Iterative properties of birational rowmotion II: Rectangles and triangles
Name
Grinberg_2015_Iterative properties.pdf
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551.66 KB
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Author(s)
Grinberg, Darij
Roby, Tom
Date Issued
2015
Journal
Electronic Journal of Combinatorics
Publisher
European Mathematical Information Service (EMIS)
Citation
Grinberg, Darij, and Tom Roby. "Iterative properties of birational rowmotion II: Rectangles and triangles." Electronic Journal of Combinatorics 22(3) (2015).
Version
Final published version
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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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.
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DOI of Published Version
https://doi.org/10.37236/4335
