{"ID":23595658,"CreatedAt":"2026-09-18T05:42:42.291333095Z","UpdatedAt":"2026-09-18T05:42:42.291333095Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20363","arxiv_id":"2609.20363","title":"Range-compatible homomorphisms on Hermitian matrices","abstract":"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.","short_abstract":"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 \\mathca...","url_abs":"https://arxiv.org/abs/2609.20363","url_pdf":"https://arxiv.org/pdf/2609.20363v1","authors":"[\"Clément de Seguins Pazzis\"]","published":"2026-09-17T13:23:29Z","proceeding":"math.RA","tasks":"[\"math.RA\"]","methods":"[]","has_code":false}