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 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.
Keywords
Cite
@article{arxiv.2004.00285,
title = {An action of the cactus group on shifted tableau crystals},
author = {Inês Rodrigues},
journal= {arXiv preprint arXiv:2004.00285},
year = {2020}
}
Comments
Extended abstract, 12 pages. To appear in Formal Power Series and Algebraic Combinatorics 2020