Rowmotion and generalized toggle groups

TL;DR

This paper generalizes toggle groups and rowmotion to broader combinatorial settings, providing structure theorems and characterizing when these actions are bijective across various mathematical structures.

Contribution

It introduces a generalized toggle group framework and characterizes when cover-closure maps are bijective, extending the concept of rowmotion beyond order ideals.

Findings

Structure theorems for finite generalized toggle groups

Cover-closure is bijective iff the set of closed sets is isomorphic to order ideals of a poset

Rowmotion is the unique bijective cover-closure map

Abstract

We generalize the notion of the toggle group, as defined in [P. Cameron-D. Fon-der-Flaass '95] and further explored in [J. Striker-N. Williams '12], from the set of order ideals of a poset to any family of subsets of a finite set. We prove structure theorems for certain finite generalized toggle groups, similar to the theorem of Cameron and Fon-der-Flaass in the case of order ideals. We apply these theorems and find other results on generalized toggle groups in the following settings: chains, antichains, and interval-closed sets of a poset; independent sets, vertex covers, acyclic subgraphs, and spanning subgraphs of a graph; matroids and convex geometries. We generalize rowmotion, an action studied on order ideals in [P. Cameron-D. Fon-der-Flaass '95] and [J. Striker-N. Williams '12], to a map we call cover-closure on closed sets of a closure operator. We show that cover-closure is bijective if and only if the set of closed sets is isomorphic to the set of order ideals of a poset, which implies rowmotion is the only bijective cover-closure map.