期刊论文

Graph homomorphisms on rectangular matrices over division rings I

中文

中国科学.数学

1674-7216

2018

48

9

1095-1120

10.1360/N012017-00179

国家自然科学基金（批准号：11371072） 湖南省教育厅（批准号：16C0037）资助项目

本校为第一机构

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 D~(m × n) be the set of m × n matrices over a division ring D. Two matrices A, B ∈ D~(m × n) are adjacent if rank(A – B) = 1. By the adjacency, D~(m × 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 graph homomorphism φ from D~(m × n) to D ' m '× n ' if φ(0) = 0 and φ preserves the dimensions of two standard maximal adjacent sets of different types in D~(m × n). As a corollary, when D is an EAS (ev...