Cyclic Sieving and Cluster Duality of Grassmannian
We introduce a decorated configuration space Confˣₙ() with a potential function . We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of (Confˣₙ(), ) canonically parametrizes a linear basis of the homogeneous coordinate ring of the Grassmannian G...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2020 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2020
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| Онлайн доступ: | 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 |
| Цитувати: | Cyclic Sieving and Cluster Duality of Grassmannian. Linhui Shen and Daping Weng. SIGMA 16 (2020), 067, 41 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862539555844915200 |
|---|---|
| author | Shen, Linhui Weng, Daping |
| author_facet | Shen, Linhui Weng, Daping |
| citation_txt | Cyclic Sieving and Cluster Duality of Grassmannian. Linhui Shen and Daping Weng. SIGMA 16 (2020), 067, 41 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We introduce a decorated configuration space Confˣₙ() with a potential function . We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of (Confˣₙ(), ) 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(), ) 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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| first_indexed | 2026-03-12T19:19:03Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-210781 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T19:19:03Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Shen, Linhui Weng, Daping 2025-12-17T14:38:01Z 2020 Cyclic Sieving and Cluster Duality of Grassmannian. Linhui Shen and Daping Weng. SIGMA 16 (2020), 067, 41 pages 1815-0659 2020 Mathematics Subject Classification: 05E10; 13F60; 14J33; 14M15; 14N35; 14T05 arXiv:1803.06901 https://nasplib.isofts.kiev.ua/handle/123456789/210781 https://doi.org/10.3842/SIGMA.2020.067 We introduce a decorated configuration space Confˣₙ() with a potential function . We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of (Confˣₙ(), ) 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(), ) 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. We are grateful to Alexander Goncharov for the inspiration on the construction of the cluster dual space, and to Jiuzu Hong for many helpful discussions on the representation theoretical aspects of the cyclic sieving problem. We would also like to thank Michael Gekhtman, Li Li, Tim Magee, Gregg Musiker, Brendon Rhoades, Bruce Sagan, Lauren Williams, Eric Zaslow, and Peng Zhou for useful conversations in the process of drafting this paper. Finally, we thank the referees for their very careful reading of this paper and for many useful suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Cyclic Sieving and Cluster Duality of Grassmannian Article published earlier |
| spellingShingle | Cyclic Sieving and Cluster Duality of Grassmannian Shen, Linhui Weng, Daping |
| title | Cyclic Sieving and Cluster Duality of Grassmannian |
| title_full | Cyclic Sieving and Cluster Duality of Grassmannian |
| title_fullStr | Cyclic Sieving and Cluster Duality of Grassmannian |
| title_full_unstemmed | Cyclic Sieving and Cluster Duality of Grassmannian |
| title_short | Cyclic Sieving and Cluster Duality of Grassmannian |
| title_sort | cyclic sieving and cluster duality of grassmannian |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210781 |
| work_keys_str_mv | AT shenlinhui cyclicsievingandclusterdualityofgrassmannian AT wengdaping cyclicsievingandclusterdualityofgrassmannian |