Algebraic Geometry Seminar, UC Davis

Daping Weng, UC Davis

Cyclic Sieving and Cluster Duality for Grassmannian

Organizer's time: 2021-11-02 11:00 America/Los_Angeles

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For any two positive integers a and b, the homogeneous coordinate ring of Gr(a,a+b) is isomorphic to a direct sum over all irreducible GL(a+b) representations associated with weights that are multiples of w_a. Following a result of Scott, the homogeneous coordinate ring of a Grassmannian has the structure of a cluster algebra. The Fock-Goncharov cluster duality conjecture states that an (upper) cluster algebra admits a cluster canonical basis parametrized by the tropical integer points of the dual cluster variety. In a joint work with L. Shen, we introduce a periodic configuration space of lines as the cluster dual for Gr(a,a+b). We equip this cluster dual with a natural potential function W and obtain a cluster canonical basis for Gr(a,a+b), parametrized by plane partitions. As an application, we prove a cyclic sieving phenomenon of plane partitions under a certain toggling sequence.

Submitted by: Eugene Gorsky