Jessica Striker - Rowmotion and generalized toggle groups

dmtcs:3962 - Discrete Mathematics & Theoretical Computer Science, May 25, 2018, Vol. 20 no. 1 -
Rowmotion and generalized toggle groups

Authors: Jessica Striker

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.

Volume: Vol. 20 no. 1
Section: Combinatorics
Published on: May 25, 2018
Accepted on: May 25, 2018
Submitted on: September 27, 2017
Keywords: Mathematics - Combinatorics,05E18, 06A75


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