Bases for cluster algebras from orbifolds

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.

39 pages, lots of figures; v3: minor changes, accepted version

References in corpus (3)

• The greedy basis equals the theta basis
• Positivity in skew-symmetric cluster algebras of finite type
• Atomic bases in cluster algebras of types A and $\tilde{A}$

Cited by in corpus (12)

• Density of $g$-vector cones from triangulated surfaces
• Dual canonical bases and quantum cluster algebras
• Bases for cluster algebras from orbifolds with one marked point
• $F$-matrices of cluster algebras from triangulated surfaces
• Snake Graphs from Triangulated Orbifolds
• Positivity for quasi-cluster algebras
• A categorification of cluster algebras of type B and C through symmetric quivers
• Positivity for quantum cluster algebras from unpunctured orbifolds
• Cluster expansion formulas and perfect matchings for type B and C
• Exchange Relations for Finite Type Cluster Algebras with Acyclic Initial Seed and Principal Coefficients
• Noncrossing partitions of a marked surface
• Denseness of $g$-vector cones from weighted orbifolds