Séminaire Lotharingien de Combinatoire, 82B.94 (2019), 12 pp.

Michael Joseph and Tom Roby

Birational antichain toggling and rowmotion

Abstract. We define a new combinatorial dynamical system on node-labelings of a poset P by rational functions called birational antichain rowmotion (BAR-motion for short). It is analogous to the previously-studied birational rowmotion action (here called birational order rowmotion, or BOR-motion). The latter action is detropicalized from an extension of combinatorial rowmotion on order ideals of P to Stanley's order polytope OP(P). Analogously BAR-motion detropicalizes the extension of rowmotion on antichains of P to the chain polytope C(P).

We study BAR-motion by defining a birational antichain toggle group generated by involutions called toggles. This lifts Striker's toggle group on the set of antichains of a poset to the birational realm. We construct an explicit isomorphism between this group and Einstein and Propp's group of birational order toggles, lifting an analogous one between the toggle groups of order ideals and antichains at the combinatorial level.

For certain nice families of posets, the order of BOR-motion is known to be finite, and can often be easily computed. We lift Stanley's transfer map between C(P) and OP(P) to a birational transfer map, which allows us to easily deduce certain properties of one kind of birational rowmotion from the other. We take advantage of this to derive the periodicity and order of BAR-motion on certain root and minuscule posets. We also lift a refined homomesy result of Propp and the second author from the combinatorial setting to the birational one, using an analogue of the "Stanley-Thomas" word, which cyclically rotates equivariantly with BAR-motion.

Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

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