The 27-dimensional module for \(E_ 6\). I. (English) Zbl 0629.20018

Aschbacher, Michael

Invent. Math. 89, 159-195 (1987).

The author initiates [in Geom. Dedicata 25, 417-465 (1988)] a study of exceptional groups of Lie type as the isometry groups of suitable forms. In particular, two presentations of a symmetric trilinear form f on the 27-dimensional module V for the universal Chevalley group \(E_ 6(F)\) of type \(E_ 6\) over an arbitrary field F are given, which were originally discovered by Dickson. In case of \(F=GF(2)\), some difficulties arise and the notion of 3-form is required.
In the paper under review the author discusses the geometry of (V,f) and proves that the isometry group \(G=O(V,f)\) is isomorphic to the universal Chevalley group of type \(E_ 6\) over the field F, where O(V,f) denotes the group consisting of the elements of GL(V) which leave f invariant. Moreover the orbits of G on points and hyperplanes are determined and it is shown that the stabilizer \(N_ G(U)\) in G of a suitable hyperplane U is isomorphic to the Chevalley group \(F_ 4(F)\) and \(N_ G(U)=O(U,f)\). When F admits a suitable involutive automorphism \(\sigma\), the subgroup of G which preserves a certain hermitian form on V is isomorphic to \({}^ 2E_ 6(K)\), where K is the fixed field of \(\sigma\). The building for each of these three groups is also described.
The work has been started under the philosophy that most of the groups of Lie type may be best described as the isometry groups of certain multilinear forms on modules of minimal dimensions for the groups over fields naturally associated to the groups. The author intends to use the geometries induced by these forms to determine all subspaces with large stabilizers and to enumerate all maximal subgroups of the groups when the field is finite and all closed maximal subgroups when the field is algebraically closed.

Reviewer: H.Yamada

MSC:

20F65	Geometric group theory
20E28	Maximal subgroups
20D06	Simple groups: alternating groups and groups of Lie type
20G15	Linear algebraic groups over arbitrary fields
20G40	Linear algebraic groups over finite fields
11E04	Quadratic forms over general fields

Keywords:

exceptional groups of Lie type; isometry groups; symmetric trilinear form; universal Chevalley group; geometry; orbits; points; hyperplanes; involutive automorphism; hermitian form; building; closed maximal subgroups

References:

[1]	Aschbacher, M.: Finite group theory. Cambridge: Cambridge University Press 1986 · Zbl 0583.20001
[2]	Aschbacher, M.: Some multilinear forms with large isometry groups. Preprint · Zbl 0646.20033
[3]	Dickson, L.: A class of groups in an arbitrary realm connected with the configuration of the 27 lines on a cubic surface. J. Math.33, 145-173 (1901) · JFM 32.0133.01
[4]	Tits, J.: A local approach to buildings. In: The Geometric Vein. Berlin-Heidelberg-New York: Springer 1982, pp. 519-547