The 2-spaces of the standard \(E_ 6(q)\)-module
For a field F, let \(\hat E\) be the universal cover of the simple Chevalley group \(E_ 6(F)\), and let \(\tilde E\) be the extension of \(\hat E\) by diagonal automorphisms. Due to Freudenthal, there is a standard module \(F^{27}\) for \(\tilde E,\) and \(\tilde E\) has 3 orbits on the set of points (one-dimensional subspaces) of \(F^{27}\) [cf. \textit{J. G. M. Mars}, Bull. Soc. Math. Fr. 94, 97-140 (1966; Zbl 0146.046)]. The authors determine the orbits of \(\tilde E\) on the set of lines (two-dimensional subspaces) in \(F^{27}\) for finite fields F. For each of these 14 orbits they give the structure of the corresponding line stabilizer, the orbit size, and a representative line. The proofs use the description of the point orbits of \(\tilde E\) and of \(E_ 6(F)\), cf. \textit{B. N. Cooperstein} [Geom. Dedicata 6, 205-258 (1977; Zbl 0385.51003)] and the authors [ibid. 15, 73-105 (1983; Zbl 0541.51010)].
scientific article; zbMATH DE number 4047949