From dimers to tensor invariants

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 SLrwebs, and is built upon the r-fold dimer model on the network. When r is 1, our map is a reformulation of Postnikov's boundary measurement used to co-ordinatize positroid strata. When r is 2 or 3, it is a reformulation of the SL2and SL3web 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 SLrwebs, reproving a result of Cautis-Kamnitzer-Morrison. We establish compatibility between our map and restriction to positroid strata, and thus between webs and total positivity. Show more