Orbits of vectors in some representations. II

TL;DR

This paper investigates the structure of vector orbits under specific group actions within certain root system representations, focusing on types A_l and D_l over fields with characteristic not 2.

Contribution

It provides a detailed description of the orbits of a particular group acting on vectors associated with maximal roots in root systems of types A_l and D_l.

Findings

Characterization of orbits for the group $G_{sc}( ext{Phi}_0, K)$

Explicit description of vectors with angle $rac{ ext{ extpi}}{3}$ to the maximal root

Results applicable to root systems of types A_l and D_l

Abstract

Let $\Phi$ be a root system of type $A_l$ or $D_l$. Let $K$ be a field of characteristic not $2$. 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 $\langle e_\alpha; \angle(\alpha, \delta) = \pi/3\rangle$.