On the linear preservers of Schur matrix functionals

Abstract

Let $\mathbb{F}$ be a field and $f:{\mathfrak{S}}_n\to \mathbb{F}\setminus \{0\}$ be an arbitrary map. The Schur matrix functional associated to $f$ is defined as $M\in {\text{M}}_n(\mathbb{F})\mapsto \tilde{f}(M):={\sum }_{\sigma \in {\mathfrak{S}}_n}f(\sigma ){\prod }_{j=1}^n{m}_{\sigma (j),j}$. 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 $\tilde{g}(U(M))=\tilde{f}(M)$ for all $M\in {\text{M}}_n(\mathbb{F})$. In particular, we give a closed form for the linear preservers of the functional $\tilde{f}$ 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.

arXiv comment60 pages