Promotion digraphs

TL;DR

This paper introduces promotion digraphs for standard and increasing tableaux, generalizing promotion permutations, and explores their properties and connections to combinatorial structures and representation theory.

Contribution

It generalizes promotion permutations to promotion digraphs for arbitrary shapes and characterizes their properties, including connections to Specht modules and flamingo webs.

Findings

Promotion digraphs uniquely determine standard and rectangular increasing tableaux.

Complete characterization of promotion digraphs for two-row rectangular increasing tableaux.

Conjectured connection between promotion digraph dynamics and flamingo webs for three-row tableaux.

Abstract

Work of Gaetz, Pechenik, Pfannerer, Striker, and Swanson (2024) introduced promotion permutations for a rectangular standard Young tableau $T$. These promotion permutations encode important features of $T$ and its orbit under Sch\"utzenberger's promotion operator. Indeed, the promotion permutations uniquely determine the tableau $T$. We introduce more general promotion digraphs for both standard and increasing tableaux of arbitrary shape. For rectangular standard tableaux, this construction recovers the functional digraphs of the promotion permutations. Among other facts, we show that promotion digraphs uniquely determine $T$ when $T$ is standard of arbitrary shape or increasing of rectangular shape, but not when $T$ is increasing and general shape. We completely characterize the promotion digraphs for two-row rectangular increasing tableaux. We use promotion digraphs for three-row rectangular increasing tableaux to conjecture a connection between their dynamics and the flamingo webs recently introduced by Kim to give a diagrammatic basis of the Specht module $S^{(k^3,1^{n-3k})}$.