The 27-dimensional module for \(E_ 6\). I
type: Item
instance of: scholarly article
title: The 27-dimensional module for \(E_ 6\). I
zbMATH Open document ID: 0629.20018
DOI: 10.1007/BF01404676
author: Michael Aschbacher
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143476
review text: 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.
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20E28
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 11E04
zbMATH DE Number: 4023561
zbMATH Keywords: exceptional groups of Lie type
zbMATH Keywords: isometry groups
zbMATH Keywords: symmetric trilinear form
zbMATH Keywords: universal Chevalley group
zbMATH Keywords: geometry
zbMATH Keywords: orbits
zbMATH Keywords: points
zbMATH Keywords: hyperplanes
zbMATH Keywords: involutive automorphism
zbMATH Keywords: hermitian form
zbMATH Keywords: building
zbMATH Keywords: closed maximal subgroups
reviewed by: Hiromichi Yamada
MaRDI profile type: Publication
cites work: Q3708174
cites work: Some multilinear forms with large isometry groups
description: scientific article; zbMATH DE number 4023561