Abstract

Using Henriques' and Kamnitzer's cactus groups, Schützenberger's  promotion and evacuation operators on standard Young tableaux can  be generalised in a very natural way to operators acting on highest  weight words in tensor products of crystals.

In several cases, the highest weight words of weight zero have  familiar representations as chord diagrams.  For example, perfect  matchings in case of the crystal for the vector representation of the  symplectic group, and permutations in case of the crystal for the  adjoint representation of the general linear group.

It is thus desirable to find bijections that map highest weight words  to chord diagrams, such that promotion is mapped to rotation.

In this talk, Stephan and I will first explain the background from  representation theory, and then focus on the case of the adjoint  representation of the general linear group and permutations.

Key ingredients are van Leeuwen's generalisation of Fomin's local  rules for jeu de taquin, connected to the action of the cactus groups  by Lenart, and variants of Fomin's growth diagrams for the  Robinson-Schensted correspondence.

This is joint work with Bruce Westbury.