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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| Vydané v: | Discrete mathematics and theoretical computer science Ročník DMTCS Proceedings, 28th... |
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| Hlavný autor: | |
| Médium: | Journal Article Konferenčný príspevok.. |
| Jazyk: | English |
| Vydavateľské údaje: |
DMTCS
22.04.2020
Discrete Mathematics & Theoretical Computer Science |
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| ISSN: | 1365-8050, 1462-7264, 1365-8050 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | 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. |
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| ISSN: | 1365-8050 1462-7264 1365-8050 |
| DOI: | 10.46298/dmtcs.6395 |