体上长方矩阵的图同态I

Graph homomorphisms on rectangular matrices over division rings I

设D^(m×n)为体D上m×n矩阵的集合.两个矩阵A,B∈D^(m×n)称为邻接的,如果rank(A-B)=1.按此邻接关系,以D^(m×n)为顶点集,本文得到一个连通图.设D和D′为两个体,|D|4,m,n,m′,n′2为整数.应用几何方法,本文刻画了从D^(m×n)到D′^(m′×n′)的非退化的图同态φ,其中φ满足条件:φ(0)=0且φ保持D^(m×n)中两个不同类型的标准极大邻接集的维数不变.作为一个推论,当D为EAS(every endomorphism to be automatically surjective)体时,本文给出了从D^(m×n)到D^(m′×n′)的非退化的图同态的代数公式.

Let Dm×n be the set of m × n matrices over a division ring D. Two matrices A, B ∈ Dm×n are adjacent if rank（A-B） = 1. By the adjacency, Dm×n is a connected graph. Suppose that D, D′ are division rings with ｜D｜≥4 and m, n, m′, n′≥2 are integers. Using the geometric method, we characterize every non-degenerate′orphism φ from Dm×n to D′m′×n′ graph homomif φ（0） = 0 and φ preserves the dimensions of two standard maximal adjacent sets of different types in Dm×n. As a corollary, when D is an EAS（every endomorphism to be automatically surjective） division ring, we get algebraic formulas of every non-degenerate graph homomorphism from D^（m×n） to Dm′×n′.