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.

References in corpus (10)

• Total positivity, Grassmannians, and networks
• Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians
• Quantum geometry of moduli spaces of local systems and representation theory
• Quiver mutation and combinatorial DT-invariants
• Cyclic Sieving for Plane Partitions and Symmetry
• Donaldson-Thomas trasnsformations of moduli spaces of G-local systems
• From Dominoes to Hexagons
• Donaldson-Thomas Transformation of Grassmannian
• Completeness of determinantal Hamiltonian flows on the matrix affine Poisson space
• Stasheff polytopes and the coordinate ring of the cluster X-variety of type A_n

Cited by in corpus (15)

• Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians
• Cyclic Sieving for Plane Partitions and Symmetry
• Infinitely many Lagrangian fillings
• Domain wall formation and magnon localization in twisted chromium trihalides
• Moment curves and cyclic symmetry for positive Grassmannians
• Skew characters and cyclic sieving
• Cluster Structures on Double Bott-Samelson Cells
• Restriction of Global Bases and Rhoades's Theorem
• Cyclic symmetry loci in Grasssmannians
• The valuation pairing on an upper cluster algebra
• Promotion and cyclic sieving on families of SSYT
• Minuscule analogues of the plane partition periodicity conjecture of Cameron and Fon-Der-Flaass
• Newton--Okounkov bodies and minimal models for cluster varieties
• Quantum affine algebras and Grassmannians
• Order polynomial product formulas and poset dynamics