An $\mathrm {A}_2$-proof of structure theorems for Chevalley groups of types $\mathrm {E}_6$ and $\mathrm {E}_7$

Abstract

In the present paper we prove the main structure theorem for Chevalley groups G = G(Φ, R) of types Φ = E 6 , E 7 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 E 6 and of the 56-dimensional module for Chevalley groups of type E 7 pave the way to much more general results such as description of subgroups normalized by some of elementary matrices. Groups of types E 8 and F 4 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. §1. IntroductionThe proofs of structure theorems are based either on reduction of the dimension of the ground ring, or on reduction of the rank of the group, or eventually on a combination of both. The early proofs for classical groups were based on reduction of rank [71], [17]-[19], and [23,24,20,3,9]; the dimension reduction proofs appeared later [60], [62]-[64], and [44, 45, 33]. At the same time for exceptional groups the first published proofs leant upon reduction of dimension (see [25,27,28,30,58,61]). The paper [30] by E. Abe and K. Suzuki was a near miss as far as the normal structure theorem for Chevalley groups is concerned (see also [25]). In fact, a somewhat weaker result was proved in [30], namely, a description of the normal subgroups of the elementary Chevalley group E(Φ, R) over a class of rings subject to a certain finiteness condition. However, as was

Decomposition of transvections: An algebro-geometric approach

“…The decomposition of transvections technique proposed by Stepanov and Vavilov in [3] is successfully used to prove the normality of the elementary subgroups in Chevalley groups and the structure theorem on subgroups normalized by an elementary subgroup, to estimate the width of Chevalley groups in various generators, to describe automorphisms, and so on. Various versions of this method were proposed in [8,9,10,11,6].…”