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题目:Cluster Duality of Grassmannian and a Cyclic Sieving Phenomenon of Plane Partitions

报告人:翁达平(Michigan State University)

摘要: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. This homogeneous coordinate ring can be decomposed into a direct sum of irreducible representations of $GL_{a+b}$ which correspond to non-negative integer multiples of the fundamental weight $w_a$. We introduce a periodic configuration space $Conf_{a+b,a}$ equipped with a natural potential function $W$ and prove that the tropicalization of $(Conf_{a+b,a},W)$ canonically parametrizes bases for the irreducible summands of 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 plan partitions of size $a\times b$. 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.



邀请人:杨东 老师