Cyclic Sieving for Plane Partitions and Symmetry
The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count the fixed points of elements of a cyclic group acting on a combinatorial set by plugging roots of unity into a polynomial related to this set. One of the most impressive instances of the cyclic sieving phenomeno...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2020 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2020
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211089 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Cyclic Sieving for Plane Partitions and Symmetry. Sam Hopkins. SIGMA 16 (2020), 130, 40 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862728019824607232 |
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| author | Hopkins, Sam |
| author_facet | Hopkins, Sam |
| citation_txt | Cyclic Sieving for Plane Partitions and Symmetry. Sam Hopkins. SIGMA 16 (2020), 130, 40 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count the fixed points of elements of a cyclic group acting on a combinatorial set by plugging roots of unity into a polynomial related to this set. One of the most impressive instances of the cyclic sieving phenomenon is a theorem of Rhoades asserting that the set of plane partitions in a rectangular box under the action of promotion exhibits cyclic sieving. In Rhoades's result, the sieving polynomial is the size generating function for these plane partitions, which has a well-known product formula due to MacMahon. We extend Rhoades's result by also considering symmetries of plane partitions: specifically, complementation and transposition. The relevant polynomial here is the size generating function for symmetric plane partitions, whose product formula was conjectured by MacMahon and proved by Andrews and Macdonald. Finally, we explain how these symmetry results also apply to the rowmotion operator on plane partitions, which is closely related to promotion.
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| first_indexed | 2026-03-21T11:25:40Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211089 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T11:25:40Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Hopkins, Sam 2025-12-23T13:12:39Z 2020 Cyclic Sieving for Plane Partitions and Symmetry. Sam Hopkins. SIGMA 16 (2020), 130, 40 pages 1815-0659 2020 Mathematics Subject Classification: 05E18;05E10;17B10;17B37 arXiv:1907.09337 https://nasplib.isofts.kiev.ua/handle/123456789/211089 https://doi.org/10.3842/SIGMA.2020.130 The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count the fixed points of elements of a cyclic group acting on a combinatorial set by plugging roots of unity into a polynomial related to this set. One of the most impressive instances of the cyclic sieving phenomenon is a theorem of Rhoades asserting that the set of plane partitions in a rectangular box under the action of promotion exhibits cyclic sieving. In Rhoades's result, the sieving polynomial is the size generating function for these plane partitions, which has a well-known product formula due to MacMahon. We extend Rhoades's result by also considering symmetries of plane partitions: specifically, complementation and transposition. The relevant polynomial here is the size generating function for symmetric plane partitions, whose product formula was conjectured by MacMahon and proved by Andrews and Macdonald. Finally, we explain how these symmetry results also apply to the rowmotion operator on plane partitions, which is closely related to promotion. I thank Chris Fraser, Gabe Frieden, Vic Reiner, Brendon Rhoades, and Jessica Striker for useful discussions related to this work. I was supported by NSF grant #1802920. I benefited from the use of Sage mathematics software [71, 72] in the course of this research. Finally, I thank all of the anonymous referees for their careful attention to the document and useful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Cyclic Sieving for Plane Partitions and Symmetry Article published earlier |
| spellingShingle | Cyclic Sieving for Plane Partitions and Symmetry Hopkins, Sam |
| title | Cyclic Sieving for Plane Partitions and Symmetry |
| title_full | Cyclic Sieving for Plane Partitions and Symmetry |
| title_fullStr | Cyclic Sieving for Plane Partitions and Symmetry |
| title_full_unstemmed | Cyclic Sieving for Plane Partitions and Symmetry |
| title_short | Cyclic Sieving for Plane Partitions and Symmetry |
| title_sort | cyclic sieving for plane partitions and symmetry |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211089 |
| work_keys_str_mv | AT hopkinssam cyclicsievingforplanepartitionsandsymmetry |