Laurent expansions in cluster algebras via quiver representations

Abstract

We study Laurent expansions of cluster variables in a cluster algebra of rank 2 associated to a generalized Kronecker quiver. In the case of the ordinary Kronecker quiver, we obtain explicit expressions for Laurent expansions of the elements of the canonical basis for the corresponding cluster algebra.

19 pages; v2: added Remark 5.7; v3: minor editorial changes, final version to appear in Moscow Math. Journal; v4: more minor editorial changes

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• Quivers with potentials and their representations I: Mutations
• Triangular bases in quantum cluster algebras
• On the quiver Grassmannian in the acyclic case