MUTATION INVARIANT FUNCTIONS ON CLUSTER ENSEMBLES ASSOCIATED WITH SURFACES

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2021

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We define the notion of an invariant function on a cluster ensemble with respect to a group action of the cluster modular group on its associated function fields. We realize many examples of previously studied functions as elements of this type of invariant ring and give many new examples. We construct invariants for cluster algebras associated with surfaces using hyperbolic geometry, Teichm'uller theory and skein algebras of surfaces. We complete a classification of them for surface ensembles for the action of Dehn twists, and generalize this classification to the non-surface mutation finite setting. We use this classification to answer some questions about the structure of affine cluster algebras, to construct a correspondence between $\A$ and $\X$ invariants, and to propose an explanation for why many different computations of canonical bases of cluster algebras agree.

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