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Combinatorial, piecewise-linear, and birational homomesy for products of two chains. (arXiv:1310.5294v2 [math.CO] UPDATED)

来源于:arXiv
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 combina 查看全文>>