Weak separation and plabic graphs
Name
Postnikov_Weak separation.pdf
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392.6 KB
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24261e039a8016a64dd04a62863a659b
Author(s)
Oh, S.
Speyer, D. E.
Postnikov, Alexander
Date Issued
February 2015
Journal
Proceedings of the London Mathematical Society
Publisher
Oxford University Press - London Mathematical Society
Citation
Oh, Suho, Alexander Postnikov, and David E. Speyer. “Weak Separation and Plabic Graphs.” Proceedings of the London Mathematical Society 110.3 (2015): 721–754.
Version
Original manuscript
Abstract
Leclerc and Zelevinsky described quasicommuting families of quantum minors in terms of a certain combinatorial condition, called weak separation. They conjectured that all inclusion-maximal weakly separated collections of minors have the same cardinality, and that they can be related to each other by a sequence of mutations. Postnikov studied total positivity on the Grassmannian. He described a stratification of the totally non-negative Grassmannian into positroid strata, and constructed theirparameterization using plabic graphs. In this paper, we link the study of weak separation to plabic graphs. We extend the notion of weak separation to positroids. We generalize the conjectures of Leclerc and Zelevinsky, and related ones of Scott, and prove them. We show that the maximal weakly separated collections in a positroid are in bijective correspondence with the plabic graphs. This correspondence allows us to use the combinatorial techniques of positroids and plabic graphs to prove the (generalized) purity and mutation connectedness conjectures.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1112/plms/pdu052
