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

Algebraic groups over the field with one element