From dimers to webs

Abstract

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 $ \textnormal {SL}_r$-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 $ \textnormal {SL}_2$- and $ \textnormal {SL}_3$-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 $ \textnormal {SL}_r$-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.