Birational and noncommutative lifts of antichain toggling and rowmotion

TL;DR

This paper extends the study of rowmotion actions on posets to birational and noncommutative labelings, revealing explicit bijections and broadening the understanding of orbit structures and periodicity.

Contribution

It introduces birational and noncommutative lifts of antichain rowmotion, providing explicit equivariant bijections and extending the framework to noncommutative rational functions.

Findings

Explicit equivariant bijections between birational toggle groups

Extension of rowmotion concepts to noncommutative rational functions

Broader context for periodicity conjectures in poset dynamics

Abstract

The rowmotion action on order ideals or on antichains of a finite partially ordered set has been studied (under a variety of names) by many authors. Depending on the poset, one finds unexpectedly interesting orbit structures, instances of (small order) periodicity, cyclic sieving, and homomesy. Many of these nice features still hold when the action is extended to $[0,1]$-labelings of the poset or (via detropicalization) to labelings by rational functions (the birational setting). In this work, we parallel the birational lifting already done for order-ideal rowmotion to antichain rowmotion. We give explicit equivariant bijections between the birational toggle groups and between their respective liftings. We further extend all of these notions to labellings by noncommutative rational functions, setting an unpublished periodicity conjecture of Grinberg in a broader context.