Frobenius extensions about centralizer matrix algebras

Abstract: This paper investigates the conditions under which the centralizer algebra $S_n(c,R)$ of a matrix $c\in M_n(R)$ is a (separable) Frobenius extension of the base algebra $R$. For an algebra $R$ over an integral domain $\mathbb{k}$, we provide necessary and sufficient conditions for $S_n(c,R)/R$ to be a (separable) Frobenius extension when $c$ is in Jordan canonical form with eigenvalues in $\mathbb{k}$. We extend this analysis to arbitrary matrices over a field and derive conditions for matrix diagonalizability through Frobenius extensions.

• The paper establishes necessary and sufficient Jordan-block criteria for the centralizer algebra S_n(c,R) to form a Frobenius extension of R, extending results from fields to integral domains.
• It constructs Frobenius functionals block by block, using the identification of a single Jordan-block centralizer with a truncated polynomial algebra and analyzing when these local structures assemble globally.
• The paper proves that, over fields, the separable Frobenius property of the centralizer extension characterizes diagonalizable matrices, turning a ring-theoretic condition into a test for linear-algebraic structure.

Context and motivation

The paper studies the centralizer algebra $S_n(c,R)$ of a matrix $c \in M_n(R)$, that is, the subalgebra of all matrices commuting with $c$, and asks when the extension $S_n(c,R)/R$ is a Frobenius extension, and when it is a separable Frobenius extension. Frobenius extensions, introduced by Kasch [62724] and developed by Nakayama–Tsuzuku [108520], occupy a position between separable and free extensions and carry a rich duality theory with applications ranging from Witt groups to topological quantum field theories [2037238, 1690111]. Centralizer matrix algebras have been intensively studied recently for their representation-theoretic properties, including structure theorems [MR4241258], connections with symmetric polynomials of partitions [MR4461655], derived and stable equivalences (Li et al., 2023), and cellularity of centrosymmetric matrix algebras [MR4053642].

The immediate predecessor of this work is the short paper of Zhu [4632826], "On Frobenius extensions of the centralizer matrix algebras," which established Frobenius criteria in special cases. The present paper substantially generalizes that analysis: it works over an integral domain $\mathbb{k}$ rather than a field, treats matrices in Jordan canonical form with eigenvalues in $\mathbb{k}$, and then removes the Jordan-form hypothesis for matrices over a field. A distinctive feature is the use of Frobenius extension techniques to derive conditions for matrix diagonalizability, connecting the ring-theoretic property of the centralizer to the linear-algebraic structure of $c$.

Preliminaries and setup

Throughout, $R$ is an algebra over an integral domain $\mathbb{k}$. A Frobenius extension $S/R$ is an extension admitting an $c \in M_n(R)$0-bimodule isomorphism $c \in M_n(R)$1; equivalently, $c \in M_n(R)$2 is split and has a dual basis with a Frobenius homomorphism. Separability adds the requirement that the multiplication map splits as an $c \in M_n(R)$3-bimodule map. The paper relies on the standard characterization of Frobenius extensions via dual bases and on the fact that separable Frobenius extensions are precisely those Frobenius extensions that are separable as ring extensions. The centralizer algebra $c \in M_n(R)$4 is viewed as an $c \in M_n(R)$5-algebra via the scalar embedding of $c \in M_n(R)$6 into $c \in M_n(R)$7.

Main results

The core contribution is a necessary and sufficient condition for $c \in M_n(R)$8 to be a Frobenius extension when $c \in M_n(R)$9 is in Jordan canonical form with all eigenvalues in $c$0. The criterion is expressed in terms of the Jordan block decomposition of $c$1: writing $c$2 in block diagonal form with Jordan blocks $c$3, the centralizer decomposes into block matrices whose $c$4-blocks are homomorphisms between the corresponding Jordan blocks. The paper shows that $c$5 is Frobenius precisely when each Jordan block of $c$6 is of a size compatible with a Frobenius condition on the associated block algebra — concretely, when the centralizer admits a suitable Frobenius functional, which the authors verify block by block using the classical description of matrices commuting with a single Jordan block as polynomials in that block.

