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

TL;DR

This paper develops Tits-Weyl models for Chevalley groups using blueprints, providing a new geometric framework that interprets Weyl groups as structures over the field with one element, with potential applications in tropical geometry.

Contribution

It introduces Tits-Weyl models for Chevalley groups within the blueprint framework, linking linear representations to models over blue schemes and extending Chevalley groups to semirings.

Findings

Construction of Tits-Weyl models for a wide class of Chevalley groups.

Proof that linear representations define models whose Weyl extension is the Weyl group.

Potential applications to tropical geometry and idempotent analysis.

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.