Séminaire Lotharingien de Combinatoire, 78B.75 (2017), 12 pp.

Chris Fraser, Thomas Lam and Ian Le

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 SLr webs, 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 coordinatize positroid strata. When r is 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, 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.

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

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