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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
, 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
comes together with a base extension to (semiring) schemes and the so-called Weyl extension to sets. We prove for
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
defines a model
whose Weyl extension is the Weyl group
. 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.
Suppose we are given a regular symmetric bilinear from on a finite-dimensional vector space V over a commutative field K of characteristic ≠ 2. We want to write given elements of the commutator subgroup ω(V) (of the orthogonal group O(V)) and also of the kernel of the spinorial norm ker(Θ) as (short) products of involutions and as products of commutators
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