Finite Weyl groupoids

Using previous results concerning the rank two and rank three cases, all connected simply connected Cartan schemes for which the real roots form a finite irreducible root system of arbitrary rank are determined. As a consequence one obtains the list of all crystallographic arrangements, a large subclass of the class of simplicial hyperplane arrangements. Supposing that the rank is at least three, the classification yields Cartan schemes of type $A$ and $B$, an infinite family of series involving the types $C$ and $D$, and $74$ sporadic examples.