Combinatorics Seminar

Fix two positive integers a and b. Scott showed that the homogeneous coordinate ring of the Grassmannian Gr(a, a+b) has the structure of a cluster algebra.

We introduce a periodic configuration space Conf(a, a+b) equipped with a natural potential function W and prove that the tropicalization of (Conf(a, a+b), W) canonically parametrizes bases for the homogeneous coordinate ring of Gr(a, a+b), as expected by the cluster duality conjecture of Fock and Goncharov. We identify the parametrizing set of each irreducible summand with a collection of plane partitions of size a x b. As an application, we use this identification to show a cyclic sieving phenomenon of plane partitions under a certain sequence of toggling operations. This is joint work with Linhui Shen. Speaker(s): Daping Weng (Michigan State University)