Combinatorial, piecewise-linear, and birational homomesy for products of two chains

Abstract: This article illustrates the dynamical concept of $homomesy$ in three kinds of dynamical systems -- combinatorial, piecewise-linear, and birational -- and shows the relationship between these three settings. In particular, we show how the rowmotion and promotion operations of Striker and Williams can be lifted to (continuous) piecewise-linear operations on the order polytope of Stanley, and then lifted to birational operations on the positive orthant in $\mathbb{R}^{|P|}$ and indeed to a dense subset of $\mathbb{C}^{|P|}$. When the poset $P$ is a product of a chain of length $a$ and a chain of length $b$, these lifted operations have order $a+b$, and exhibit the homomesy phenomenon: the time-averages of various quantities are the same in all orbits. One important tool is a concrete realization of the conjugacy between rowmotion and promotion found by Striker and Williams; this $recombination$ $map$ allows us to use homomesy for promotion to deduce homomesy for rowmotion. NOTE: An earlier draft showed that Stanley's transfer map between the order polytope and the chain polytope arises as the tropicalization of an analogous map in the bilinear realm; in 2020 we removed this material for the sake of brevity, especially after Joseph and Roby generalized our proof to the noncommutative realm (see arXiv:1909.09658v3). Readers who nonetheless wish to see our proof can find the September 2018 draft of this preprint through the arXiv.