Adjacency preservers on invertible hermitian matrices II

TL;DR

This paper characterizes maps that preserve adjacency among invertible hermitian matrices over finite fields, revealing their group structure and applications to finite Minkowski space-time, without assuming bijectivity.

Contribution

It provides a complete description of adjacency-preserving maps on invertible hermitian matrices, including their generation by specific transformations and applications to finite Minkowski space.

Findings

Maps form a group generated by conjugation, field automorphisms, and inversion.

Bijectivity is derived, not assumed.

Applications to finite Minkowski space-time and light cone structures.

Abstract

Maps that preserve adjacency on the set of all invertible hermitian matrices over a finite field are characterized. It is shown that such maps form a group that is generated by the maps $A\mapsto PAP^{\ast}$, $A\mapsto A^{\sigma}$, and $A\mapsto A^{-1}$, where $P$ is an invertible matrix, $P^{\ast}$ is its conjugate transpose, and $\sigma$ is an automorphism of the underlying field. Bijectivity of maps is not an assumption but a conclusion. Moreover, adjacency is assumed to be preserved in one directions only. The main result and author's previous result [16] are applied to characterize maps that preserve the `speed of light' on (a) finite Minkowski space-time and (b) the complement of the light cone in finite Minkowski space-time.