Equations determining the orbit of the highest weight vector in the adjoint representation

TL;DR

This paper explicitly constructs quadratic equations defining the orbit of the highest weight vector in adjoint representations of certain Chevalley groups, aiding calculations especially for exceptional groups like E_8.

Contribution

It provides a new set of quadratic equations for the highest weight vector orbit in adjoint representations of specific Chevalley groups, linking combinatorics of root systems.

Findings

Quadratic equations explicitly constructed for types D_l, E_6, E_7, E_8

Framework facilitates calculations with exceptional groups

Equations relate to A_3 root system embeddings

Abstract

We explicitly construct a set of quadratic equations defining the highest weight vector orbit for adjoint representations of Chevalley groups of types D_l, E_6, E_7, and E_8. The combinatorics of these equations is related to the combinatorics of embeddings of the root system of type A_3. We believe that the constructed equations provide a prominent framework for calculations with exceptional groups in adjoint representations, which is particularly interesting for groups of type E_8.