Two disguises of the linear representation of a subgeometry

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 $\text{PG}(n,q)$,...

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Bibliographic Details
Main Author Denaux, Lins
Format Journal Article
LanguageEnglish
Published 23.12.2021
Subjects
Mathematics - Combinatorics
Online AccessGet full text
DOI10.48550/arxiv.2112.12452

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Summary: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.
DOI:10.48550/arxiv.2112.12452