Cluster Structures and Subfans in Scattering Diagrams

Zhou, Yan

doi:10.3842/sigma.2020.013

Cited by 7 publications

(11 citation statements)

References 24 publications

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“…The X φ scattering diagram D X φ s defined in [8, Section 2.2.3] can be described as the quotient of the D X s diagram in N ⊗ R. This construction is well-defined as laid out in [8, Section 2.2.3]. Besides, Zhou [57] also gave a description of quotient scattering diagrams.…”

mentioning

confidence: 99%

Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces

Bardwell-Evans¹,

Cheung²,

Hong³

et al. 2021

Preprint

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We construct special Lagrangian fibrations for log Calabi-Yau surfaces, and scattering diagrams from Lagrangian Floer theory of the fibres. Then we prove that the scattering diagrams recover the scattering diagrams of Gross-Pandharipande-Siebert [28] and the canonical scattering diagrams of Gross-Hacking-Keel [24]. With an additional assumption on the non-negativity of boundary divisors, we compute the disc potentials of the Lagrangian torus fibres via a holomorphic/tropical correspondence. As an application, we provide a version of mirror symmetry for rank two cluster varieties.

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mentioning

confidence: 99%

Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces

Bardwell-Evans¹,

Cheung²,

Hong³

et al. 2021

Preprint

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“…So the source-labeled seed of G can be realized as a relabeled plabic graph seed satisfying the assumptions of Theorem 4. 21 We now state our main result in the direction of Conjecture 1.3. As preparation, let G ρ and H σ be reduced plabic graphs with trip permutation π, both satisfying Theorem 4.21(1) (i.e.…”

Section: Positroid Cluster Structures From Relabeled Plabic Graphsmentioning

confidence: 95%

“…We note that in general, seeds Σ and Σ giving two different cluster structures on a variety may not be related by quasi-cluster transformations. Zhou [21] gives an example of this for the cluster algebra of the Markov quiver.…”

Section: Introductionmentioning

confidence: 99%

Positroid cluster structures from relabeled plabic graphs

Fraser¹,

Sherman-Bennett²

2022

Algebraic Combinatorics

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The Grassmannian is a disjoint union of open positroid varieties Π • µ , certain smooth irreducible subvarieties whose definition is motivated by total positivity. The coordinate ringµ ] is a cluster algebra, and each reduced plabic graph G for Π • µ determines a cluster. We study the effect of relabeling the boundary vertices of G by a permutation ρ. Under suitable hypotheses on the permutation, we show that the relabeled graph G ρ determines a cluster for a different open positroid variety Π • π . As a key step in the proof, we show that Π • π and Π • µ are isomorphic by a nontrivial twist isomorphism. Our constructions yield a family of cluster structures on each open positroid variety, given by plabic graphs with appropriately permuted boundary labels. We conjecture that the seeds in all of these cluster structures are related by a combination of mutations and rescalings by Laurent monomials in frozen variables. We establish this conjecture for (open) Schubert and opposite Schubert varieties. As an application, we also show that for certain reduced plabic graphs G, the "source" cluster and the "target" cluster are related by mutation and Laurent monomial rescalings.

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“…We are grateful to all these institutions for their hospitality and outstanding working conditions they provided. Special thanks are due to Peigen Cao and Fang Li who in response to our request provided a generalization [3] of their previous results, to Linhui Shen for pointing out to us the preprint [21], to Alexander Shapiro and Gus Schrader for many fruitful discussions, and to the reviewers for constructive suggestions.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…Remark In a recent preprint [21], a technique of scattering diagrams is used to construct a log Calabi–Yau variety with two non‐equivalent cluster structures both associated with the Markov quiver. This variety is obtained by a certain augmentation of a cluster A‐variety with principal coefficients in the sense of [15].…”

Section: Two Generalized Cluster Structures On Dfalse(gl4false)$d(gl_4)$mentioning

confidence: 99%

Generalized cluster structures related to the Drinfeld double of GLn$GL_n$

Gekhtman

Shapiro

Vainshtein

2022

Journal of London Math Soc

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We prove that the regular generalized cluster structure on the Drinfeld double of 𝐺𝐿 𝑛 constructed in Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) is complete and compatible with the standard Poisson-Lie structure on the double. Moreover, we show that for 𝑛 = 4 this structure is distinct from a previously known regular generalized cluster structure on the Drinfeld double, even though they have the same compatible Poisson structure and the same collection of frozen variables. Further, we prove that the regular generalized cluster structure on band periodic matrices constructed in Gekhtman, Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) possesses similar compatibility and completeness properties.

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Cluster Structures and Subfans in Scattering Diagrams

Cited by 7 publications

References 24 publications

Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces

Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces

Positroid cluster structures from relabeled plabic graphs

Generalized cluster structures related to the Drinfeld double of GLn$GL_n$

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