  Remarks in a paper by Jacques Tits from 1956 led to a philosophy how a theory
of split reductive groups over $\F_1$, the so-called field with one element,
should look like. Namely, every split reductive group over $\Z$ should descend
to $\F_1$, and its group of $\F_1$-rational points should be its Weyl group. We
connect the notion of a torified variety to the notion of $\F_1$-schemes as
introduced by Connes and Consani. This yields models of toric varieties,
Schubert varieties and split reductive groups as $\Fun$-schemes. We endow the
class of $\F_1$-schemes with two classes of morphisms, one leading to a
satisfying notion of $\F_1$-rational points, the other leading to the notion of
an algebraic group over $\F_1$ such that every split reductive group is defined
as an algebraic group over $\F_1$. Furthermore, we show that certain
combinatorics that are expected from parabolic subgroups of $\GL(n)$ and
Grassmann varieties are realized in this theory.
