On Cluster Algebras Arising from Unpunctured Surfaces

By Ralf Schiffler, Hugh Thomas. 2007

92 citations recorded by the source corpus.

We study cluster algebras that are associated to unpunctured surfaces, with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. In the special case where the cluster algebra is acyclic, we also give a formula for the expansion of cluster variables as a polynomial whose indeterminates are the cluster variables contained in the union of an arbitrary acyclic cluster and all its neighboring clusters in the mutation graph.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f10a490d28bff1a492c4ca2800fa752692ba7fc3

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Citations in the corpus

Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths (2012, background): “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).”
Positivity for cluster algebras from surfaces (2009, methodology): “simply-connected interior containing γas a diagonal, whose other diagonal and sides (denoted γ′,α 1,α 2,α 3,α 4) cross Tfewer times than γdoes. To do so we use (a slight generalization of) a lemma of [ST], which will be stated and proved in Section 9. Step 6. We check that φγ(x ˜γ) = xγ, by inductio”
Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings (2012, unknown): “In [18], building on earlier work [22, 24, 23, 17], the authors used this geometric interpretation to obtain a direct combinatorial formula for the cluster variables in cluster algebras from surfaces.”
Representations of Algebras and Related Topics (2011, background): “ntation of the graphs Dn, A˜n and D˜n. In these cases, the surface with marked points is respectively an n-gon with a puncture, an annulus without puncture, and an n-gon with two punctures (see [86], [88], [87]). QT = Qτ oo FZ−mutation //Q τ′ = QT′ τ ideal triang. oo ﬂip // OO OO 1:1 τ ideal triang”
Cluster expansion formulas and perfect matchings (2008, methodology): “Following [33], we will consider paths α in S that are concatenations of arcs and boundary arcs in the triangulation T , more precisely, α = (α1, α2, .”
Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs (2014, result): “In [23], building on earlier work [22,27–29], the authors gave a combinatorial formula for the cluster variables in cluster algebras of surface type.”
Positivity for cluster algebras (2013, background): “In this case, positivity has been shown in [24] building on [28], [30], [29], using the fact that each cluster variable in such a cluster algebra corresponds to a curve in an oriented Riemann surface, and the Laurent expansion of the cluster variable is determined by the crossing pattern of the curv”
Skein algebras and cluster algebras of marked surfaces (2012, unknown): “s a commutative cluster algebra A 1(). Commutative cluster algebras associated to marked surfaces have already been introduced ([GSV05], [FG06]) and extensively studied ([FST08a],[FST08b] and [Sch08],[ST09],[MSW11]). The relation of A 1() to skein algebras was noticed in [FG06, Section 12.3], and th”
Cluster algebras and derived categories (2012, background): “• combinatorics and in particular the study of polyhedra like the Stasheff associahedra [21] [22] [42] [43] [71] [90] [102] [104] [103] [130] .”