2013/10/19 by David J. Einstein, James Propp, Einstein, David +1 · 1 citation
Mathematics · #05E18 #06A11 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.1310.5294
openalex publication_date 2013/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This article illustrates the dynamical concept of homomesy in three kinds\nof dynamical systems -- combinatorial, piecewise-linear, and birational -- and\nshows the relationship between these three settings. In particular, we show how\nthe rowmotion and promotion operations of Striker and Williams can be lifted to\n(continuous) piecewise-linear operations on the order polytope of Stanley, and\nthen lifted to birational operations on the positive orthant in\n\ℝ|P| and indeed to a dense subset of \ℂ|P|. When the\nposet P is a product of a chain of length a and a chain of length b,\nthese lifted operations have order a+b, and exhibit the homomesy phenomenon:\nthe time-averages of various quantities are the same in all orbits. One\nimportant tool is a concrete realization of the conjugacy between rowmotion and\npromotion found by Striker and Williams; this recombination map allows us\nto use homomesy for promotion to deduce homomesy for rowmotion.\n NOTE: An earlier draft showed that Stanley's transfer map between the order\npolytope and the chain polytope arises as the tropicalization of an analogous\nmap in the bilinear realm; in 2020 we removed this material for the sake of\nbrevity, especially after Joseph and Roby generalized our proof to the\nnoncommutative realm (see arXiv:1909.09658v3). Readers who nonetheless wish to\nsee our proof can find the September 2018 draft of this preprint through the\narXiv.\n