Finite Weyl groupoids

TL;DR

This paper classifies all connected simply connected Cartan schemes with finite irreducible root systems, extending previous results to arbitrary rank and providing a comprehensive list of crystallographic arrangements and examples.

Contribution

It generalizes the classification of finite Weyl groupoids to arbitrary rank, identifying all possible types and examples, including infinite series and sporadic cases.

Findings

Classification of Cartan schemes of type A and B

Identification of infinite series involving types C and D

Discovery of 74 sporadic examples

Abstract

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.