Polarities of symplectic quadrangles
\textit{J. Tits} [Arch. Math. 13, 187-198 (1962; Zbl 0109.39402)] showed that the symplectic quadrangle \(W(F)\) is self-dual if and only if the field \(F\) is perfect of characteristic 2 and that the existence of a polarity is equivalent to the existence of a square root of the Frobenius automorphism \(x\mapsto x^2\) of \(F\). These results were obtained in the context of ovoids related to the Suzuki simple groups. In the paper under review the author gives an elementary proof of these facts using simple computations in the elation and translation groups in the automorphism group of \(W(F)\) thus avoiding the use of simple groups. (The groups considered are nilpotent subgroups of the symplectic group and actually even abelian in the interesting characteristic 2 case.) Moreover, the author determines all polarities, characterizes the conjugacy classes of polarities and gives a system of representatives. Using these results he also shows that the centralizer of a polarity acts 2-transitively on the corresponding ovoid.
scientific article; zbMATH DE number 2031035