Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 067, 41 pages      arXiv:1803.06901      https://doi.org/10.3842/SIGMA.2020.067
Contribution to the Special Issue on Cluster Algebras

Cyclic Sieving and Cluster Duality of Grassmannian

Linhui Shen and Daping Weng
Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824, USA

Received January 07, 2020, in final form July 14, 2020; Published online July 25, 2020

Abstract
We introduce a decorated configuration space $\mathcal{C}{\rm onf}_n^\times(a)$ with a potential function $\mathcal{W}$. We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of $\big(\mathcal{C}{\rm onf}_n^\times(a), \mathcal{W}\big)$ 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 $\big(\mathcal{C}{\rm onf}_n^\times(a), \mathcal{W}\big)$ 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.

Key words: cluster algebra; cluster duality; mirror symmetry; Grassmannian; cyclic sieving phenomenon.

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