Classification of nilpotent and semisimple fourvectors of a real eight-dimensional space
Emanuele Di Bella, Willem A. de Graaf, Andrea Santi
TL;DR
This work extends Antonyan's complex classification of SL$(8,\mathbb{C})$-orbits on $\bigwedge^4 \mathbb{C}^8$ to the real setting by leveraging Vinberg's $\theta$-group framework and Galois cohomology for SL$(8,\mathbb{R})$ acting on $\bigwedge^4 \mathbb{R}^8$. The authors develop and implement a substantial computational pipeline to determine stabilizer component groups, compute Galois cohomology sets, and enumerate real nilpotent and semisimple orbits, including the Cartan-subspace structure in the real graded component $\mathfrak{g}_1^{\mathbb{R}}$. They obtain 258 nonzero real nilpotent orbits and 1470 real semisimple orbit classes (including zero), with semisimple representatives depending on up to seven parameters and ten Cartan subspaces up to real conjugacy. The results rely on a combination of Gröbner-basis computations, GAP/Magma tools, and the Galois-cohomology machinery to translate complex orbit data into real orbit data, while carefully treating outer automorphisms that relate various real forms. The work provides comprehensive tables and explicit representatives that illuminate the real orbit structure and its geometric implications, with significant connections to Cartan subspaces and holonomy contexts.
Abstract
In 1981 Antonyan classified the orbits of SL$(8,\mathbb{C})$ on $\bigwedge^4 \mathbb{C}^8$. This is an example of a $θ$-group action as introduced and studied by Vinberg. The orbits of a $θ$-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.