The geometry of blueprints. Part II: Tits-Weyl models of algebraic groups

Abstract

This paper is dedicated to a problem raised by Jacquet Tits in 1956: the Weyl group of a Chevalley group should find an interpretation as a group over what is nowadays called $\mathbb{F}_1$, \emph{the field with one element}. Based on Part I of The geometry of blueprints, we introduce the class of \emph{Tits morphisms} between blue schemes. The resulting \emph{Tits category} $\textup{Sch}_\mathcal{T}$ comes together with a base extension to (semiring) schemes and the so-called \emph{Weyl extension} to sets. We prove for $\mathcal{G}$ in a wide class of Chevalley groups---which includes the special and general linear groups, symplectic and special orthogonal groups, and all types of adjoint groups---that a linear representation of $\mathcal{G}$ defines a model $G$ in $\textup{Sch}_\mathcal{T}$ whose Weyl extension is the Weyl group $W$ of $\mathcal{G}$. We call such models \emph{Tits-Weyl models}. The potential of Tits-Weyl models lies in \textit{(a)} their intrinsic definition that is given by a linear representation; \textit{(b)} the (yet to be formulated) unified approach towards thick and thin geometries; and \textit{(c)} the extension of a Chevalley group to a functor on blueprints, which makes it, in particular, possible to consider Chevalley groups over semirings. This opens applications to idempotent analysis and tropical geometry.