Séminaire Lotharingien de Combinatoire, 78B.71 (2017), 12 pp.
Stephan Pfannerer, Martin Rubey and Bruce W. Westbury
Promotion, Evacuation and Cactus Groups
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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