Bases for cluster algebras from orbifolds

TL;DR

This paper extends the construction of certain bases in cluster algebras to those arising from orbifolds, proving positivity and atomicity properties for these bases in various cases.

Contribution

It generalizes known bases to orbifold cluster algebras and proves their positivity and atomicity, including for affine and finite type cases.

Findings

Bracelet bases are positive in orbifold cluster algebras.

Bracelet basis for affine type $C_n^{(1)}$ is atomic.

Cluster monomial bases of finite type are atomic.

Abstract

We generalize the construction of the bracelet and bangle bases defined by Musiker, Schiffler and Williams, and the band basis defined by D. Thurston to cluster algebras arising from orbifolds. We prove that the bracelet bases are positive, and the bracelet basis for the affine cluster algebra of type $C_n^{(1)}$ is atomic. We also show that cluster monomial bases of all skew-symmetrizable cluster algebras of finite type are atomic.