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Journal of Lie Theory 32 (2022), No. 1, 075--086
Copyright Heldermann Verlag 2022

Quadratic Forms on the 27-Dimensional Modules for E6 in Characteristic Two

Mashhour Bani-Ata
Dept. of Mathematics, Public Authority for Applied Education and Training, Kuwait


The purpose of this paper is to study the Chevalley group $E$ of type $E_6(\mathbb{K})$ over fields $\mathbb{K}$ of characteristic two. We use the generalized quadrangle $(\mathbb{P},\l)$ over $\mathbb{K}$ of type $O^-_6(2)$ to construct a trilinear form $T$ on a 27-dimensional vector space $A$, this form preserves the action of $E$. We introduce an involution\\[2mm] \centerline{$g\to g^\alpha=g^*=(g^t)^{-1}$}\\[2mm] on $E$, algebra structure on $A$ and a quadratic map $\hat{Q}:A\to A$. Then we prove the following results:\\[1mm] (a)\ \ \ $\hat{Q}(x^g)=\hat{Q}(x)^{g^*}$ for all $x\in A$ and $g\in E$.\\[1mm] (b)\ \ \ For $x,y,z\in A$ and $g\in E$, the following holds true:\\ \hspace*{8mm}(1) $x^g\,y^g=(xy)^{g^*}$,\\ \hspace*{8mm}(2) $T(x^g,y^g,z^g)=T(x,y,z)$.\\[1mm] (c)\ \ \ The main results:\\ \hspace*{8mm}(1) The group $G$ of isometries of $T$ coincides with the group\\ \hspace*{14mm}$G^* = \{g \in GL(A)\;|\;a^gb^g = (ab)^{g^*}\}$.\\ \hspace*{8mm}(2) The group $G_0=\{g\in GL(A)\;|\;\hat{Q}(a^g)=\hat{Q}(a)^{g^*}\}$ is intermediate\\ \hspace*{14mm}between $E$ and $G$.\\ \hspace*{8mm}(3) The group $E=E^*=\{g^*=(g^t)^{-1}\;|\;g\in E\}$.

Keywords: Quadratic forms, generalized quadrangles, groups of Lie type.

MSC: 17A75, 17A45.

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