regular map Y_Q×Z_Q→M_n(𝔽_p) (where Y_Q parametrises distinct eigenvalues and Z_Q encodes the subspace data via Grassmannians), they compute dim X_diag^Q. The maximal dimension occurs for two families of quivers, leading to the dominant term

|X_diag(𝔽_q)| = c_diag(p,n)·q^{⌊n²/3⌋+1} + O(q^{⌊n²/3⌋+½})

with explicit constants: c_diag(p,n)=p/2 for n=2, 2 for n=4, and 1 otherwise.

Full Frobenius orbit (X_∞).
If M∈X_∞, the algebra generated by its Frobenius orbit is commutative; when M is diagonalizable it is simultaneously diagonalizable, i.e. a torus of diagonal matrices. This forces a strong restriction on the possible dimensions of the orbit algebra, yielding a maximal dimension of ⌊n²/4⌋+1. Using classical results of Steinberg on commuting semisimple elements and of Schur on maximal tori, the authors obtain

|X_∞(𝔽_q)| = c_∞(p,n)·q^{⌊n²/4⌋+1} + O(q^{⌊n²/4⌋})

where c_∞(p,n) is a piecewise polynomial in p: for n=2, c_∞=p²+p+1; for n=3, a longer polynomial; for even n≥4, c_∞ = n·