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 (for a finite poset). We also focus on the rowmotion action on antichains of a poset 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 the 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.
Highlights
In [CF95], Cameron and Fon-Der-Flaass defined a group consisting of permutations on the set J (P ) of order ideals of a poset P
In Subsection 2.4, we show that antichain rowmotion can be expressed as the composition of every toggle, each used exactly once, in a specified order (Proposition 2.24)
Many properties of rowmotion on order ideals extend to the order polytope, and we show here that the same is true between antichain toggles and chain polytope toggles
Summary
In [CF95], Cameron and Fon-Der-Flaass defined a group ( called the toggle group) consisting of permutations on the set J (P ) of order ideals of a poset P. This group is generated by #P simple maps called toggles each of which correspond to an element of the poset. These correspond to antichain toggles when restricted to the vertices This follows work of Einstein and Propp [EP18] who generalized the notion of toggles from order ideals to the order polytope of a poset, defined by Stanley [Sta86]. The other sections detail the necessary background material and framework as well as the notation we use, much of which varies between sources
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