From dimers to webs

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

paper · pdf · doi:10.48550/arxiv.1705.09424

fixed typos in bibliography

arxiv created 2017/06/04 · arxiv updated 2017/06/06

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.

Cited by

• Clasped web bases from hourglass plabic graphs
• 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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• Higher-rank dimer models
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• Edge vectors on plabic networks in the disk and amalgamation of totally non-negative Grassmannians
• Rank-N Dimer Models on Surfaces