Rotation-invariant web bases from hourglass plabic graphs

The Grassmannian is a disjoint union of open positroid varieties Π μ ∘ , certain smooth irreducible subvarieties whose definition is motivated by total positivity. The coordinate ring ℂ [ Π μ ∘ ] is a cluster algebra, and each reduced plabic graph G for Π μ ∘ determines a cluster. We study the effect of relabeling the boundary vertices of G by a permutation ρ . Under suitable hypotheses on the permutation, we show that the relabeled graph G ρ determines a cluster for a different open positroid variety Π π ∘ . As a key step in the proof, we show that Π π ∘ and Π μ ∘ are isomorphic by a nontrivial twist isomorphism. Our constructions yield a family of cluster structures on each open positroid variety, given by plabic graphs with appropriately permuted boundary labels. We conjecture that the seeds in all of these cluster structures are related by a combination of mutations and rescalings by Laurent monomials in frozen variables. We establish this conjecture for (open) Schubert and opposite Schubert varieties. As an application, we also show that for certain reduced plabic graphs G , the “source” cluster and the “target” cluster are related by mutation and Laurent monomial rescalings.

Inspired by the BCFW recurrence for tilings of the amplituhedron, we introduce the general framework of `plabic tangles' that utilizes plabic graphs to define rational maps between products of Grassmannians called `promotions'. The central conjecture of the paper is that promotion maps are quasi-cluster homomorphisms, which we prove for several classes of promotions. In order to define promotion maps, we utilize $m$-vector-relation configurations ($m$-VRCs) on plabic graphs. We relate $m$-VRCs to the degree (a.k.a `intersection number') of the amplituhedron map on positroid varieties and characterize all plabic trees with intersection number one and their VRCs. Finally, we show that promotion maps admit an operad structure and, supported by the class of `$4$-mass box' promotions, we point at new positivity properties for non-rational maps beyond cluster algebras. Promotion maps have important connections to the geometry and cluster structure of the amplituhedron and singularities of scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory.

We study the dimer model for a planar bipartite graph N embedded in a disk,\nwith boundary vertices on the boundary of the disk. Counting dimer\nconfigurations with specified boundary conditions gives a point in the totally\nnonnegative Grassmannian. Considering pairing probabilities for the\ndouble-dimer model gives rise to Grassmann analogues of Rhoades and Skandera's\nTemperley-Lieb immanants. The same problem for the (probably novel)\ntriple-dimer model gives rise to the combinatorics of Kuperberg's webs and\nGrassmann analogues of Pylyavskyy's web immanants. This draws a connection\nbetween the square move of plabic graphs (or urban renewal of planar bipartite\ngraphs), and Kuperberg's square reduction of webs. Our results also suggest\nthat canonical-like bases might be applied to the dimer model.\n We furthermore show that these functions on the Grassmannian are compatible\nwith restriction to positroid varieties. Namely, our construction gives bases\nfor the degree two and degree three components of the homogeneous coordinate\nring of a positroid variety that are compatible with the cyclic group action.\n