Links in the complex of weakly separated collections
Plabic graphs are combinatorial objects used to study the totally nonnegative Grassmannian. Faces of plabic graphs are labeled by k-element sets of positive integers, and a collection of such k-element sets are the face labels of a plabic graph if that collection forms a maximal weakly separated col...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2020-04-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/6389/pdf |
| _version_ | 1797270217915629568 |
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| author | Suho Oh David Speyer |
| author_facet | Suho Oh David Speyer |
| author_sort | Suho Oh |
| collection | DOAJ |
| description | Plabic graphs are combinatorial objects used to study the totally nonnegative Grassmannian. Faces of plabic graphs are labeled by k-element sets of positive integers, and a collection of such k-element sets are the face labels of a plabic graph if that collection forms a maximal weakly separated collection. There are moves that one can apply to plabic graphs, and thus to maximal weakly separated collections, analogous to mutations of seeds in cluster algebras. In this short note, we show if two maximal weakly separated collections can be mutated from one to another, then one can do so while freezing the face labels they have in common. In particular, this provides a new, and we think simpler, proof of Postnikov's result that any two reduced plabic graphs with the same decorated permutations can be mutated to each other. |
| first_indexed | 2024-04-25T02:00:46Z |
| format | Article |
| id | doaj.art-94a231e20b884a58bef9b810f0c5b6d6 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:00:46Z |
| publishDate | 2020-04-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-94a231e20b884a58bef9b810f0c5b6d62024-03-07T14:55:20ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502020-04-01DMTCS Proceedings, 28th...10.46298/dmtcs.63896389Links in the complex of weakly separated collectionsSuho Oh0David Speyer1Texas State UniversityUniversity of Michigan [Ann Arbor]Plabic graphs are combinatorial objects used to study the totally nonnegative Grassmannian. Faces of plabic graphs are labeled by k-element sets of positive integers, and a collection of such k-element sets are the face labels of a plabic graph if that collection forms a maximal weakly separated collection. There are moves that one can apply to plabic graphs, and thus to maximal weakly separated collections, analogous to mutations of seeds in cluster algebras. In this short note, we show if two maximal weakly separated collections can be mutated from one to another, then one can do so while freezing the face labels they have in common. In particular, this provides a new, and we think simpler, proof of Postnikov's result that any two reduced plabic graphs with the same decorated permutations can be mutated to each other.https://dmtcs.episciences.org/6389/pdfcombinatorics[math.math-co]mathematics [math]/combinatorics [math.co] |
| spellingShingle | Suho Oh David Speyer Links in the complex of weakly separated collections Discrete Mathematics & Theoretical Computer Science combinatorics [math.math-co]mathematics [math]/combinatorics [math.co] |
| title | Links in the complex of weakly separated collections |
| title_full | Links in the complex of weakly separated collections |
| title_fullStr | Links in the complex of weakly separated collections |
| title_full_unstemmed | Links in the complex of weakly separated collections |
| title_short | Links in the complex of weakly separated collections |
| title_sort | links in the complex of weakly separated collections |
| topic | combinatorics [math.math-co]mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/6389/pdf |
| work_keys_str_mv | AT suhooh linksinthecomplexofweaklyseparatedcollections AT davidspeyer linksinthecomplexofweaklyseparatedcollections |