Promotion for fans of Dyck paths

Abstract

We construct an injection from the set of r-fans of Dyck paths 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 matrices and in terms of Fomin growth diagrams. Our analysis uses the fact that r-fans of Dyck paths can be viewed as highest weight elements of weight zero in crystals of type Br, 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. Our construction generalizes to vacillating tableaux as well. We give a cyclic sieving phenomenon on r-fans of Dyck paths using the promotion action.