Classification of nilpotent and semisimple fourvectors of a real eight-dimensional space

TL;DR

This paper classifies nilpotent and semisimple fourvector orbits in an 8-dimensional real space using Galois cohomology, extending previous complex classifications to the real case.

Contribution

It provides a detailed classification of nilpotent and semisimple orbits of SL(8,R) on 4-vectors, including 1441 semisimple classes, using Galois cohomology methods.

Findings

1441 semisimple orbit classes identified

Nilpotent orbits classified in the real setting

Complex orbit classification extended to real case

Abstract

In 1981 Antonyan classified the orbits of SL$(8,\mathbb{C})$ on $\bigwedge^4 \mathbb{C}^8$. This is an example of a $\theta$-group action as introduced and studied by Vinberg. The orbits of a $\theta$-group are divided into three classes: nilpotent, semisimple and mixed. We consider the action of SL$(8,\mathbb{R})$ on $\bigwedge^4 \mathbb{R}^8$ and classify the nilpotent and semisimple orbits as well as the Cartan subspaces. The semisimple orbits are divided into 1441 parametrized classes. Due to this high number a classification of the mixed orbits does not seem feasible. Our methods are based on Galois cohomology.