关于Schur矩阵泛函的线性映射
On the linear preservers of Schur matrix functionals
摘要：
设$\mathbb{F}$是一个域，$f : \mathfrak{S}_n \rightarrow \mathbb{F} \setminus \{0\}$是一个任意映射。与$f$相关的Schur矩阵泛函定义为$M \in \text{M}_n(\mathbb{F}) \mapsto \widetilde{f}(M):=\sum_{σ\in \mathfrak{S}_n} f(σ) \prod_{j=1}^n m_{σ(j),j}$。这种泛函的典型例子是行列式（其中$f$是签名态射）和永久式（其中$f$是常数，值为$1$）。给定两个这样的映射$f$和$g$，我们研究了向量空间$\text{M}_n(\mathbb{F})$的满足$\widetilde{g}(U(M))=\widetilde{f}(M)$（对所有$M \in \text{M}_n(\mathbb{F})$）的自同态$U$.特别地，当$f$为中心时，我们给出了泛函$\widetilde{f}$的线性代数的封闭形式，并作为特例，我们将Botta对积和式的线性代数的刻画推广到任意域.
英文摘要：
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 $M \in \text{M}_n(\mathbb{F}) \mapsto \widetilde{f}(M):=\sum_{σ\in \mathfrak{S}_n} f(σ) \prod_{j=1}^n m_{σ(j),j}$. 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.
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