Algebraic groups over the field with one element

Abstract

In this paper we provide a first realization of an idea of Jacques Tits from a 1956 paper, which first mentioned that there should be a field of charactéristique une, which is now called \({\mathbb{F}_1}\), the field with one element. This idea was that every split reductive group scheme over \({\mathbb{Z}}\) should descend to \({\mathbb{F}_1}\), and its group of \({\mathbb{F}_1}\)-rational points should be its Weyl group. We connect the notion of a torified scheme to the notion of \({\mathbb{F}_1}\)-schemes as introduced by Connes and Consani. This yields models of toric varieties, Schubert varieties and split reductive group schemes as \({\mathbb{F}_1}\)-schemes. We endow the class of \({\mathbb{F}_1}\)-schemes with two classes of morphisms, one leading to a satisfying notion of \({\mathbb{F}_1}\)-rational points, the other leading to the notion of an algebraic group over \({\mathbb{F}_1}\) such that every split reductive group is defined as an algebraic group over \({\mathbb{F}_1}\). Furthermore, we show that certain combinatorics that are expected from parabolic subgroups of GL(n) and Grassmann varieties are realized in this theory.