Séminaire Lotharingien de Combinatoire, 78B.71 (2017), 12 pp.

Stephan Pfannerer, Martin Rubey and Bruce W. Westbury

Promotion, Evacuation and Cactus Groups

Abstract. The promotion operator on rectangular standard tableaux can be generalised to an operator acting on the invariant highest weight words in the tensor power of a crystal. For the vector representation of a symplectic group the Sundaram correspondence is an injective map to perfect matchings. We show that this map intertwines promotion and rotation. For the adjoint representation of a general linear group we construct a similar map to permutations. We show that this map also intertwines promotion and rotation. These results are proved using an approach to the action of the cactus group using a generalisation of local rules and growth diagrams.

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

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