Piecewise-linear and birational toggling

TL;DR

This paper introduces piecewise-linear and birational analogues of toggle-involutions on posets, connecting them to rowmotion and promotion, and explores their properties, symmetries, and orbit structures.

Contribution

It defines new analogues of toggle-involutions and rowmotion in piecewise-linear and birational settings, linking them to classical combinatorial operations and proving their order and homomesy properties.

Findings

Birational rowmotion has order a+b for posets of form [a]×[b].

Piecewise-linear and birational operations relate via tropicalization.

Homomesy results show orbit averages are orbit-independent.

Abstract

We define piecewise-linear and birational analogues of the toggle-involutions on order ideals of posets studied by Striker and Williams and use them to define corresponding analogues of rowmotion and promotion that share many of the properties of combinatorial rowmotion and promotion. Piecewise-linear rowmotion (like birational rowmotion) admits an alternative definition related to Stanley's transfer map for the order polytope; piecewise-linear promotion relates to Sch\"utzenberger promotion for semistandard Young tableaux. The three settings for these dynamical systems (combinatorial, piecewise-linear, and birational) are intimately related: the piecewise-linear operations arise as tropicalizations of the birational operations, and the combinatorial operations arise as restrictions of the piecewise-linear operations to the vertex-set of the order polytope. In the case where the poset is of the form $[a] \times [b]$, we exploit a reciprocal symmetry property recently proved by Grinberg and Roby to show that birational rowmotion (and consequently piecewise-linear rowmotion) is of order $a+b$. This yields a new proof of a theorem of Cameron and Fon-der-Flaass. Our proofs make use of the correspondence between rowmotion and promotion orbits discovered by Striker and Williams, which we make more concrete. We also prove some homomesy results, showing that for certain functions $f$, the average value of $f$ over each rowmotion/promotion orbit is independent of the orbit chosen.