Structure of centralizer algebras

TL;DR

This paper studies the structure of centralizer algebras of matrices over rings, proving conditions under which these algebras are separable Frobenius extensions or cellular algebras, with implications for algebraic representation theory.

Contribution

It characterizes when principal centralizer matrix rings are separable Frobenius extensions or cellular algebras, extending known results to matrices over rings and algebraically closed fields.

Findings

$M_n(R)$ is a separable Frobenius extension of $S_n(c,R)$ under certain conditions.

$S_n(c,R)$ is a cellular algebra when $R$ is an integral domain and $c$ is Jordan-similar.

$S_n(c,R)$ is always a cellular algebra over algebraically closed fields for any matrix $c$.

Abstract

Given an $n\times n$ matrix $c$ over a unitary ring $R$, the centralizer of $c$ in the full $n\times n$ matrix ring $M_n(R)$ is called a principal centralizer matrix ring, denoted by $S_n(c,R)$. We investigate its structure and prove: $(1)$ If $c$ is an invertible matrix with a $c$-free point, or if $R$ has no zero-divisors and $c$ is a Jordan-similar matrix with all eigenvalues in the center of $R$, then $M_n(R)$ is a separable Frobenius extension of $S_{n}(c,R)$ in the sense of Kasch. $(2)$ If $R$ is an integral domain and $c$ is a Jordan-similar matrix, then $S_n(c,R)$ is a cellular $R$-algebra in the sense of Graham and Lehrer. In particular, if $R$ is an algebraically closed field and $c$ is an arbitrary matrix in $M_n(R)$, then $S_n(c,R)$ is always a cellular algebra, and the extension $S_n(c,R)\subseteq M_n(R)$ is always a separable Frobenius extension.