发表机构
Université de Versailles Saint-Quentin-en-Yvelines（凡尔赛圣康坦昂伊夫林大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文完整刻画了Hermitian矩阵（及其反对称变体）到列空间的群同态，推广至任意对合，为非交换情形提供新结果，并服务于后续有界秩子空间研究。
AI 中文摘要
设$\mathbb{D}$为带对合$x \mapsto x^\star$的除环，$n \geq 2$为整数。记$\mathcal{H}_n(\mathbb{D})$为$\mathbb{D}$上所有$n \times n$ Hermitian矩阵的集合，记$\mathcal{A}\mathcal{H}_n(\mathbb{D})$为所有形如$A-A^\star$（其中$A \in \mathcal{M}_n(\mathbb{D})$）的矩阵的集合。本文给出以下问题的完整解答：确定所有从$\mathcal{H}_n(\mathbb{D})$到$\mathbb{D}^n$（以及当$(-)^\star$不是恒等映射时，从$\mathcal{A}\mathcal{H}_n(\mathbb{D})$到$\mathbb{D}^n$）的群同态，这些同态将每个矩阵映射为其列的右线性组合。当$(-)^\star$为恒等映射时，该问题的解已知，本文的新颖之处在于推广到任意对合，特别是非交换情形。这些结果将用于后续关于有界秩Hermitian矩阵子空间以及大规模可对角化矩阵空间的论文中。
英文摘要
Let $\mathbb{D}$ be a division ring with an involution $x \mapsto x^\star$, and $n \geq 2$ be an integer. Denote by $\mathcal{H}_n(\mathbb{D})$ the set of all $n$-by-$n$ Hermitian matrices with entries in $\mathbb{D}$, and by $\mathcal{A}\mathcal{H}_n(\mathbb{D})$ the set of all matrices $A-A^\star$ with $A \in \mathcal{M}_n(\mathbb{D})$. Here, we give a complete solution to the following problem: Determine all group homomorphisms from $\mathcal{H}_n(\mathbb{D})$ to $\mathbb{D}^n$ (respectively, from $\mathcal{A}\mathcal{H}_n(\mathbb{D})$ to $\mathbb{D}^n$ unless $(-)^\star$ is the identity) that take every matrix to a right linear combination of its columns. The solution to this problem was already known when $(-)^\star$ is the identity, and the novelty here lies in the generalization to arbitrary involutions, and in particular in the noncommutative case. These results are to be used in a subsequent article on subspaces of Hermitian matrices of bounded rank, and on large spaces of diagonalisable matrices.
Comments20 pages
