Two disguises of the linear representation of a subgeometry

TL;DR

This paper explores two different representations of the linear geometry associated with a subgeometry in projective space, providing an explicit isomorphism between them using field reduction techniques.

Contribution

It establishes an explicit isomorphism between two known disguises of the linear representation of a subgeometry in projective space.

Findings

Identifies two different representations of the linear geometry of a subgeometry.

Provides an explicit isomorphism between these representations.

Uses field reduction to establish the isomorphism.

Abstract

Let $\text{PG}(n,q)$ be the Desarguesian projective space of dimension $n$ over the finite field of order $q$. The \emph{linear representation} of a point set $\mathcal{K}$ in a hyperplane at infinity of $\text{PG}(n,q)$ is the point-line geometry consisting of the affine points of $\text{PG}(n,q)$, together with the union of the parallel classes of affine lines corresponding to the points of $\mathcal{K}$. This type of point-line geometry has been widely investigated in the literature. Curiously, if $\mathcal{K}$ is a subgeometry, two disguises of its linear representation occur in two separate works. In this short note, we give an explicit isomorphism between these two disguises by making use of field reduction.