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Iterative properties of birational rowmotion II: rectangles and triangles
Journal article   Open access   Peer reviewed

Iterative properties of birational rowmotion II: rectangles and triangles

Darij Grinberg and Tom Roby
The Electronic journal of combinatorics, v 22(3), 3
20 Sep 2015
DOI: https://doi.org/10.37236/4335
url
https://doi.org/10.37236/4335View
Published, Version of Record (VoR) Open

Abstract

Mathematics Mathematics, Applied Physical Sciences Science & Technology
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-Fq 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.

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Mathematics
Mathematics, Applied
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