Cyclic Sieving and Cluster Duality of Grassmannian

TL;DR

This paper proves a cluster duality conjecture for Grassmannians, linking a decorated configuration space with a potential to a canonical basis, and demonstrates a cyclic sieving phenomenon involving plane partitions.

Contribution

It establishes the cluster duality for Grassmannians and connects it to mirror symmetry and cyclic sieving phenomena, providing new insights into the structure of Grassmannian coordinate rings.

Findings

Proves the cluster duality conjecture for Grassmannians.

Shows the configuration space with potential models the mirror Landau-Ginzburg of Grassmannians.

Demonstrates a cyclic sieving phenomenon involving plane partitions.

Abstract

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