Title: Twisting and Mixing,
Abstract: We present a framework that connects three interesting classes of groups: the
twisted groups (also known as Suzuki-Ree groups), the mixed groups and the
exotic pseudo-reductive groups.
For a given characteristic p, we construct categories of twisted and mixed
schemes. Ordinary schemes are a full subcategory of the mixed schemes. Mixed
schemes arise from a twisted scheme by base change, although not every mixed
scheme arises this way. The group objects in these categories are called
twisted and mixed group schemes.
Our main theorems state: (1) The twisted Chevalley groups ${}^2\mathsf B_2$,
${}^2\mathsf G_2$ and ${}^2\mathsf F_4$ arise as rational points of twisted
group schemes. (2) The mixed groups in the sense of Tits arise as rational
points of mixed group schemes over mixed fields. (3) The exotic
pseudo-reductive groups of Conrad, Gabber and Prasad are Weil restrictions of
mixed group schemes.