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Symmetry, Integrability and Geometry: Methods and Applications, 2020, Volume 16, 067, 41 pp.
DOI: https://doi.org/10.3842/SIGMA.2020.067
(Mi sigma1604)
 

This article is cited in 8 scientific papers (total in 8 papers)

Cyclic Sieving and Cluster Duality of Grassmannian

Linhui Shen, Daping Weng

Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824, USA
Full-text PDF (697 kB) Citations (8)
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Abstract: We introduce a decorated configuration space $\mathscr{C}\!\mathrm{onf}_n^\times(a)$ with a potential function $\mathcal{W}$. We prove the cluster duality conjecture of Fock–Goncharov for Grassmannians, that is, the tropicalization of $\big(\mathscr{C}\!\mathrm{onf}_n^\times(a), \mathcal{W}\big)$ canonically parametrizes a linear basis of the homogeneous coordinate ring of the Grassmannian $\operatorname{Gr}_a(n)$ with respect to the Plücker embedding. We prove that $\big(\mathscr{C}\!\mathrm{onf}_n^\times(a), \mathcal{W}\big)$ is equivalent to the mirror Landau–Ginzburg model of the Grassmannian considered by Eguchi–Hori–Xiong, Marsh–Rietsch and Rietsch–Williams. As an application, we show a cyclic sieving phenomenon involving plane partitions under a sequence of piecewise-linear toggles.
Keywords: cluster algebra, cluster duality, mirror symmetry, Grassmannian, cyclic sieving phenomenon.
Received: January 7, 2020; in final form July 14, 2020; Published online July 25, 2020
Bibliographic databases:
Document Type: Article
Language: English
Citation: Linhui Shen, Daping Weng, “Cyclic Sieving and Cluster Duality of Grassmannian”, SIGMA, 16 (2020), 067, 41 pp.
Citation in format AMSBIB
\Bibitem{SheWen20}
\by Linhui~Shen, Daping~Weng
\paper Cyclic Sieving and Cluster Duality of Grassmannian
\jour SIGMA
\yr 2020
\vol 16
\papernumber 067
\totalpages 41
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\crossref{https://doi.org/10.3842/SIGMA.2020.067}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85090620122}
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  • This publication is cited in the following 8 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Symmetry, Integrability and Geometry: Methods and Applications
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    References:17
     
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