Suzuki-Ree groups and Tits mixed groups over rings

TL;DR

The paper introduces a unified approach to defining Suzuki-Ree groups and Tits mixed groups over rings by comparing fundamental representations, simplifying their construction and extending their applicability beyond perfect fields.

Contribution

It provides a natural definition of Suzuki-Ree groups over rings and constructs polynomial maps between B_n and C_n groups in characteristic 2, defining Tits mixed groups over rings.

Findings

Unified definition of Suzuki-Ree groups over rings

Explicit polynomial maps between B_n and C_n groups in characteristic 2

Simplified construction of 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 analogues 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 B_n and C_n in characteristic 2 that compose to the Frobenius endomorphism. This, in turn, provides a simple definition for the Tits mixed groups over rings.