The order of birational rowmotion
Various authors have studied a natural operation (under various names) on the order ideals (equivalently antichains) of a finite poset, here called \emphrowmotion. For certain posets of interest, the order of this map is much smaller than one would naively expect, and the orbits exhibit unexpected p...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Discrete Mathematics & Theoretical Computer Science
2014-01-01
|
| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/2439/pdf |
| _version_ | 1797270282080092160 |
|---|---|
| author | Darij Grinberg Tom Roby |
| author_facet | Darij Grinberg Tom Roby |
| author_sort | Darij Grinberg |
| collection | DOAJ |
| description | Various authors have studied a natural operation (under various names) on the order ideals (equivalently antichains) of a finite poset, here called \emphrowmotion. For certain posets of interest, the order of this map is much smaller than one would naively expect, and the orbits exhibit unexpected properties. In very recent work (inspired by discussions with Berenstein) Einstein and Propp describe how rowmotion can be generalized: first to the piecewise-linear setting of order polytopes, then via detropicalization to the birational setting. In the latter setting, it is no longer \empha priori clear even that birational rowmotion has finite order, and for many posets the order is infinite. However, we are able to show that birational rowmotion has the same order, p+q, for the poset P=[p]×[q] (product of two chains), as ordinary rowmotion. We also show that birational (hence ordinary) rowmotion has finite order for some other classes of posets, e.g., the upper, lower, right and left halves of the poset above, and trees having all leaves on the same level. Our methods are based on those used by Volkov to resolve the type AA (rectangular) Zamolodchikov Periodicity Conjecture. |
| first_indexed | 2024-04-25T02:01:47Z |
| format | Article |
| id | doaj.art-62cd0fe22d01473bbd5c3fdfe6e94e24 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:01:47Z |
| publishDate | 2014-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-62cd0fe22d01473bbd5c3fdfe6e94e242024-03-07T14:53:18ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502014-01-01DMTCS Proceedings vol. AT,...Proceedings10.46298/dmtcs.24392439The order of birational rowmotionDarij Grinberg0Tom Roby1Massachusetts Institute of TechnologyUniversity of ConnecticutVarious authors have studied a natural operation (under various names) on the order ideals (equivalently antichains) of a finite poset, here called \emphrowmotion. For certain posets of interest, the order of this map is much smaller than one would naively expect, and the orbits exhibit unexpected properties. In very recent work (inspired by discussions with Berenstein) Einstein and Propp describe how rowmotion can be generalized: first to the piecewise-linear setting of order polytopes, then via detropicalization to the birational setting. In the latter setting, it is no longer \empha priori clear even that birational rowmotion has finite order, and for many posets the order is infinite. However, we are able to show that birational rowmotion has the same order, p+q, for the poset P=[p]×[q] (product of two chains), as ordinary rowmotion. We also show that birational (hence ordinary) rowmotion has finite order for some other classes of posets, e.g., the upper, lower, right and left halves of the poset above, and trees having all leaves on the same level. Our methods are based on those used by Volkov to resolve the type AA (rectangular) Zamolodchikov Periodicity Conjecture.https://dmtcs.episciences.org/2439/pdfantichainsbirational actionsbirational rowmotionbrouwer-schrijver mapcluster algebrasorbitorder idealspanyushev complementationposetproduct of chainspromotionrowmotiontoggle groupzamolodchikov conjecture.[info.info-dm] computer science [cs]/discrete mathematics [cs.dm][math.math-co] mathematics [math]/combinatorics [math.co] |
| spellingShingle | Darij Grinberg Tom Roby The order of birational rowmotion Discrete Mathematics & Theoretical Computer Science antichains birational actions birational rowmotion brouwer-schrijver map cluster algebras orbit order ideals panyushev complementation poset product of chains promotion rowmotion toggle group zamolodchikov conjecture. [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| title | The order of birational rowmotion |
| title_full | The order of birational rowmotion |
| title_fullStr | The order of birational rowmotion |
| title_full_unstemmed | The order of birational rowmotion |
| title_short | The order of birational rowmotion |
| title_sort | order of birational rowmotion |
| topic | antichains birational actions birational rowmotion brouwer-schrijver map cluster algebras orbit order ideals panyushev complementation poset product of chains promotion rowmotion toggle group zamolodchikov conjecture. [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/2439/pdf |
| work_keys_str_mv | AT darijgrinberg theorderofbirationalrowmotion AT tomroby theorderofbirationalrowmotion AT darijgrinberg orderofbirationalrowmotion AT tomroby orderofbirationalrowmotion |