Cyclic Sieving and Cluster Duality of Grassmannian

Abstract

We introduce a decorated configuration space Conf × n (a) with a potential function W. We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of Conf × n (a), W canonically parametrizes a linear basis of the homogeneous coordinate ring of the Grassmannian Gr a (n) with respect to the Plücker embedding. We prove that Conf × n (a), 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.

Cyclic Sieving for Plane Partitions and Symmetry

“…In particular, he employed the dual canonical basis for representations of the general linear group [41,49]. More recently, Shen and Weng [79] gave a different proof of Theorem 1.2. Their approach was from the perspective of cluster algebras and the "cluster duality" conjecture of Fock and Goncharov [18].…”

“…In both the Rhoades [63] and Shen-Weng [79] proofs the vector space in question actually carries more structure: it is a GL(a + b) representation. And in both proofs the linear operator corresponding to promotion is the action of a particular lift to the general linear group of the long cycle (a.k.a.…”

“…Remark 3.5. As mentioned in Section 1, there is another proof of cyclic sieving for plane partitions under promotion (Theorem 1.2) due to Shen and Weng [79]. The main difference from Rhoades's proof is that, rather than use the Lusztig/Kashiwara dual canonical basis, Shen-Weng used a basis of the coordinate ring of the Grassmannian coming from its cluster algebra structure.…”

“…Understanding the behavior of the Dynkin diagram automorphism φ B seems tractable. Indeed, in the case a = b = n, the quiver Γ a,a+b (see [79,Section 2.4]) defining the Grassmannian cluster ensemble has a transposition symmetry A i,j → A j,i (ignoring arrows between frozen vertices, which are largely irrelevant). Moreover, this A i,j → A j,i symmetry exactly corresponds to the Plücker coordinate map ∆ I → ∆ −w 0 (I) which Lemma 3.1 says defines the action of φ B .…”

“…(We do not mean to suggest that it is totally hopeless to work in other types. For instance, the theory of plabic graphs [56,76] is ultimately the combinatorial underpinning of the Shen-Weng [79] proof of cyclic sieving, and the work of Karpman [38,39,40] extends much of the theory of plabic graphs to the Lagrangian Grassmannian.) Our results in this paper point the way to an alternative but ultimately complementary approach to Conjectures 5.1 and 5.2: stay in the type A world but impose symmetries.…”