An action of the cactus group of shifted tableau crystals

TL;DR

This paper introduces a shifted version of crystal reflection operators on shifted tableau crystals, establishing a cactus group action that generalizes previous involutions without forming a symmetric group action.

Contribution

It defines a new shifted crystal reflection operator and demonstrates a natural cactus group action on shifted tableau crystals, extending the structure of involutions.

Findings

Introduces shifted crystal reflection operators.

Establishes a cactus group action on shifted crystals.

Shows operators do not satisfy braid relations.

Abstract

Recently, Gillespie, Levinson and Purbhoo introduced a crystal-like structure for shifted tableaux, called the shifted tableau crystal. We introduce, on this structure, a shifted version of the crystal reflection operators, which coincide with the restrictions 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 since the braid relations do not need to hold. Following a similar approach as Halacheva, we exhibit a natural internal action of the cactus group on this crystal, realized by the restrictions of the shifted Sch\"utzenberger involution to all primed intervals of the underlying crystal alphabet, containing, in particular, the aforesaid action of the shifted crystal reflection operator analogues.