Cyclic Sieving and Cluster Duality for Grassmannian

Linhui Shen, Daping Weng

Published 2018-03-19Version 1

We introduce a decorated configuration space $\mathscr{C}onf_n^\times(a)$ with a potential function $\mathcal{W}$. We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of $(\mathscr{C}onf_n^\times(a), \mathcal{W})$ canonically parametrizes a linear basis of the homogenous coordinate ring of the Grassmannian ${\rm Gr}_a(n)$. We prove that $(\mathscr{C}onf_n^\times(a), \mathcal{W})$ is equivalent to the mirror Landau-Ginzburg model of Grassmannian considered by Marsh-Rietsch and Rietsch-Williams. As an application, we show a cyclic sieving phenomenon involving plane partitions under a sequence of piecewise-linear toggles.