From dimers to webs

Chris Fraser, Thomas Lam, Ian Tuan-Yen Le

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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, 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.

Original languageEnglish (US)
Pages (from-to)6087-6124
Number of pages38
JournalTransactions of the American Mathematical Society
Volume371
Issue number9
DOIs
StatePublished - 2019

Keywords

  • Boundary measurement
  • Dimer
  • Grassmannian
  • Positroid
  • Web

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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