Cluster expansion formulas and perfect matchings

Abstract

We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph $G_{T,\gamma }$ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph $G_{T,\gamma }$.

19 pages, 8 figures

References in corpus (7)

• Alternating sign matrices and domino tilings
• Cluster characters for triangulated 2-Calabi--Yau categories
• Denominators of cluster variables
• On cluster algebras with coefficients and 2-Calabi-Yau categories
• Cluster algebras and triangulated surfaces. Part I: Cluster complexes
• On cluster algebras arising from unpunctured surfaces II
• Combinatorial interpretations for rank-two cluster algebras of affine type

Cited by in corpus (2)

• Quivers with potentials associated to triangulated surfaces, Part II: Arc representations
• Cluster Variables and Perfect Matchings of Subgraphs of the $d{P}_3$ Lattice