Superunitary regions of cluster algebras

TL;DR

This paper introduces the superunitary region of a cluster algebra, showing it forms a regular CW complex homeomorphic to the generalized associahedron, with implications for positive integral friezes.

Contribution

It defines the superunitary region and proves its topological structure for finite type cluster algebras, linking it to generalized associahedra.

Findings

Superunitary region is a regular CW complex.

Homeomorphic to the generalized associahedron.

Finiteness of positive integral friezes for each Dynkin diagram.

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.