Geometry Of Alternate Matrices Over Principal Ideal Domain

The study of the geometry of matrices was initiated by L. K. Hua in the mid forties of the last century. In the geometry of matrices, the points of the associated space are a certain kind of matrices, there is an arithmetic distance of two points in this space, and there is a transformation group acting on this space. The fundamental problem of the geometry of matrices is to characterize the transformation group of the geometry by as few geometric invariants as possible. The answer to this problem is often called the fundamental theorem of the geometry of matrices. In this paper, we discuss the fundamental theorem of the geometry of alternate matrices over a commutative principal ideal domain.Let R be a commutative principal domain (PID) which its characteristic is not2. Denote by K_n(R) the set of all n×n alternate matrices over R. Firstly, we discussthe construction of maximal sets in K_n(R) under the transformation of form X(?)^tPXP + K₀,(?)X∈G K_n(R), where P∈GL_n(R),K₀∈K_n(R). Secondly, by the method of maximal sets, we prove the fundamental theorem of the geometry of alternatematrices over R as follows: Let R be a PID which is Jacobson semisimple and char(R)≠2. Letφ:K_n(R)→K_n(R) be a bijective map such thatφandφ^-1preserve the adjacency and weak unimodular. Then when n > 4,φis of the formφ(X) =α^tPX^σP + K₀,(?)X∈K_n(R), whereα∈R*,P∈GL_n(R), K₀∈K_n(R), andσis an automorphism of R. When n = 4,φis of the formφ(X) =α^tP(X*)^σP+K₀,(?)X∈K₄(R).Finally, we discuss the application of the fundamental theorem of the geometry of alternate matrices over R.