Cyclic Sieving and Cluster Duality of Grassmannian - Shen, Linhui; Weng, Daping (Symmetry, Integrability and Geometry: Methods and Applications, 2020)
We introduce a decorated configuration space Confˣₙ() with a potential function . We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of (Confˣₙ(), ) canonically parametrizes a linear basis of the homogeneous coordinate ring of the Grassmannian Grₐ(n) with respect to the Plücker embedding. We prove that (Confˣₙ(), ) 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.