2018/01/01 by Oliver Lorscheid · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology #Advanced Algebra and Geometry #Mathematics #Weyl group #Group (periodic table) #Algebraic group #Morphism #Symplectic geometry #Extension (predicate logic) #Symplectic group #Pure mathematics #Functor #Algebraic number #Combinatorics #Algebra over a field #Mathematical analysis #Physics
paper · pdf · doi:10.1017/fms.2018.17
openalex publication_date 2018/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 \mathbbF1 , the field with one element . Based on Part I of The geometry of blueprints, we introduce the class of Tits morphisms between blue schemes. The resulting Tits category Sch_T comes together with a base extension to (semiring) schemes and the so-called Weyl extension to sets. We prove for 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 G defines a model G in Sch_T whose Weyl extension is the Weyl group W of G . We call such models Tits–Weyl models . The potential of Tits–Weyl models lies in (a) their intrinsic definition that is given by a linear representation; (b) the (yet to be formulated) unified approach towards thick and thin geometries; and (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.