Frobenius extensions about centralizer matrix algebras

TL;DR

This paper characterizes when the centralizer algebra of a matrix forms a Frobenius extension over the base algebra, providing necessary and sufficient conditions related to matrix form and eigenvalues.

Contribution

It offers new criteria for Frobenius extensions in centralizer algebras, extending analysis to arbitrary matrices and diagonalizability conditions.

Findings

Conditions for $S_n(c,R)/R$ to be Frobenius extensions

Characterization of Frobenius extensions for matrices in Jordan form

Criteria for matrix diagonalizability via Frobenius extensions

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.