Reflection groupoids of rank two and cluster algebras of type A

摘要

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 2 n objects.