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

Einstein, David; Propp, James

Algebraic Combinatorics, Tome 4 (2021) no. 2, pp. 201-224.

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 ${ℝ}^{\left|P\right|}$ and indeed to a dense subset of ${ℂ}^{\left|P\right|}$. 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.

Classification : 05E18, 06A07
Mots-clés : Dynamics, homomesy, order ideal, order polytope, piecewise-linear, promotion, recombination, rowmotion, toggle group, tropicalization.

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