Bases for cluster algebras from surfaces

Musiker, Gregg; Schiffler, Ralf; Williams, Lauren

doi:10.1112/s0010437x12000450

Cited by 120 publications

(220 citation statements)

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“…11). The bracelets basis was considered by Musiker, Schiffler, and Williams (12), who proved a weaker form of positivity. This weaker positivity and explicit combinatorial formulas have been well studied (13)(14)(15)(16).…”

Section: Introductionmentioning

confidence: 99%

Positive basis for surface skein algebras

Thurston

2014

Proc. Natl. Acad. Sci. U.S.A.

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Section: Introductionmentioning

confidence: 99%

Positive basis for surface skein algebras

Thurston

2014

Proc. Natl. Acad. Sci. U.S.A.

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“…Dimer configurations have also been used as a method for computing Laurent expansions for cluster variables for cluster algebras of finite classical type [10,31], and for cluster algebras associated to triangulations of surfaces [9,32,33,34]. Both cases involve cluster algebras of finite mutation type, the homogeneous coordinate ring of the Grassmannian Gr 2,n (up to coefficients) being common to both cases.…”

Section: Introductionmentioning

confidence: 99%

Twists of Plücker Coordinates as Dimer Partition Functions

Marsh

Scott

2015

Commun. Math. Phys.

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Abstract. The homogeneous coordinate ring of the Grassmannian Gr k,n has a cluster structure defined in terms of planar diagrams known as Postnikov diagrams. The cluster corresponding to such a diagram consists entirely of Plücker coordinates. We introduce a twist map on Gr k,n , related to the Berenstein-Fomin-Zelevinsky-twist, and give an explicit Laurent expansion for the twist of an arbitrary Plücker coordinate in terms of the cluster variables associated with a fixed Postnikov diagram. The expansion arises as a (scaled) dimer partition function of a weighted version of the bipartite graph dual to the Postnikov diagram, modified by a boundary condition determined by the Plücker coordinate. We also relate the twist map to a maximal green sequence. IntroductionFor positive integers k ≤ n let Gr k,n denote the Grassmannian of all k-dimensional vector subspaces of C n . The results of [39] (see also [17,18]) prove that its homogeneous coordinate ring C[Gr k,n ] has the structure of a cluster algebra which possesses a distinguished finite family of seeds ( x P , Q P ) constructed from certain planar diagrams P , known as alternating strand diagrams or Postnikov diagrams.The extended cluster x P of each seed of this kind consists entirely of Plücker coordinates which, in addition to the associated quiver Q P , can be read off directly from the Postnikov diagram. Moreover, every Plücker coordinate occurs as an element of x P for some Postnikov diagram P and thus every Plücker coordinate is either a cluster variable or a coefficient. When k = 2 every seed is of this form and consequently every cluster variable is a Plücker coordinate. In general the homogeneous coordinate ring is of wild type -possessing infinitely many seeds and infinitely many cluster variables, which in general will not be Plücker coordinates and which are, at present, unclassified.In this paper we consider a certain rational map from the Grassmannian to itself which we call the twist. This map may be pre-composed with any regular function f in C[Gr k,n ] to form a twisted version ← − f ; here, we consider twisted Plücker coordinates. Up to coefficients 1 , the twist of any cluster variable is a cluster variable (see Proposition 8.10). 2By the Laurent Phenomenon [15], each cluster variable in C[Gr k,n ] can be expressed uniquely as a Laurent polynomial in the extended cluster of any seed. In view of this, we compute Laurent

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“…The coefficients are now known to be non-negative [25,30], but this had been an open problem for more than ten years. Several different bases have been constructed for cluster algebras of different types; see [4,22,25,29,33]. …”

Section: What Is a Cluster Algebra?mentioning

confidence: 99%

“…The Laurent polynomial expansion of the cluster variable is given as a sum over all perfect matchings of the weight and the height of the matching. For surfaces without punctures these formulas were used in [33] to define canonical bases for the cluster algebra in terms of curves in the surface. Moreover, the relations in the cluster algebra are given geometrically by locally replacing a crossing × of two segments of curves with the sum of segments and ⊃⊂, respectively; see [34].…”

Section: Relations To Other Areasmentioning

confidence: 99%