Dimers, webs, and positroids

Abstract

We study the dimer model for a planar bipartite graph N embedded in a disk, with boundary vertices on the boundary of the disk. Counting dimer configurations with specified boundary conditions gives a point in the totally non-negative Grassmannian. Considering pairing probabilities for the double-dimer model gives rise to Grassmann analogs of Rhoades and Skandera's Temperley–Lieb immanants. The same problem for the (probably novel) triple-dimer model gives rise to the combinatorics of Kuperberg's webs and Grassmann analogs of Pylyavskyy's web immanants. This draws a connection between the square move of plabic graphs (or urban renewal of planar bipartite graphs), and Kuperberg's square reduction of webs. Our results also suggest that canonical-like bases might be applied to the dimer model. We furthermore show that these functions on the Grassmannian are compatible with restriction to positroid varieties. Namely, our construction gives bases for the degree 2 and degree 3 components of the homogeneous coordinate ring of a positroid variety that are compatible with the cyclic group action.