Abstract
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.
Highlights
Numerous authors have studied a natural operation on the order ideals of a finite poset, here called rowmotion, following Striker and Williams [StWi11]
We have found that birational rowmotion has finite order for several other classes of posets, e.g., the upper, lower, right and left halves of the poset above, and trees which are graded
Our methods are based on those used by Volkov to resolve the type AA Zamolodchikov Periodicity Conjecture [Volk06]
Summary
Numerous authors have studied a natural operation (under various names) on the order ideals (equivalently antichains) of a finite poset, here called rowmotion, following Striker and Williams [StWi11]. Einstein and Propp [EiPr13] describe how rowmotion can be generalized: first to the piecewise-linear setting of an order polytope [Stan, Definition 1.1], via detropicalization to the birational setting; they find applications to classical combinatorics of tableaux. Much of their paper focuses on the case where the poset P is [p] × [q], a product of two chains, generalizing earlier work of Propp and Roby in the combinatorial setting [PrRo13]. In the case of a rectangle, we demonstrate a symmetry property of birational rowmotion conjectured by James Propp and Tom Roby. In this extended abstract, proofs are only roughly outlined. Applications of our results ( Theorems 33 and 34) are found in [EiPr13]
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