Weak separation and plabic graphs
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 b...
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| Format: | Article |
| Language: | en_US |
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Oxford University Press - London Mathematical Society
2017
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| Online Access: | http://hdl.handle.net/1721.1/106463 https://orcid.org/0000-0002-3964-8870 |
| _version_ | 1826196386241052672 |
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| author | Oh, S. Speyer, D. E. Postnikov, Alexander |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Oh, S. Speyer, D. E. Postnikov, Alexander |
| author_sort | Oh, S. |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T10:26:03Z |
| format | Article |
| id | mit-1721.1/106463 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T10:26:03Z |
| publishDate | 2017 |
| publisher | Oxford University Press - London Mathematical Society |
| record_format | dspace |
| spelling | mit-1721.1/1064632022-09-30T21:06:17Z Weak separation and plabic graphs Oh, S. Speyer, D. E. Postnikov, Alexander Massachusetts Institute of Technology. Department of Mathematics Postnikov, Alexander 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. National Science Foundation (U.S.) (CAREER Award DMS-0504629) 2017-01-12T19:36:14Z 2017-01-12T19:36:14Z 2015-02 2014-03 Article http://purl.org/eprint/type/JournalArticle 0024-6115 1460-244X http://hdl.handle.net/1721.1/106463 Oh, Suho, Alexander Postnikov, and David E. Speyer. “Weak Separation and Plabic Graphs.” Proceedings of the London Mathematical Society 110.3 (2015): 721–754. https://orcid.org/0000-0002-3964-8870 en_US http://dx.doi.org/10.1112/plms/pdu052 Proceedings of the London Mathematical Society Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Oxford University Press - London Mathematical Society arXiv |
| spellingShingle | Oh, S. Speyer, D. E. Postnikov, Alexander Weak separation and plabic graphs |
| title | Weak separation and plabic graphs |
| title_full | Weak separation and plabic graphs |
| title_fullStr | Weak separation and plabic graphs |
| title_full_unstemmed | Weak separation and plabic graphs |
| title_short | Weak separation and plabic graphs |
| title_sort | weak separation and plabic graphs |
| url | http://hdl.handle.net/1721.1/106463 https://orcid.org/0000-0002-3964-8870 |
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