Hyperplanes of $DW(5,{\mathbb{K}})$ with ${\mathbb{K}}$ a perfect field of characteristic 2

De Bruyn, Bart

Journal of Algebraic Combinatorics, Tome 30 (2009) no. 4, pp. 567-584.

Résumé

Summary: Let $\mathbb K$ mathbbK be a perfect field of characteristic 2. In this paper, we classify all hyperplanes of the symplectic dual polar space $D$W$(5,\mathbb K) DW(5,{\mathbb{K}})$ that arise from its Grassmann embedding. We show that the number of isomorphism classes of such hyperplanes is equal to $5+ N$, where $N$ is the number of equivalence classes of the following equivalence relation $R$ on the set l Ĩ $\mathbb K | X ^{2}+ l X+1$ isirreducible {lambda$\in $mathbbK | X^2+$\lambda $X+1mbox isirreducible in $\mathbb K[ X]$ mboxin mathbbK[X]} : $( \lambda _{1}, \lambda _{2})\in R$ whenever there exists an automorphism $\sigma $ of $\mathbb K$ mathbbK and an $a$ Ĩ $\mathbb K$ a$\in $mathbbK such that $( \lambda _{2} ^{ \sigma }) ^{ - 1}= \lambda _{1} ^{ - 1}+ a ^{2}+ a$.

Détail

Keywords: keywords symplectic dual polar space, hyperplane, perfect field