Quasi-isomorphisms of cluster algebras and the combinatorics of webs (extended abstract)
We provide bijections between the cluster variables (and clusters) in two families of cluster algebras which have received considerable attention. These cluster algebras are the ones associated with certain Grassmannians of k-planes, and those associated with certain spaces of decorated SLk-local sy...
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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/6395/pdf |
| _version_ | 1797270272606208000 |
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| author | Chris Fraser |
| author_facet | Chris Fraser |
| author_sort | Chris Fraser |
| collection | DOAJ |
| description | We provide bijections between the cluster variables (and clusters) in two families of cluster algebras which have received considerable attention. These cluster algebras are the ones associated with certain Grassmannians of k-planes, and those associated with certain spaces of decorated SLk-local systems in the disk in the work of Fock and Goncharov. When k is 3, this bijection can be described explicitly using the combinatorics of Kuperberg's basis of non-elliptic webs. Using our bijection and symmetries of these cluster algebras, we provide evidence for conjectures of Fomin and Pylyavskyy concerning cluster variables in Grassmannians of 3-planes. We also prove their conjecture that there are infinitely many indecomposable nonarborizable webs in the Grassmannian of 3-planes in 9-dimensional space. |
| first_indexed | 2024-04-25T02:01:38Z |
| format | Article |
| id | doaj.art-411f73187bd54343901ce16fd839445c |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:01:38Z |
| publishDate | 2020-04-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-411f73187bd54343901ce16fd839445c2024-03-07T14:55:20ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502020-04-01DMTCS Proceedings, 28th...10.46298/dmtcs.63956395Quasi-isomorphisms of cluster algebras and the combinatorics of webs (extended abstract)Chris Fraser0Department of Mathematics [Ann Arbor]We provide bijections between the cluster variables (and clusters) in two families of cluster algebras which have received considerable attention. These cluster algebras are the ones associated with certain Grassmannians of k-planes, and those associated with certain spaces of decorated SLk-local systems in the disk in the work of Fock and Goncharov. When k is 3, this bijection can be described explicitly using the combinatorics of Kuperberg's basis of non-elliptic webs. Using our bijection and symmetries of these cluster algebras, we provide evidence for conjectures of Fomin and Pylyavskyy concerning cluster variables in Grassmannians of 3-planes. We also prove their conjecture that there are infinitely many indecomposable nonarborizable webs in the Grassmannian of 3-planes in 9-dimensional space.https://dmtcs.episciences.org/6395/pdf[math.math-co]mathematics [math]/combinatorics [math.co] |
| spellingShingle | Chris Fraser Quasi-isomorphisms of cluster algebras and the combinatorics of webs (extended abstract) Discrete Mathematics & Theoretical Computer Science [math.math-co]mathematics [math]/combinatorics [math.co] |
| title | Quasi-isomorphisms of cluster algebras and the combinatorics of webs (extended abstract) |
| title_full | Quasi-isomorphisms of cluster algebras and the combinatorics of webs (extended abstract) |
| title_fullStr | Quasi-isomorphisms of cluster algebras and the combinatorics of webs (extended abstract) |
| title_full_unstemmed | Quasi-isomorphisms of cluster algebras and the combinatorics of webs (extended abstract) |
| title_short | Quasi-isomorphisms of cluster algebras and the combinatorics of webs (extended abstract) |
| title_sort | quasi isomorphisms of cluster algebras and the combinatorics of webs extended abstract |
| topic | [math.math-co]mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/6395/pdf |
| work_keys_str_mv | AT chrisfraser quasiisomorphismsofclusteralgebrasandthecombinatoricsofwebsextendedabstract |