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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



From dimers to webs

Authors: Chris Fraser, Thomas Lam and Ian Le
Journal: Trans. Amer. Math. Soc. 371 (2019), 6087-6124
MSC (2010): Primary 05E10; Secondary 14M15, 20C30, 05C10
Published electronically: January 24, 2019
MathSciNet review: 3937319
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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 $ \textnormal {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 $ \textnormal {SL}_2$- and $ \textnormal {SL}_3$-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 $ \textnormal {SL}_r$-webs, re-proving 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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Additional Information

Chris Fraser
Affiliation: Department of Mathematical Sciences, Indiana Unversity–Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana 46202

Thomas Lam
Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109

Ian Le
Affiliation: Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada, N2L 2Y5

Keywords: Dimer, web, boundary measurement, positroid, Grassmannian
Received by editor(s): June 23, 2017
Published electronically: January 24, 2019
Additional Notes: Some of the work took place at the Perimeter Institute. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation.
The second author acknowledges support from the Simons Foundation under award number 341949 and from the NSF under agreement No. DMS-1464693.
Article copyright: © Copyright 2019 American Mathematical Society