The 27-dimensional module for \(E_ 6\). I
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.
scientific article; zbMATH DE number 4023561