A key structural observation is that for a single Jordan block $c$7, the centralizer is the commutative algebra $c$8 (tensored with $c$9), which is a Frobenius algebra over $S_n(c,R)/R$0 with Frobenius functional extracting the coefficient of $S_n(c,R)/R$1. The difficulty lies in assembling these local Frobenius structures across multiple blocks of different sizes, and the paper identifies exactly when this gluing succeeds.

The separable case is treated separately: separability of $S_n(c,R)/R$2 imposes stronger requirements, essentially forcing the Jordan structure to be as degenerate as possible (blocks of size one, i.e., diagonalizability, in the field case). This yields the second main result, for arbitrary matrices over a field $S_n(c,R)/R$3: after passing to the algebraic closure or an extension over which $S_n(c,R)/R$4 admits a Jordan form, the Frobenius (respectively separable) property of the centralizer extension is invariant under similarity, and the criteria reduce to conditions on the Jordan form of $S_n(c,R)/R$5. The paper derives from this a characterization of diagonalizable matrices: $S_n(c,R)/R$6 is diagonalizable over $S_n(c,R)/R$7 if and only if the corresponding centralizer extension satisfies the separable Frobenius condition. This is a notable claim, as it inverts the usual direction of reasoning — rather than computing a centralizer from a known matrix structure, the Frobenius property detects the matrix structure.

Relation to prior work

Compared with Zhu's result [4632826], which handled Frobenius extensions of centralizer algebras in a restricted setting, the present paper extends the integral base from a field to an arbitrary integral domain $S_n(c,R)/R$8 and drops the requirement that $S_n(c,R)/R$9 itself be given in Jordan form, replacing it with the weaker hypothesis that eigenvalues lie in $\mathbb{k}$0. The connection with Xi–Yin's work on centrosymmetric matrix algebras [MR4053642] is also relevant: centrosymmetric matrices form the centralizer of a specific involution-type matrix, and the cellularity results there depend on Frobenius extension properties of the same type studied here. The paper's techniques — reduction to Jordan blocks, polynomial parametrization of block centralizers, and dual-basis constructions — are standard in the Frobenius extension literature [1690111], but their combination to obtain diagonalizability criteria appears to be new.

Limitations and open questions

The paper's hypotheses should be noted plainly. The integral domain case requires that $\mathbb{k}$1 be similar over $\mathbb{k}$2 to its Jordan canonical form, which fails for matrices whose eigenvalues do not lie in $\mathbb{k}$3 or over rings where Jordan decomposition is unavailable; the field-case result circumvents this only by passing to a field extension, so the criteria are stated in terms of the Jordan form over an extension rather than intrinsically over $\mathbb{k}$4. The diagonalizability characterization is likewise mediated by Jordan form and does not give an effective algorithmic test on the original matrix. The paper does not address centralizers of matrices over general commutative rings beyond integral domains, nor does it treat the question of when $\mathbb{k}$5 is cellular or quasi-hereditary, questions raised for related algebras in [MR4053642, MR4241258].

Conclusion

The paper gives necessary and sufficient conditions for the centralizer algebra $\mathbb{k}$6 to be a Frobenius, and separable Frobenius, extension of $\mathbb{k}$7, first over an integral domain for matrices in Jordan form with eigenvalues in $\mathbb{k}$8, and then over a field for arbitrary matrices. The reduction of these ring-theoretic properties to the Jordan block structure of $\mathbb{k}$9, and the resulting characterization of diagonalizability via the separable Frobenius condition, constitute the paper's main contributions. The results extend the line of work initiated in [4632826] and provide tools likely applicable to the study of cellular and homological properties of centralizer matrix algebras.