Rowmotion and Increasing Labeling Promotion

TL;DR

This paper generalizes the connection between rowmotion and promotion-like actions from specific posets to arbitrary finite posets with label restrictions, revealing new conjugacy relations and promoting a unified framework.

Contribution

It introduces a generalized toggle-promotion for arbitrary posets with label restrictions and establishes conjugacy to rowmotion under certain conditions.

Findings

Toggle-promotion is conjugate to rowmotion for certain label restrictions.

A natural toggle-promotion action is defined for posets embedded in Cartesian products.

The framework unifies previous results on promotion and rowmotion for specific poset classes.

Abstract

In 2012, N. Williams and the second author showed that on order ideals of ranked partially ordered sets (posets), rowmotion is conjugate to (and thus has the same orbit structure as) a different toggle group action, which in special cases is equivalent to promotion on linear extensions of posets constructed from two chains. In 2015, O. Pechenik and the first and second authors extended these results to show that increasing tableaux under K-promotion naturally corresponds to order ideals in a product of three chains under a toggle group action conjugate to rowmotion they called hyperplane promotion. In this paper, we generalize these results to the setting of arbitrary increasing labelings of any finite poset with given restrictions on the labels. We define a generalization of K-promotion in this setting and show it corresponds to a toggle group action we call toggle-promotion on order ideals of an associated poset. When the restrictions on labels are particularly nice (for example, specifying a global bound on all labels used), we show that toggle-promotion is conjugate to rowmotion. Additionally, we show that any poset that can be nicely embedded into a Cartesian product has a natural toggle-promotion action conjuate to rowmotion.