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 [16] can be lifted to (continuous) piecewise-linear operations on the order polytope of Stanley [14], and then lifted to birational operations on the positive orthant in ${\mathbb{R}}^{\left|P\right|}$ and indeed to a dense subset of ${\u2102}^{\left|P\right|}$. 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.

Revised:

Accepted:

Published online:

Classification: 05E18, 06A07

Keywords: Dynamics, homomesy, order ideal, order polytope, piecewise-linear, promotion, recombination, rowmotion, toggle group, tropicalization.

@article{ALCO_2021__4_2_201_0, author = {Einstein, David and Propp, James}, title = {Combinatorial, piecewise-linear, and birational homomesy for products of two chains}, journal = {Algebraic Combinatorics}, pages = {201--224}, publisher = {MathOA foundation}, volume = {4}, number = {2}, year = {2021}, doi = {10.5802/alco.139}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.139/} }

Einstein, David; Propp, James. Combinatorial, piecewise-linear, and birational homomesy for products of two chains. Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 201-224. doi : 10.5802/alco.139. https://alco.centre-mersenne.org/articles/10.5802/alco.139/

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