Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets
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Grinberg-2016-Iterative properties.pdf
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535.08 KB
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Author(s)
Grinberg, Darij
Roby, Tom
Date Issued
February 2016
Journal
Electronic Journal of Combinatorics
Publisher
European Mathematical Information Service (EMIS)
Citation
Grinberg, Darij, and Tom Roby. "Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets." Electronic Journal OF Combinatorics, 23.1 (2016).
Version
Final published version
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.37236/4334
