Rotation and PromotionAlgebra & Discrete Mathematics
|Stephan Pfannerer-Mittas, University of Vienna
|Fri, Apr 1 2022, 12:10PM
We present a construction to obtain pictorial descriptions of bases of invariant spaces of representations of Lie groups in terms of certain graphs, which we call chord diagrams. These are graphs whose vertices, arranged in a circle, correspond to the tensor positions of the invariant. In particular, our approach yields a map that intertwines rotation of chord diagrams and the action of the long cycle of the symmetric group naturally acting on tensor powers.
Our approach uses generalizations of Schützenberger's promotion operator on highest weight vertices in tensor products of Kashiwara's crystal graphs. Using van Leeuwen's local rules we define promotion-evacuation diagrams. Similarly to Fomin's growth diagrams we want to decorate the cells with positive integers depending only on the labels of their corners. The filling of the promotion-evacuation diagram can then be interpreted as the adjacency matrix of a chord diagram. It remains to define these filling rules, such that the construction is injective.