Graph homomorphisms on rectangular matrices over division rings I

Abstract

Let $\mathbb{D}$ be a division ring, and let ${\mathbb{D}}^{m\times n}$ be the set of $m\times n$ matrices over $\mathbb{D}$. Two matrices $A,B\in {\mathbb{D}}^{m\times n}$ are adjacent if $\mathrm{rank}(A-B)=1$. By the adjacency, ${\mathbb{D}}^{m\times n}$ is a connected graph. Suppose that $m,n,m^{\prime },n^{\prime }\ge 2$ are integers and ${\mathbb{D}}^{\prime }$ is a division ring. Using the weighted semi-affine map and algebraic method, we characterize graph homomorphisms from ${\mathbb{D}}^{m\times n}$ to ${{\mathbb{D}}^{\prime }}^{m^{\prime }\times n^{\prime }}$ (where $|\mathbb{D}|\ge 4$) under some weaker conditions.

arXiv comment27 pages