On the linear preservers of Schur matrix functionals

TL;DR

This paper characterizes linear transformations that preserve Schur matrix functionals, including the determinant and permanent, providing a unified framework and extending known results to arbitrary fields.

Contribution

It offers a comprehensive description of linear preservers of Schur functionals, generalizing previous results and including new cases for arbitrary fields.

Findings

Closed form for linear preservers when $f$ is central

Extension of Botta's characterization of permanent preservers

Unified framework for Schur matrix functional preservers

Abstract

Let $\mathbb{F}$ be a field and $f : \mathfrak{S}_n \rightarrow \mathbb{F} \setminus \{0\}$ be an arbitrary map. The Schur matrix functional associated to $f$ is defined as $M \in \text{M}_n(\mathbb{F}) \mapsto \widetilde{f}(M):=\sum_{\sigma \in \mathfrak{S}_n} f(\sigma) \prod_{j=1}^n m_{\sigma(j),j}$. Typical examples of such functionals are the determinant (where $f$ is the signature morphism) and the permanent (where $f$ is constant with value $1$). Given two such maps $f$ and $g$, we study the endomorphisms $U$ of the vector space $\text{M}_n(\mathbb{F})$ that satisfy $\widetilde{g}(U(M))=\widetilde{f}(M)$ for all $M \in \text{M}_n(\mathbb{F})$. In particular, we give a closed form for the linear preservers of the functional $\widetilde{f}$ when $f$ is central, and as a special case we extend to an arbitrary field Botta's characterization of the linear preservers of the permanent.