The correlations with identity companion automorphism, of finite Desarguesian planes
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.
scientific article; zbMATH DE number 1420029