From dimers to webs

2017/05/26 by Chris Fraser, Thomas Lam, Fraser, Chris +3 · 7 citations
Physics and Astronomy · Mathematics · #Theoretical and Computational Physics #Stochastic processes and statistical mechanics #Noncommutative and Quantum Gravity Theories

paper · pdf · doi:10.48550/arxiv.1705.09424

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 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 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 SL_r-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.

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• A quantum N-dimer model
• Webification of symmetry classes of plane partitions
• Local Statistics of the M_n-Dimer Model
• Plabic Tangles and Cluster Promotion Maps
• Twists, Higher Dimer Covers, and Web Duality for Grassmannian Cluster Algebras
• Web Diagrams of Cluster Variables for Grassmannian Gr(4,8)

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