On the action of the toggle group of the Dynkin diagram of type A

TL;DR

This paper investigates the action of toggle involutions on independent sets of type A Dynkin diagrams, demonstrating that the generated subgroup is the entire symmetric group, revealing a rich symmetry structure.

Contribution

It proves that the subgroup generated by toggle involutions on independent sets of type A Dynkin diagrams equals the full symmetric group, highlighting a novel symmetry property.

Findings

The toggle group acts transitively on independent sets.

The toggle group is isomorphic to the symmetric group.

Symmetries of the independent sets are fully characterized.

Abstract

In this article, we consider involutions, called togglings, on the set of independent sets of the Dynkin diagram of type A, or a path graph. We are interested in the action of the subgroup of the symmetric group of the set of independent sets generated by togglings. We show that the subgroup coincides with the symmetric group.