Antichain toggling and rowmotion

TL;DR

This paper studies toggle groups on antichains of posets, establishing their relationship with order ideals, and explores the dynamical action of rowmotion, including a piecewise-linear analogue related to Stanley's chain polytope.

Contribution

It constructs an explicit isomorphism between toggle groups of antichains and order ideals, and extends toggle concepts to a piecewise-linear setting.

Findings

Established an explicit isomorphism between toggle groups of antichains and order ideals.

Described rowmotion as a composition of antichain toggles.

Extended toggle results to a piecewise-linear analogue of the chain polytope.

Abstract

In this paper, we analyze the toggle group on the set of antichains of a poset. Toggle groups, generated by simple involutions, were first introduced by Cameron and Fon-Der-Flaass for order ideals of posets. Recently Striker has motivated the study of toggle groups on general families of subsets, including antichains. This paper expands on this work by examining the relationship between the toggle groups of antichains and order ideals, constructing an explicit isomorphism between the two groups. We also focus on the rowmotion action that has been well-studied in dynamical algebraic combinatorics, describing it as the composition of antichain toggles. We also describe a piecewise-linear analogue of toggling to Stanley's chain polytope. We examine the connections with the piecewise-linear toggling Einstein and Propp introduced for order polytopes and prove that almost all of our results for antichain toggles extend to the piecewise-linear setting.