Superunitary regions of cluster algebras

Abstract

This note introduces the superunitary region of a cluster algebra, the subspace of the totally positive region on which each cluster variable is at least 1. Our main result is that the superunitary region of a finite type cluster algebra is a regular CW complex which is homeomorphic to the generalized associahedron of the cluster algebra. As an application, the compactness of the superunitary region implies that each Dynkin diagram admits finitely many positive integral friezes.