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Algebraic groups over the field with one element

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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.

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  1. University of Wuppertal, Gaußstr. 20, 42097, Wuppertal, Germany

    Oliver Lorscheid

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  1. Oliver Lorscheid
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Lorscheid, O. Algebraic groups over the field with one element. Math. Z. 271, 117–138 (2012). https://doi.org/10.1007/s00209-011-0855-1

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