In the present paper we prove the main structure theorem for Chevalley groups G = G(Φ, R) of types Φ = E6, E7 over a commutative ring R. More precisely, we describe subgroups in G normalized by the elementary subgroup E(Φ, R). This result is not new, since structure theorems are known for all Chevalley groups [25, 27, 28, 30], [38]–[40], and [58, 61] (see [42, 65, 34, 56] for further references). The gist of the present paper resides not in the results themselves, but rather in the method of their proof based on the geometry of exceptional groups. We believe that this method is novel and of significant interest. Actually the Schwerpunkt of the present paper abides in a new descent procedure, which enables reduction to groups of smaller rank. This procedure is both simpler and more powerful than any other method known today. Our results on the geometry of the 27-dimensional module for Chevalley groups of type E6 and of the 56- dimensional module for Chevalley groups of type E7 pave the way to much more general results such as description of subgroups normalized by some of elementary matrices. Groups of types E8 and F4 can be handled in essentially the same style, and we intend to return to these cases in our subsequent publications. However, from a technical viewpoint the proofs in these cases are noticeably more involved because these groups do not have microweight representations.

Original languageEnglish
Pages (from-to)649-672
Number of pages24
JournalSt. Petersburg Mathematical Journal
Volume16
Issue number4
DOIs
StatePublished - 2005

    Research areas

  • Chevalley groups, Decomposition of unipotents, Elementary subgroups, Minimal module, Normal subgroups, Orbit of the highest weight vector, Parabolic subgroups, Root elements, Standard description, The proof from the Book

    Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

ID: 76484292