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Cyclic Sieving and Cluster Duality of Grassmannian
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@article{SIGMA_2020_16_a66,
     author = {Linhui Shen and Daping Weng},
     title = {Cyclic {Sieving} and {Cluster} {Duality} of {Grassmannian}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2020},
     volume = {16},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a66/}
}
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%A Linhui Shen
%A Daping Weng
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%J Symmetry, integrability and geometry: methods and applications
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Linhui Shen; Daping Weng. Cyclic Sieving and Cluster Duality of Grassmannian. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a66/

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