Matrix Generators for Exceptional Groups of Lie Type

Until recently it was impractical to use general purpose computer algebra systems to investigate Chevalley groups except for those of small rank over small fields. But computer algebra systems (such as Magma (Bosma et al. 1997) and GAP (Schönert et al. 1994)) now have the power to deal with some aspects of all finite groups of Lie type. A natural way to represent these groups is via matrices over the defining field. Thus, for computational purposes, there is a need to provide matrix generators for these groups. This has been done for the classical groups (Taylor 1987, Rylands and Taylor 1998) and now it remains to extend this to all finite groups of Lie type. It is the purpose of the present paper to give a uniform method of constructing generators for groups of Lie type with particular emphasis on the exceptional groups. The constructions described here could be carried out within any computer algebra system and, in particular, have been implemented in Magma. This completes the determination of matrix generators for all groups of Lie type, including the twisted groups of Steinberg, Suzuki and Ree (and the Tits group). The Lie algebras and related Chevalley groups of types An, Bn, Cn and Dn can be identified with classical groups (Carter (1972), §11.3), and in Taylor (1987) and Rylands and Taylor (1998) this identification was used to translate the generators given by Steinberg (1962) to matrix forms. The constructions in this paper rely on an investigation of the root systems of Lie algebras, providing a uniform approach and avoiding case by case discussions of nonassociative algebras. In each case (except E8) we obtain the lowest dimensional module for the Lie algebra via an embedding in a Lie algebra of higher rank.