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arXiv · 2004.00285

An action of the cactus group on shifted tableau crystals

Inês Rodrigues
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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 Sch\"utzenberger involution on primed intervals of the underlying crystal alphabet.

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Inês Rodrigues. 2020-04-01. An action of the cactus group on shifted tableau crystals. https://arxiv.org/abs/2004.00285

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