From dimers to webs

We formulate a higher-rank version of the boundary measurement map for weighted planar bipartite networks in the disk. It sends a network to a linear combination of SLr-webs and is built upon the r-fold dimer model on the network. When r equals 1, our map is a reformulation of Postnikov's boundary measurement used to coordinatize positroid strata. When r equals 2 or 3, it is a reformulation of the SL2- and SL3-web immanants defined by the second author. The basic result is that the higher-rank map factors through Postnikov's map. As an application, we deduce generators and relations for the space of SLr-webs, re-proving a result of Cautis-Kamnitzer-Morrison. We establish compatibility between our map and restriction to positroid strata and thus between webs and total positivity.