Crystallographic arrangements: Weyl groupoids and simplicial arrangements

TL;DR

This paper introduces crystallographic arrangements, establishing a correspondence with certain Cartan schemes, simplifying the classification of a large subclass of simplicial arrangements with finite root systems.

Contribution

It provides a new, accessible definition for crystallographic arrangements and links them to connected simply connected Cartan schemes, aiding classification.

Findings

One-to-one correspondence between crystallographic arrangements and Cartan schemes.

Simplified classification of simplicial arrangements with finite root systems.

Enhanced understanding of the structure of crystallographic arrangements.

Abstract

We introduce the simple notion of a "crystallographic arrangement" and prove a one-to-one correspondence between these arrangements and the connected simply connected Cartan schemes for which the real roots are a finite root system (up to equivalence on both sides). We thus obtain a more accessible definition for this very large subclass of the class of simplicial arrangements for which a complete classification is known.