Denominators of cluster variables

Abstract

Associated to any acyclic cluster algebra is a corresponding triangulated category known as the cluster category. It is known that there is a one-to-one correspondence between cluster variables in the cluster algebra and exceptional indecomposable objects in the cluster category inducing a correspondence between clusters and cluster-tilting objects. Fix a cluster-tilting object T and a corresponding initial cluster. By the Laurent phenomenon, every cluster variable can be written as a Laurent polynomial in the initial cluster. We give conditions on T equivalent to the fact that the denominator in the reduced form for every cluster variable in the cluster algebra has exponents given by the dimension vector of the corresponding module over the endomorphism algebra of T.

22 pages; one figure

References in corpus (2)

• From triangulated categories to cluster algebras
• On cluster algebras with coefficients and 2-Calabi-Yau categories

Cited by in corpus (14)

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• On cluster algebras with coefficients and 2-Calabi-Yau categories
• On cluster algebras arising from unpunctured surfaces II
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• On indecomposable $\tau$-rigid modules over cluster-tilted algebras of tame type
• The number of elements in the mutation class of a quiver of type $D_n$
• Cluster Expansions: T-walks, Labeled Posets and Matrix Calculations
• C-vectors and dimension vectors for cluster-finite quivers
• Denominator vectors and dimension vectors from triangulated surfaces
• Friezes of type D-tilde