Range-compatible homomorphisms on Hermitian matrices
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.