Cluster algebras and triangulated orbifolds

Abstract

We construct geometric realization for non-exceptional mutation-finite cluster algebras by extending the theory of Fomin and Thurston to skew-symmetrizable case. Cluster variables for these algebras are renormalized lambda lengths on certain hyperbolic orbifolds. We also compute growth rate of these cluster algebras, provide positivity of Laurent expansions of cluster variables, and prove sign-coherence of c-vectors.

58 pages, lots of figures; v4: the accepted version with further corrections (not included in the journal version): the definition of an arc on an orbifold is corrected (thanks to T. Nakanishi who has told us about the mistake), typos corrected, references to the paper of Fomin-Thurston updated (including the numbers of statements). arXiv admin note: text overlap with arXiv:1210.5569 by other authors

References in corpus (2)

• Cluster algebras and triangulated surfaces. Part II: Lambda lengths
• Mutations of group species with potentials and their representations. Applications to cluster algebras

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• Continuous cluster categories of type D
• Expansion Posets for Polygon Cluster Algebras
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• Bases for cluster algebras from orbifolds with one marked point
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• Quantum cluster algebras from unpunctured triangulated surfaces: arbitrary coefficients and quantization
• Cyclic symmetry loci in Grasssmannians
• Note on character varieties and cluster algebras
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• Acyclic cluster algebras with dense $g$-vector fans
• A categorification of cluster algebras of type B and C through symmetric quivers
• Special Folding of Quivers and Cluster Algebras
• Positivity for quantum cluster algebras from unpunctured orbifolds
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• Exchange Relations for Finite Type Cluster Algebras with Acyclic Initial Seed and Principal Coefficients
• Quivers with potentials and actions of finite abelian groups
• Dominance phenomena: mutation, scattering and cluster algebras
• Snake Graphs and Caldero-Chapoton Functions from Triangulated Orbifolds
• Cluster expansion formulas and perfect matchings for type B and C
• Exchange graphs for mutation-finite non-integer quivers of rank 3
• Denseness of $g$-vector cones from weighted orbifolds
• Noncrossing partitions of a marked surface