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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
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
Citation:
Linhui Shen, Daping Weng, “Cyclic Sieving and Cluster Duality of Grassmannian”, SIGMA, 16 (2020), 067, 41 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1604 https://www.mathnet.ru/eng/sigma/v16/p67
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