Adjacency preservers on invertible hermitian matrices I

TL;DR

This paper characterizes adjacency-preserving maps on invertible hermitian matrices over finite fields, showing such maps are necessarily bijective and the associated graph is a core, with implications for finite Minkowski space-time.

Contribution

It provides a new characterization of adjacency-preserving maps on invertible hermitian matrices over finite fields, establishing their bijectivity and core property.

Findings

Maps preserving adjacency are necessarily bijective.

The graph of invertible hermitian matrices is a core.

Results have applications to finite Minkowski space-time.

Abstract

Hua's fundamental theorem of geometry of hermitian matrices characterizes all bijective maps on the space of all hermitian matrices, which preserve adjacency in both directions. In this and subsequent paper we characterize maps on the set of all invertible hermitian matrices over a finite field, which preserve adjacency in one direction. This and author's previous result are used to obtain two new results related to maps that preserve the `speed of light' on finite Minkowski space-time. In this first paper it is shown that maps that preserve adjacency on the set of all invertible hermitian matrices over a finite field are necessarily bijective, so the corresponding graph on invertible hermitian matrices, where edges are defined by the adjacency relation, is a core. Besides matrix theory, the proof relies on results from several other mathematical areas, including spectral and chromatic graph theory, and projective geometry.