Positivity for cluster algebras from surfaces

By Gregg Musiker, Ralf Schiffler, Lauren K. Williams. 2009

251 citations recorded by the source corpus.

We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our formulas is a proof of the positivity conjecture of Fomin and Zelevinsky for cluster algebras from surfaces, in geometric type.

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

View Paper: https://www.semanticscholar.org/paper/8a4907c2c0f4ae593e75a3a9c1d1abc5788feaff

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Citations in the corpus

Cluster categories for marked surfaces: punctured case (2013, background): “on type, that they are all from marked surfaces expect for few cases. • Musiker-Schiﬄer-Williams [37] constructed two canonical bases by two types of collections of curves. • Musiker-Schiﬄer-Williams [36], Musiker-Williams [38] and Canakci-Schiﬄer [9, 10, 11] gave combinatorial formulas for cluster ”
Cluster algebras and continued fractions (2016, methodology): “arc . Each edge eof G is labeled by an arc ˝(e) of the triangulation T. We dene the weight x(e) of the edge eto be cluster variable associated to the arc ˝(e). Thus x(e) = x ˝(e). The main result of [MSW] is a combinatorial formula for the cluster variables in A(S;M). Let be an arc and x the corresp”
Cluster algebras and derived categories (2012, background): “…[37] [38] [39] . . . ; • combinatorics and in particular the study of polyhedra like the Stasheff associahedra [21] [22] [42] [43] [71] [90] [102] [104] [103] [130] . . . ; • commutative and non commutative algebraic geometry and in particular the study of stability conditions in the sense of…”
Tensor diagrams and cluster algebras (2012, background): “ [15] predicts that the Laurent polynomial expressing any cluster variable in terms of any given seed has positive coeﬃcients. This conjecture has been proved in many special cases, see in particular [4,45,46]. It is well known, and easy to see, that Conjecture 9.10 is stronger than Laurent positivi”
Cluster algebras and representation theory (2010, methodology): “ positivity for a certain class of cluster algebras. More positivity results, based on combinatorial or geometric descriptions of these coeﬃcients, have been obtained by Musiker, Schiﬄer and Williams [48], and by Nakajima [52]. 3. The cluster structure of C[N] To attack Problem 1.1 we adopt the foll”
Total positivity and cluster algebras (2010, background): “—all of them are in fact Laurent polynomials! We conjectured [26] that these Laurent polynomials always have positive coeﬃcients; many instances of this conjecture have been proved (see in particular [4, 14, 48, 50]) but the general case seems out of reach at the moment. 12 Sergey Fomin Another basi”
On cluster algebras arising from unpunctured surfaces II (2008, background): “In a sequel to the present paper [34], the authors give expansion formulas for the cluster variables in cluster algebras from arbitrary surfaces (allowing punctures) and prove the positivity conjecture for these cluster algebras.”
Cluster expansion formulas and perfect matchings (2008, background): “Note added in print In a sequel to the present paper [ 27 ], the authors give expansion formulas for the cluster variables in cluster algebras from arbitrary surfaces (allowing punctures) and prove the positivity conjecture for these cluster algebras.”
Positivity for cluster algebras (2013, background): “• An elementary combinatorial formula for cluster algebras from surfaces given in [24].”
Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths (2012, background): “h [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 of this kind, for elements of certain explicit”
Matrix formulae and skein relations for cluster algebras from surfaces (2011, methodology): “2]. Each element will be deﬁned as a sum over matchings of a graph. When γ is an ordinary arc, XT γ recovers the cluster expansion formula for the cluster variable associated to γ, with respect to ΣT [MSW]. Deﬁnition 3.1 (Generalized arcs). A generalized arc in (S,M) is a curve γ in S such that: the”
Cluster algebras and triangulated orbifolds (2011, methodology): “ion-ﬁnite matrices. We then use unfoldings to compute the growth rate of all cluster algebras originating from orbifolds, and for generalization of positivity results by Musiker, Schiﬄer and Williams [MSW] to Laurent expansions of corresponding cluster variables. Another application of the construct”
Bases for cluster algebras from surfaces (2011, methodology): “s, a (tagged) arc with a self-crossing. See Figure 22. For this reason we need to make sense of the element of the (fraction ﬁeld of the) cluster algebra associated to a generalized tagged arc. As in [MSW], in order to deduce the positivity of such elements with respect to all clusters, it suﬃces to”
Skein algebras and cluster algebras of marked surfaces (2012, unknown): “mutative 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 the equal”
Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs (2014, background): “If the curve is an arc, these Laurent polynomials are cluster variables and are given by the formula of [23] in terms of perfect matchings of snake graphs.”