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Birational rowmotion is an action on the space of assignments of rational functions to the elements of a finite partially-ordered set (poset). It is lifted from the well-studied rowmotion map on order ideals (equivariantly on antichains) of a poset $P$, which when iterated on special posets, has unexpectedly nice properties in terms of periodicity, cyclic sieving, and homomesy (statistics whose averages over each orbit are constant) [AST11, BW74, CF95, Pan09, PR13, RuSh12,RuWa15+,SW12, ThWi17, Yil17. In this context, rowmotion appears to be related to Auslander-Reiten translation on certain quivers, and birational rowmotion to $Y$-systems of type $A_m times A_n$ described in Zamolodchikov periodicity. We give a formula in terms of families of non-intersecting lattice paths for iterated actions of the birational rowmotion map on a product of two chains. This allows us to give a much simpler direct proof of the key fact that the period of this map on a product of chains of lengths $r$ and $s$ is $r+s+2$ (first proved by D.~Grinberg and the second author), as well as the first proof of the birational analogue of homomesy along files for such posets.
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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 prove that birational rowmotion 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 furthermore compute its orders on some triangle-shaped posets and on a class of posets which we call skeletal (this class includes all graded forests). In all cases mentioned, birational rowmotion turns out to have a 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. We also make a digression to study classical rowmotion on skeletal posets, since this case has seemingly been overlooked so far.
The dynamics of certain combinatorial actions and their liftings to actions at the piecewise-linear and birational level have been studied lately with an eye towards questions of periodicity, orbit structure, and invariants. One key property enjoyed by the rowmotion operator on certain finite partially-ordered sets is homomesy, where the average value of a statistic is the same for all orbits. To prove refin
The rowmotion action on order ideals or on antichains of a finite partially ordered set has been studied (under a variety of names) by many authors. Depending on the poset, one finds unexpectedly interesting orbit structures, instances of (small order) periodicity, cyclic sieving, and homomesy. Many of these nice features still hold when the action is extended to $[0,1]$-labelings of the poset or (via detropicalization) to labelings by rational functions (the birational setting). In this work, we parallel the birational lifting already done for order-ideal rowmotion to antichain rowmotion. We give explicit equivariant bijections between the birational toggle groups and between their respective liftings. We further extend all of these notions to labellings by noncommutative rational functions, setting an unpublished periodicity conjecture of Grinberg in a broader context.
Many invertible actions $tau$ on a set ${mathcal{S}}$ of combinatorial objects, along with a natural statistic $f$ on ${mathcal{S}}$, exhibit the following property which we dub textbf{homomesy}: the average of $f$ over each $tau$-orbit in ${mathcal{S}}$ is the same as the average of $f$ over the whole set ${mathcal{S}}$. This phenomenon was first noticed by Panyushev in 2007 in the context of the rowmotion action on the set of antichains of a root poset; Armstrong, Stump, and Thomas proved Panyushevs conjecture in 2011. We describe a theoretical framework for results of this kind that applies more broadly, giving examples in a variety of contexts. These include linear actions on vector spaces, sandpile dynamics, Suters action on certain subposets of Youngs Lattice, Lyness 5-cycles, promotion of rectangular semi-standard Young tableaux, and the rowmotion and promotion actions on certain posets. We give a detailed description of the latter situation for products of two chains.
A fence is a poset with elements F = {x_1, x_2, ..., x_n} and covers x_1 < x_2 < ... < x_a > x_{a+1} > ... > x_b < x_{b+1} < ... where a, b, ... are positive integers. We investigate rowmotion on antichains and ideals of F. In particular, we show that orbits of antichains can be visualized using tilings. This permits us to prove various homomesy results for the number of elements of an antichain or ideal in an orbit. Rowmotion on fences also exhibits a new phenomenon, which we call orbomesy, where the value of a statistic is constant on orbits of the same size. Along the way, we prove a homomesy result for all self-dual posets and show that any two Coxeter elements in certain toggle groups behave similarly with respect to homomesies which are linear combinations of ideal indicator functions. We end with some conjectures and avenues for future research.
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