Normalizer of the Chevalley group of type ${\mathrm E}_6$

Abstract

This is true over an arbitrary commutative ring R, all normalizers and transporters being taken in GL(27, R). Moreover, G(E 6 , R) is characterized as the stabilizer of a system of quadrics. This result is classically known over algebraically closed fields; in the paper it is established that the corresponding scheme over Z is smooth, which implies that the above characterization is valid over an arbitrary commutative ring. As an application of these results, we explicitly list equations a matrix g ∈ GL(27, R) must satisfy in order to belong to G(E 6 , R). These results are instrumental in a subsequent paper of the authors, where overgroups of exceptional groups in minimal representations will be studied.

Automorphisms of Chevalley groups of different types over commutative rings

“…Note that for the simply-connected Chevalley group of the type E 6 Luzgarev and Vavilov in [55] proved that the normalizers of the Chevalley group and its elementary subgroup coincide. Then in [56] they proved the same theorem for the root system E 7 .…”

Overgroups of exterior powers of an elementary group. levels

“…Next we present an alternative description of m GL n (R) as a stabilizer of a form. Analogous forms are well known for classical and exceptional groups in the standard representation over an arbitrary ring, see [17,22,21,18,19]. Conveniently for the reader, a general approach was developed by Skip Garibaldi and Robert Guralnick [7].…”