Suzuki–Ree groups and Tits mixed groups over rings

Abstract

It is shown that Suzuki–Ree groups can be easily defined by means of comparing two fundamental representations of the ambient Chevalley group in characteristic 2 or 3. This eliminates the distinction between the Suzuki–Ree groups over perfect and imperfect fields and gives a natural definition for the analogs of such groups over commutative rings. As an application of the same idea, we explicitly construct a pair of polynomial maps between the groups of types and in characteristic 2 that compose to the Frobenius endomorphism. This, in turn, provides a simple definition for the Tits mixed groups over rings.