Toric Promotion

TL;DR

This paper introduces toric promotion, a cyclic operator on graph labelings, and explores its orbit structure, especially for forests, connecting it to toric posets and friends-and-strangers graphs.

Contribution

It defines toric promotion as a cyclic analogue of Schützenberger's promotion and characterizes its orbit structure on forests, linking it to existing combinatorial concepts.

Findings

Orbit structure of toric promotion on forests is explicitly described.

Toric promotion relates to toric posets and friends-and-strangers graphs.

Main theorem simplifies understanding of the cyclic action on graph labelings.

Abstract

We introduce toric promotion as a cyclic analogue of Sch\"utzenberger's promotion operator. Toric promotion acts on the set of labelings of a graph $G$. We discuss connections between toric promotion and previously-studied notions such as toric posets and friends-and-strangers graphs. Our main theorem provides a surprisingly simple description of the orbit structure of toric promotion when $G$ is a forest.