Cellularity of centrosymmetric matrix algebras and Frobenius extensions

TL;DR

This paper investigates the algebraic structure of centrosymmetric matrix algebras over an arbitrary algebra, establishing their Morita equivalences, Frobenius extension properties, and cellularity when the base ring is commutative.

Contribution

It proves Morita equivalences for centrosymmetric matrix algebras, shows they form Frobenius extensions of full matrix algebras, and demonstrates their cellularity over commutative rings.

Findings

$S_n(R)$ is Morita equivalent to $S_2(R)$ if $n$ is even.

$S_n(R)$ is Morita equivalent to $S_3(R)$ if $n extgreater 2$ is odd.

The full matrix algebra is a separable Frobenius extension of $S_n(R)$.

Abstract

Centrosymmetric matrices of order $n$ over an arbitrary algebra $R$ form a subalgebra of the full $n\times n$ matrix algebra over $R$. It is called the centrosymmetric matrix algebra of order $n$ over $R$ and denoted by $S_n(R)$. We prove (1) $S_n(R)$ is Morita equivalent to $S_2(R)$ if $n$ is even, and to $S_3(R)$ if $n\ge 3$ is odd; (2) the full $n\times n$ matrix algebra over $R$ is a separable Frobenius extension of $S_n(R)$; and (3) if $R$ is a commutative ring, then $S_n(R)$ is a cellular $R$-algebra in the sense of Graham-Lehrer for all $n\ge 1$.