Frieze patterns as root posets and affine triangulations

TL;DR

This paper explores the connection between frieze patterns, root posets, and affine triangulations, establishing new classifications and properties of these mathematical structures within the context of Weyl groupoids.

Contribution

It introduces a classification of frieze patterns that can be used to construct affine simplicial arrangements and proves the existence of maximal elements in their root posets.

Findings

Existence of maximal elements in root posets of frieze patterns

Classification of frieze patterns suitable for affine triangulations

Connection between frieze patterns and affine Weyl groupoids

Abstract

The entries of frieze patterns may be interpreted as coordinates of roots of a finite Weyl groupoid of rank two. We prove the existence of maximal elements in their root posets and classify those frieze patterns which can be used to build an affine simplicial arrangement.