Invariant algebras of matrices and symmetric polynomials of partitions

TL;DR

This paper investigates the algebraic structure of centralizer algebras of matrices over fields, revealing their Frobenius-finite and Gorenstein properties, and characterizes when invariant matrix algebras are semisimple and isomorphic.

Contribution

It establishes structural properties of matrix centralizer algebras and provides criteria for their semisimplicity and isomorphism based on permutation cycle types.

Findings

Centralizer algebras are Frobenius-finite, 1-Auslander-Gorenstein, and gendo-symmetric.

Conditions for invariant algebra semisimplicity are identified.

A combinatorial criterion for isomorphism of semisimple invariant algebras is provided.

Abstract

For a field $R$ of characteristic $p\ge 0$ and a matrix $c$ in the full $n\times n$ matrix algebra $M_n(R)$ over $R$, let $S_n(c,R)$ be the centralizer algebra of $c$ in $M_n(R)$. We show that $S_n(c,R)$ is a Frobenius-finite, $1$-Auslander-Gorenstein, and gendo-symmetric algebra, and that the extension $S_n(c,R)\subseteq M_n(R)$ is separable and Frobenius. Further, we study the isomorphism problem of invariant matrix algebras. Let $\sigma$ be a permutation in the symmetric group $\Sigma_n$ and $c_{\sigma}$ the corresponding permutation matrix in $M_n(R)$. We give sufficient and necessary conditions for the invariant algebra $S_n(c_{\sigma},R)$ to be semisimple. If $R$ is an algebraically closed field, we establish a combinatoric characterization of when two semisimple invariant $R$-algebras are isomorphic in terms of the cycle types of permutations.