Cluster algebras and triangulated orbifolds

Felikson, Anna; Shapiro, Michael; Tumarkin, Pavel

doi:10.1016/j.aim.2012.07.032

Cited by 54 publications

(143 citation statements)

References 11 publications

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“…Caution 3.5. This should not be confused with a tagged triangulation of an orbifold also studied in [FST11]. We do not use tagged triangulations in this paper.…”

Section: Definition 34 a Labeled Ideal Triangulation T Of An Orbifomentioning

confidence: 99%

“…It was explained in [FST11] that any labeled ideal triangulation of an orbifold is built by glueing the edges of four kinds of "puzzle pieces" depicted in Figure 11. We call them type i, type ii, type iii, and type iv pieces, respectively, as specified in Figure 11.…”

Section: Definition 34 a Labeled Ideal Triangulation T Of An Orbifomentioning

confidence: 99%

“…Instead of using the surface realization of cluster algebras, we need to use the orbifold realization of cluster algebras by Felikson, Shapiro, and Tumarkin [FST11]. The simple poles correspond to the orbifold points therein.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Exact WKB Analysis and Cluster Algebras II: Simple Poles, Orbifold Points, and Generalized Cluster Algebras

Iwaki

Наканиши

2015

Int Math Res Notices

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Abstract. This is a continuation of developing mutation theory in exact WKB analysis using the framework of cluster algebras. Here we study the Schrödinger equation on a compact Riemann surface with turning points of simple-pole type. We show that the orbifold triangulations by Felikson, Shapiro, and Tumarkin provide a natural framework of describing the mutation of Stokes graphs, where simple poles correspond to orbifold points. We then show that under the mutation of Stokes graphs around simple poles the Voros symbols mutate as the variables of generalized cluster algebras introduced by Chekhov and Shapiro.

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“…Caution 3.5. This should not be confused with a tagged triangulation of an orbifold also studied in [FST11]. We do not use tagged triangulations in this paper.…”

Section: Definition 34 a Labeled Ideal Triangulation T Of An Orbifomentioning

confidence: 99%

Section: Definition 34 a Labeled Ideal Triangulation T Of An Orbifomentioning

confidence: 99%

See 1 more Smart Citation

Exact WKB Analysis and Cluster Algebras II: Simple Poles, Orbifold Points, and Generalized Cluster Algebras

Iwaki

Наканиши

2015

Int Math Res Notices

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“…Moreover, they determine in [10] (for the orbifold cases) and in [11] (with H. Thomas for the remaining exceptional cases) the growth rate for all mutation finite cluster algebras. In particular, [11] provides an independent proof for the exponential growth of all tubular cluster algebras which is based on a direct study of the corresponding cluster modular groups.…”

Section: Introductionmentioning

confidence: 99%

Tubular cluster algebras II: Exponential growth

Barot

Geiß

Jasso

2013

Journal of Pure and Applied Algebra

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“…From a classification point of view, this class is very important, since it has been shown in [6] that almost all (skew-symmetric) mutation finite cluster algebras are of surface type. For generalizations to the skew-symmetrizable case see [7,8]. The closely related surface skein algebras were studied in [21,30].…”

Section: Introductionmentioning

confidence: 99%

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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Cluster algebras and triangulated orbifolds

Cited by 54 publications

References 11 publications

Exact WKB Analysis and Cluster Algebras II: Simple Poles, Orbifold Points, and Generalized Cluster Algebras

Exact WKB Analysis and Cluster Algebras II: Simple Poles, Orbifold Points, and Generalized Cluster Algebras

Tubular cluster algebras II: Exponential growth

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

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