Iterative properties of birational rowmotion II: Rectangles and triangles
Birational rowmotion — a birational map associated to any finite poset P — has been introduced by Einstein and Propp as a far-reaching generalization of the (well-studied) classical rowmotion map on the set of order ideals of P. Continuing our exploration of this birational rowmotion, we prove that...
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| Format: | Article |
| Language: | en_US |
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European Mathematical Information Service (EMIS)
2016
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| Online Access: | http://hdl.handle.net/1721.1/100753 https://orcid.org/0000-0002-9661-8432 |
| _version_ | 1826212106031071232 |
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| author | Grinberg, Darij Roby, Tom |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Grinberg, Darij Roby, Tom |
| author_sort | Grinberg, Darij |
| collection | MIT |
| description | Birational rowmotion — a birational map associated to any finite poset P — has been introduced by Einstein and Propp as a far-reaching generalization of the (well-studied) classical rowmotion map on the set of order ideals of P. Continuing our exploration of this birational rowmotion, we prove that it has order p + q on the (p,q)-rectangle poset (i.e., on the product of a p-element chain with a q-element chain); we also compute its orders on some triangle-shaped posets. In all cases mentioned, it turns out to have finite (and explicitly computable) order, a property it does not exhibit for general finite posets (unlike classical rowmotion, which is a permutation of a finite set). Our proof in the case of the rectangle poset uses an idea introduced by Volkov (arXiv:hep-th/0606094) to prove the AA case of the Zamolodchikov periodicity conjecture; in fact, the finite order of birational rowmotion on many posets can be considered an analogue to Zamolodchikov periodicity. We comment on suspected, but so far enigmatic, connections to the theory of root posets. |
| first_indexed | 2024-09-23T15:16:28Z |
| format | Article |
| id | mit-1721.1/100753 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T15:16:28Z |
| publishDate | 2016 |
| publisher | European Mathematical Information Service (EMIS) |
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| spelling | mit-1721.1/1007532022-10-02T01:50:17Z Iterative properties of birational rowmotion II: Rectangles and triangles Grinberg, Darij Roby, Tom Massachusetts Institute of Technology. Department of Mathematics Grinberg, Darij Birational rowmotion — a birational map associated to any finite poset P — has been introduced by Einstein and Propp as a far-reaching generalization of the (well-studied) classical rowmotion map on the set of order ideals of P. Continuing our exploration of this birational rowmotion, we prove that it has order p + q on the (p,q)-rectangle poset (i.e., on the product of a p-element chain with a q-element chain); we also compute its orders on some triangle-shaped posets. In all cases mentioned, it turns out to have finite (and explicitly computable) order, a property it does not exhibit for general finite posets (unlike classical rowmotion, which is a permutation of a finite set). Our proof in the case of the rectangle poset uses an idea introduced by Volkov (arXiv:hep-th/0606094) to prove the AA case of the Zamolodchikov periodicity conjecture; in fact, the finite order of birational rowmotion on many posets can be considered an analogue to Zamolodchikov periodicity. We comment on suspected, but so far enigmatic, connections to the theory of root posets. National Science Foundation (U.S.) (Grant 1001905) 2016-01-07T16:41:03Z 2016-01-07T16:41:03Z 2015 Article http://purl.org/eprint/type/JournalArticle 1077-8926 http://hdl.handle.net/1721.1/100753 Grinberg, Darij, and Tom Roby. "Iterative properties of birational rowmotion II: Rectangles and triangles." Electronic Journal of Combinatorics 22(3) (2015). https://orcid.org/0000-0002-9661-8432 en_US http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p40 Electronic Journal of Combinatorics Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf European Mathematical Information Service (EMIS) European Mathematical Information Service (EMIS) |
| spellingShingle | Grinberg, Darij Roby, Tom Iterative properties of birational rowmotion II: Rectangles and triangles |
| title | Iterative properties of birational rowmotion II: Rectangles and triangles |
| title_full | Iterative properties of birational rowmotion II: Rectangles and triangles |
| title_fullStr | Iterative properties of birational rowmotion II: Rectangles and triangles |
| title_full_unstemmed | Iterative properties of birational rowmotion II: Rectangles and triangles |
| title_short | Iterative properties of birational rowmotion II: Rectangles and triangles |
| title_sort | iterative properties of birational rowmotion ii rectangles and triangles |
| url | http://hdl.handle.net/1721.1/100753 https://orcid.org/0000-0002-9661-8432 |
| work_keys_str_mv | AT grinbergdarij iterativepropertiesofbirationalrowmotioniirectanglesandtriangles AT robytom iterativepropertiesofbirationalrowmotioniirectanglesandtriangles |