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

By David M. Einstein, James Gary Propp. Algebraic Combinatorics, 2013

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The purpose of this article is to illustrate the dynamical concept of {\em homomesy} in three kinds of dynamical systems -- combinatorial, piecewise-linear, and birational -- and to show the relationship between these three settings. In particular, we show how the rowmotion and promotion operations of Striker and Williams (2012), in the case where the poset $P$ is a product of a chain of length $a$ and a chain of length $b$, can be lifted to (continuous) piecewise-linear operations on the order polytope of Stanley (1986), and then lifted to birational operations on the positive orthant in ${\mathbb{R}}^{|P|}$ and indeed to a dense subset of ${\mathbb{C}}^{|P|}$. We prove that these lifted operations, like their combinatorial counterparts, have order $a+b$. In the birational setting, we prove a multiplicative homomesy theorem that by tropicalization yields an additive homomesy result in the piecewise-linear setting, which in turn specializes to an additive homomesy result in the combinatorial setting already proved by the second author and Roby. Here the term "additive homomesy" refers to the phenomenon wherein the arithmetic means of some quantity over the various orbits of a dynamical system are all equal to one another, and multiplicative homomesy is an analogous multiplicative notion. In the final section, we describe results of experiments indicating directions for future work.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/06868cbcc63a66a28e2b197e2a150e60251a8341