Séminaire Lotharingien de Combinatoire, 78B.75 (2017), 12 pp.
Chris Fraser, Thomas Lam and Ian Le
From Dimers to Tensor Invariants
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
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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