Cyclic Sieving and Cluster Duality of Grassmannian

Abstract

We introduce a decorated configuration space $\mathcal{C}{\mathrm{onf}}_n^{\times }(a)$ with a potential function $\mathcal{W}$. We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of $(\mathcal{C}{\mathrm{onf}}_n^{\times }(a),\mathcal{W})$ canonically parametrizes a linear basis of the homogeneous coordinate ring of the Grassmannian ${\mathrm{Gr}}_a(n)$ with respect to the Plücker embedding. We prove that $(\mathcal{C}{\mathrm{onf}}_n^{\times }(a),\mathcal{W})$ 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.