The correlations with identity companion automorphism, of finite Desarguesian planes
type: Item
instance of: scholarly article
title: The correlations with identity companion automorphism, of finite Desarguesian planes
zbMATH Open document ID: 0948.51002
DOI: 10.1016/S0024-3795(99)00140-8
author: Barbu C. Kestenband
published in: Linear Algebra and its Applications
publication date: 2000-11-12T00:00:00Z
review text: The simplest correlations of finite Desarguesian planes are polarities whose classification is folklore. When it comes to correlations which are not polarities, only partial results have been known up to now. The author gives a first systematic investigation of this problem by determining all correlations of finite Desarguesian planes which have identity companion automorphism and which are not polarities. For \(P = \text{ PG}(2,q)\) with \(q\) an odd prime power up to isomorphism any correlation of \(P\) which is not a polarity is defined by one of the matrices \[ \left(\begin{smallmatrix} 1 & 2 & 2 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{smallmatrix}\right),\;\left(\begin{smallmatrix} 0 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \end{smallmatrix}\right),\;\left(\begin{smallmatrix} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \omega\end{smallmatrix}\right),\;\left(\begin{smallmatrix} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{smallmatrix}\right),\;\left(\begin{smallmatrix} 1 & \rho & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{smallmatrix}\right)\text{ with }\rho \neq -2,0,2,\text{ and }\left(\begin{smallmatrix} 1 & \rho & 0 \\ 0 & \omega & 0 \\ 0 & 0 & 1\end{smallmatrix}\right)\text{ with }\rho \neq 0, \] where \(\omega\) is a primitive root of \(\text{ GF}(q)\). For \(q\) a power of \(2\), such correlations are defined by one of the matrices \(\left(\begin{smallmatrix} 1 & \rho & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{smallmatrix}\right)\) with \(\rho \neq 0\), \(\left(\begin{smallmatrix} 1 & v & 0 \\ 0 & 1 & v \\ 0 & 0 & 1 \end{smallmatrix}\right)\) with \(v \in \text{ GF}(q)\) such that the polynomial \(x^2+vx+1\) is irreducible, and \(\left(\begin{smallmatrix} 1 & \omega + 1 / \omega & 2 \\ 0 & 1 & \omega + 1 / \omega \\ 0 & 0 & 1 \end{smallmatrix}\right)\), where \(\omega\) is again a primitive root of \(\text{ GF}(q)\). Moreover, the author completely answers the question which of these matrices are congruent and which are not.
Mathematics Subject Classification ID: 51A30
Mathematics Subject Classification ID: 51A10
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 1420029
zbMATH Keywords: finite Desarguesian planes
zbMATH Keywords: correlations
zbMATH Keywords: absolute set
MaRDI profile type: Publication
cites work: Q5824372
cites work: Q5545191
cites work: Q3251484
cites work: Q3851217
cites work: Q5683869
cites work: Generalizing a non-existence theorem in finite projective planes
description: scientific article; zbMATH DE number 1420029