Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets
We study a birational map associated to any finite poset P. This map is a far-reaching generalization (found by Einstein and Propp) of classical rowmotion, which is a certain permutation of the set of order ideals of P. Classical rowmotion has been studied by various authors (Fon-der-Flaass, Cameron...
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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/1721.1/105231 https://orcid.org/0000-0002-9661-8432 |
| _version_ | 1811085004219875328 |
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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 | We study a birational map associated to any finite poset P. This map is a far-reaching generalization (found by Einstein and Propp) of classical rowmotion, which is a certain permutation of the set of order ideals of P. Classical rowmotion has been studied by various authors (Fon-der-Flaass, Cameron, Brouwer, Schrijver, Striker, Williams and many more) under different guises (Striker-Williams promotion and Panyushev complementation are two examples of maps equivalent to it). In contrast,
birational rowmotion is new and has yet to reveal several of its mysteries. In this paper, we set up the tools for analyzing the properties of iterates of this map, and prove that it has finite order for a certain class of posets which we call “skeletal”. Roughly speaking, these are graded posets constructed from one-element posets by repeated disjoint union and “grafting onto an antichain”; in particular, any forest having its leaves all on the same rank is such a poset. We also make a parallel analysis of classical rowmotion on this kind of posets, and prove that the order in this case equals the order of birational rowmotion. |
| first_indexed | 2024-09-23T13:01:27Z |
| format | Article |
| id | mit-1721.1/105231 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T13:01:27Z |
| publishDate | 2016 |
| publisher | European Mathematical Information Service (EMIS) |
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| spelling | mit-1721.1/1052312022-10-01T12:35:33Z Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets Grinberg, Darij Roby, Tom Massachusetts Institute of Technology. Department of Mathematics Grinberg, Darij We study a birational map associated to any finite poset P. This map is a far-reaching generalization (found by Einstein and Propp) of classical rowmotion, which is a certain permutation of the set of order ideals of P. Classical rowmotion has been studied by various authors (Fon-der-Flaass, Cameron, Brouwer, Schrijver, Striker, Williams and many more) under different guises (Striker-Williams promotion and Panyushev complementation are two examples of maps equivalent to it). In contrast, birational rowmotion is new and has yet to reveal several of its mysteries. In this paper, we set up the tools for analyzing the properties of iterates of this map, and prove that it has finite order for a certain class of posets which we call “skeletal”. Roughly speaking, these are graded posets constructed from one-element posets by repeated disjoint union and “grafting onto an antichain”; in particular, any forest having its leaves all on the same rank is such a poset. We also make a parallel analysis of classical rowmotion on this kind of posets, and prove that the order in this case equals the order of birational rowmotion. National Science Foundation (U.S.) (Grant 1001905) 2016-11-07T19:41:10Z 2016-11-07T19:41:10Z 2016-02 2014-05 Article http://purl.org/eprint/type/JournalArticle 1077-8926 1097-1440 http://hdl.handle.net/1721.1/105231 Grinberg, Darij, and Tom Roby. "Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets." Electronic Journal OF Combinatorics, 23.1 (2016). https://orcid.org/0000-0002-9661-8432 en_US http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p33 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 I: Generalities and Skeletal Posets |
| title | Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets |
| title_full | Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets |
| title_fullStr | Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets |
| title_full_unstemmed | Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets |
| title_short | Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets |
| title_sort | iterative properties of birational rowmotion i generalities and skeletal posets |
| url | http://hdl.handle.net/1721.1/105231 https://orcid.org/0000-0002-9661-8432 |
| work_keys_str_mv | AT grinbergdarij iterativepropertiesofbirationalrowmotionigeneralitiesandskeletalposets AT robytom iterativepropertiesofbirationalrowmotionigeneralitiesandskeletalposets |