Cyclic Sieving and Cluster Duality of Grassmannian
We introduce a decorated configuration space Confˣₙ(𝑎) with a potential function 𝒲. We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of (Confˣₙ(𝑎), 𝒲) canonically parametrizes a linear basis of the homogeneous coordinate ring of the Grassmanni...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2020 |
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Інститут математики НАН України
2020
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/210781 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Cyclic Sieving and Cluster Duality of Grassmannian. Linhui Shen and Daping Weng. SIGMA 16 (2020), 067, 41 pages |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| Zusammenfassung: | We introduce a decorated configuration space Confˣₙ(𝑎) with a potential function 𝒲. We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of (Confˣₙ(𝑎), 𝒲) canonically parametrizes a linear basis of the homogeneous coordinate ring of the Grassmannian Grₐ(n) with respect to the Plücker embedding. We prove that (Confˣₙ(𝑎), 𝒲) is equivalent to the mirror Landau-Ginzburg model of the Grassmannian considered by Eguchi-Hori-Xiong, Marsh-Rietsch, and Rietsch-Williams. As an application, we show a cyclic sieving phenomenon involving plane partitions under a sequence of piecewise-linear toggles.
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| ISSN: | 1815-0659 |