Promotion and growth diagrams for fans of Dyck paths and vacillating tableaux

TL;DR

This paper introduces a new injection linking fans of Dyck paths and vacillating tableaux to chord diagrams, revealing their symmetry properties and cyclic sieving phenomena through innovative diagrammatic and crystal analysis methods.

Contribution

It constructs an injection connecting fans of Dyck paths and vacillating tableaux to chord diagrams, intertwining promotion and rotation using novel diagrammatic and crystal techniques.

Findings

Established an injection from fans of Dyck paths and vacillating tableaux to chord diagrams.

Demonstrated the cyclic sieving phenomenon using promotion action.

Connected crystal theory with diagrammatic bases for combinatorial objects.

Abstract

We construct an injection from the set of $r$-fans of Dyck paths (resp. vacillating tableaux) of length $n$ into the set of chord diagrams on $[n]$ that intertwines promotion and rotation. This is done in two different ways, namely as fillings of promotion-evacuation diagrams and in terms of Fomin growth diagrams. Our analysis uses the fact that $r$-fans of Dyck paths and vacillating tableaux can be viewed as highest weight elements of weight zero in crystals of type $B_r$ and $C_r$, respectively, which in turn can be analyzed using virtual crystals. On the level of Fomin growth diagrams, the virtualization process corresponds to the Roby-Krattenthaler blow up construction. One of the motivations for finding rotation invariant diagrammatic bases such as chord diagrams is the cyclic sieving phenomenon. Indeed, we give a cyclic sieving phenomenon on $r$-fans of Dyck paths and vacillating tableaux using the promotion action.