Daping Weng, Yale University – Geometric Methods in Representation Theory Seminar

Title: Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian

Abstract: Fix two positive integers $a$ and $b$. Scott showed that a homogeneous coordinate ring of the Grassmannian $Gr_{a, a+b}$ has the structure of a cluster algebra. This homogeneous coordinate ring can be decomposed into a direct sum of irreducible representations of $GL_{a+b}$ which correspond to integer multiples of the fundamental weight $w_a$. By proving the Fock-Goncharov cluster duality conjecture for the Grassmannian using a sufficient condition found by Gross, Hacking, Keel, and Kontsevich, we obtain bases parametrized by plane partitions for these irreducible representations. As an application, we use these bases to show a cyclic sieving phenomenon of plane partitions under a certain sequence of toggling operations. This is joint work with Linhui Shen.