Cyclic Sieving and Cluster Duality of Grassmannian

Symmetry Integrability and Geometry-Methods and Applications

article

Linhui ShenDaping Weng¹ 

[1] Department of Mathematics, Michigan State University

关键词: cluster algebra;    cluster duality;    mirror symmetry;    Grassmannian;    cyclic sieving phenomeno;   

DOI  :  10.3842/SIGMA.2020.067

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown