Reflection groupoids of rank two and Cluster algebras of type $A$

TL;DR

This paper extends the classification of rank-two Weyl groupoids, introduces reflection groupoids with non-integral Cartan matrices, and reveals a surprising connection between these groupoids and the spectrum of type A cluster algebras.

Contribution

It generalizes finite Weyl groupoids to reflection groupoids with non-integral entries and uncovers a novel link to cluster algebra spectra.

Findings

Classification of finite reflection groupoids of rank two with 2n objects

Connection between reflection groupoids and spectra of type A cluster algebras

Extension of Weyl groupoid theory to non-integral Cartan matrices

Abstract

We extend the classification of finite Weyl groupoids of rank two. Then we generalize these Weyl groupoids to `reflection groupoids' by admitting non-integral entries of the Cartan matrices. This leads to the unexpected observation that the spectrum of the cluster algebra of type $A_{n-3}$ completely describes the set of finite reflection groupoids of rank two with $2n$ objects.