The Symmetries of the $\pi$-metric

TL;DR

This paper characterizes the full symmetry group of an n-dimensional vector space over a finite field under the recently introduced π-metric, providing a complete description of its automorphisms.

Contribution

It offers the first comprehensive description of the symmetry group of the π-metric on vector spaces over finite fields.

Findings

Complete characterization of the symmetry group of the π-metric.

Identification of automorphisms preserving the π-metric.

Extension of known symmetries to a full group description.

Abstract

Let V be an n-dimensional vector space over a finite field F_q. We consider on V the $\pi$-metric recently introduced by K. Feng, L. Xu and F. J. Hickernell. In this short note we give a complete description of the group of symmetries of V under the $\pi$-metric.