Triples of long root subgroups

N. A. Vavilov ; I. M. Pevzner

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 15, Tome 343 (2007), pp. 54-83

Résumé

Let $G=G(\Phi,K)$ be a Chevalley group over a field $K$ of characteristic $\ne 2$. In the present paper, we classify subgroups of $G$ generated by triples of long root subgroups, two of which are opposite, up to conjugacy. For finite fields this result is contained in the papers by B. Cooperstein on geometry of root subgroups, whereas for $\mathrm{SL}\,(n,K)$ it is proven in a paper by L. Di Martino and the first-named author. All interesting items of our list appear in the deep geometric results on abstract root subgroups and quadratic actions by F. Timmesfeld and A. Steinbach, and also by E. Bashkirov. However, for applications to the groups of type $\mathrm{E}_l$, we need detailed justification of this list, which we could not extract from the published works. This is why in the present paper, we produce a direct elementary proof based on reduction to $\mathrm{D}_4$ where the question is settled by straightforward matrix calculations.