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

This article illustrates the dynamical concept of homomesy in three kinds systems – combinatorial, piecewise-linear, and birational shows relationship between these settings. In particular, we show how rowmotion promotion operations Striker Williams [16] can be lifted to (continuous) piecewise-linear on order polytope Stanley [14], then positive orthant ? |P| indeed a dense subset ? . When poset P is product chain length b, have a+b, exhibit phenomenon: time-averages various quantities are same all orbits. One important tool concrete realization conjugacy found by Williams; this recombination map allows us use for deduce rowmotion.