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 [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 ℝ |P| and indeed to a dense subset of 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.
Highlights
Many authors [2, 3, 6, 12, 16] have studied an operation ρ on the set of order ideals of a poset P that, following Striker and Williams, we call rowmotion
In exploring the properties of rowmotion, Striker and Williams introduced and studied a closely related operation π they call promotion on account of its ties with promotion of Young tableaux, which depends on the choice of an rc embedding
In this article we mostly focus on a very particular case, where P is of the form [a] × [b] and the rc embedding sends (i, j) ∈ P to (j − i, i + j − 2) ∈ Z2, and we explore how the cardinality of an order ideal I behaves as one iterates rowmotion and promotion
Summary
Many authors [2, 3, 6, 12, 16] have studied an operation ρ on the set of order ideals of a poset P that, following Striker and Williams, we call rowmotion. Homomesy, order ideal, order polytope, piecewise-linear, promotion, recombination, rowmotion, toggle group, tropicalization. That is, ignoring the use of recombination for passing back and forth between rowmotion and promotion, the logic of the argument is that we first prove birational homomesy, we deduce piecewise-linear homomesy by tropicalization, and we deduce combinatorial homomesy by specialization. The authors are grateful to Arkady Berenstein, Darij Grinberg, Michael Joseph, Tom Roby, Richard Stanley, and Jessica Striker for helpful conversations and detailed comments on the manuscript
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