Séminaire Lotharingien de Combinatoire, 84B.51 (2020), 12 pp.

Inês Rodrigues

An Action of the Cactus Group on Shifted Tableau Crystals

Abstract. Recently, Gillespie, Levinson and Purbhoo introduced a crystal-like structure for shifted tableaux, called the shifted tableau crystal. We introduce a shifted analogue of the crystal reflection operators, which coincides with the restriction of the shifted Sch\"utzenberger involution to any primed interval of two adjacent letters. Unlike type A Young tableau crystals, these operators do not realize an action of the symmetric group on the shifted tableau crystal because braid relations do not hold. We exhibit a natural internal action of the cactus group, realized by restrictions of the shifted Schützenberger involution on primed intervals of the underlying crystal alphabet.


Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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