Cluster expansion formulas and perfect matchings
By Gregg Musiker, Ralf Schiffler. 2008
101 citations recorded by the source corpus.
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 GT,γ 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 GT,γ.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/5e0f8bd7eb300870c7536ef86c031cae97827994
How researchers see this paper
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In Algebra and Number Theory
- Ralf Schiffler
- Ilke Çanakçi
- Dylan Thurston
- Greg Muller
- Sergey Fomin
- B. Keller
- I. Reiten
- Bernard Leclerc
In Discrete Mathematics and Combinatorics
- Ralf Schiffler
- Dylan Thurston
- Ilke Çanakçi
- Greg Muller
- Sergey Fomin
- B. Keller
- Bernard Leclerc
- Andrei Zelevinsky
In Geometry and Topology
- Ralf Schiffler
- Ilke Çanakçi
- Dylan Thurston
- Greg Muller
- B. Keller
- Sergey Fomin
- I. Reiten
- Bernard Leclerc
Citations in the corpus
- Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths (2012, background): “cation of cluster algebras of polynomial growth [38]. Manifestly positive combinatorial formulas for Laurent expansions of cluster variables in cluster algebras associated with surfaces were given in [43, 48] (for surfaces without punctures) and [44] (in full generality). Even more general formulas ”
- Skein algebras and cluster algebras of marked surfaces (2012, methodology): “ein relations in [GSV10, Section 2.1.5]. For general marked surfaces, a related method for producing Laurent expansions in terms of T-paths and snake graphs has been developed in [Sch10], [ST09], and [MS10]. One consequence of this formula is a denominator [ ([x])] for the skew-Laurent expression. ”
- Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs (2014, background): “Abstract snake graphs have been introduced in [3] motivated by the snake graphs appearing in the combinatorial formulas for cluster variables in cluster algebras of surface type in [22,23,26].”