On cluster algebras from unpunctured surfaces with one marked point

Çanakçı, ̛İlke; Lee, Kyungyong; Schiffler, Ralf

doi:10.1090/bproc/21

Cited by 24 publications

(23 citation statements)

References 32 publications

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“…Furthermore, in all known cases quivers admitting a maximal green sequence, a particular type of a green-to-red sequence, also have the property that their cluster algebra agrees with their upper cluster algebra. See [5] for a detailed overview, however a precise relationship between the two concepts remains unclear. Later, it was shown in [17] that the existence of such sequence is not invariant under mutation, even though the cluster algebra remains the same.…”

Section: Introductionmentioning

confidence: 99%

Green-to-red sequences for positroids

Ford

Serhiyenko

2018

Journal of Combinatorial Theory, Series A

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L-diagrams are combinatorial objects that parametrize cells of the totally nonnegative Grassmannian, called positroid cells, and each L-diagram gives rise to a cluster algebra which is believed to be isomorphic to the coordinate ring of the corresponding positroid variety. We study quivers arising from these diagrams and show that they can be constructed from the well-behaved quivers associated to Grassmannians by deleting and merging certain vertices. Then, we prove that quivers coming from arbitrary L-diagrams, and more generally reduced plabic graphs, admit a particular sequence of mutations called a green-to-red sequence.

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Section: Introductionmentioning

confidence: 99%

Green-to-red sequences for positroids

Ford

Serhiyenko

2018

Journal of Combinatorial Theory, Series A

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“…In the sequel [24], the formula was the key ingredient for the construction of two bases for the cluster algebra, in the case where the surface has no punctures and has at least 2 marked points. As an application of the computational tools developed in [3] and in the present paper, it is proved in [2] that the basis construction of [24] also applies to surfaces with non-empty boundary and with exactly one marked point.…”

Section: Introductionmentioning

confidence: 67%

“…Proof The bijection ϕ is the same as in the grafting case in Theorem 4.4 (2). It is given explicitly by the operation described in [3, Figure 11].…”

Section: Theorem 413 There Is a Bijectionmentioning

confidence: 99%

“…This theory has already found the following applications. In [5], the authors use snake graphs to study extensions of modules over Jacobian algebras and triangles in the cluster category associated to triangulations of unpunctured surfaces, and in [2], the snake graph calculus is used to show that for unpunctured surfaces with exactly one marked point, the upper cluster algebra coincides with the cluster algebra, which was one of the last cases of the mutation finite cluster algebras for which the question was still open.…”

Section: Introductionmentioning

confidence: 99%

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Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs

Çanakçı

Schiffler

2015

Math. Z.

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Snake graphs appear naturally in the theory of cluster algebras. For cluster algebras from surfaces, each cluster variable is given by a formula whose terms are parametrized by the perfect matchings of a snake graph. In this paper, we continue our study of snake graphs from a combinatorial point of view. We advance the study of abstract snake graphs by introducing the notions of abstract band graphs, self-crossings of abstract snake graphs as well as their resolutions. We show that there is a bijection between the set of perfect matchings of crossing or self-crossing snake graphs and the set of perfect matchings of the resolution of the crossing. In the situation where the snake and band graphs are coming from arcs and loops in a surface without punctures, we obtain a new proof of skein relations in the corresponding cluster algebra.

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“…It has been observed in the literature that the existence of a maximal green sequence or reddening sequence seems to coincide with equality of the cluster algebra and upper cluster algebra [CLS15]. The existence of such a sequence depends only on the mutable part of a quiver.…”

Section: Relationship With Reddeningmentioning

confidence: 97%

Upper cluster algebras and choice of ground ring

2019

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We initiate a study of the dependence of the choice of ground ring on the question on whether a cluster algebra is equal to its upper cluster algebra. A condition for when there is equality of the cluster algebra and upper cluster algebra is given by using a variation of Muller's theory of cluster localization. An explicit example exhibiting dependence on the ground ring is provided. We also present a maximal green sequence for this example.2010 Mathematics Subject Classification. 13F60.

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On cluster algebras from unpunctured surfaces with one marked point

Abstract: We extend the construction of canonical bases for cluster algebras from unpunctured surfaces to the case where the number of marked points on the boundary is one, and we show that the cluster algebra is equal to the upper cluster algebra in this case. arXiv:1407.5060v2 [math.RT]

Cited by 24 publications

References 32 publications

Green-to-red sequences for positroids

Green-to-red sequences for positroids

Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs

Upper cluster algebras and choice of ground ring

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