Orbits of vectors in some representations. III

TL;DR

This paper studies the orbit structure of certain group actions on vectors associated with root systems of types E6, E7, and E8 over fields of characteristic 2, extending understanding of their algebraic and geometric properties.

Contribution

It provides a detailed description of the orbits of a specific group acting on vectors related to maximal roots in these root systems, focusing on characteristic 2 fields.

Findings

Classification of orbits for the group action.

Explicit descriptions of orbit representatives.

Insights into the structure of representations in characteristic 2.

Abstract

Let $\Phi$ be a root system of type $E_6$, $E_7$, or $E_8$. Let $K$ be a field of characteristic $2$. Let $\delta$ be the maximal root of $\Phi$ and set $\Phi_0 = \{\alpha\in\Phi; \delta\perp\alpha\}$. We describe orbits of the group $G_{\text{\rm sc}}(\Phi_0, K)$ acting on the set $\langle e_\alpha; \angle(\alpha, \delta) = \pi/3\rangle$.