Promotion and cyclic sieving via webs

TL;DR

This paper connects Schützenberger's promotion on Young tableaux with cyclic rotation of planar graphs, demonstrating that this action exhibits the cyclic sieving phenomenon, thus linking combinatorial and graphical representations.

Contribution

It introduces a novel realization of promotion as cyclic rotation of planar graphs and proves the cyclic sieving phenomenon for this action.

Findings

Promotion corresponds to cyclic rotation of planar graphs

The cyclic sieving phenomenon is established for this action

Provides a new graphical perspective on Young tableaux operations

Abstract

We show that Sch\"utzenberger's promotion on two and three row rectangular Young tableaux can be realized as cyclic rotation of certain planar graphs introduced by Kuperberg. Moreover, following work of the third author, we show that this action admits the cyclic sieving phenomenon.