A cactus group action on shifted tableau crystals and a shifted Berenstein-Kirillov group

Abstract: Gillespie, Levinson and Purbhoo introduced a crystal-like structure for shifted tableaux, called the shifted tableau crystal. Following a similar approach as Halacheva, we exhibit a natural internal action of the cactus group on this structure, realized by the restrictions of the shifted Schützenberger involution to all primed intervals of the underlying crystal alphabet. This includes the shifted crystal reflection operators, which agree with the restrictions of the shifted Schützenberger involution to single-coloured connected components, but unlike the case for type A crystals, these do not need to satisfy the braid relations of the symmetric group.

In addition, we define a shifted version of the Berenstein-Kirillov group, by considering shifted Bender-Knuth involutions. Paralleling the works of Halacheva and Chmutov, Glick and Pylyavskyy for type A semistandard tableaux of straight shape, we exhibit another occurrence of the cactus group action on shifted tableau crystals of straight shape via the action of the shifted Berenstein-Kirillov group. We conclude that the shifted Berenstein-Kirillov group is isomorphic to a quotient of the cactus group. Not all known relations that hold in the classic Berenstein-Kirillov group need to be satisfied by the shifted Bender-Knuth involutions, namely the one equivalent to the braid relations of the type A crystal reflection operators, but the ones implying the relations of the cactus group are verified, thus we have another presentation for the cactus group in terms of shifted Bender-Knuth involutions.

mathematical physicsalgebraic geometrydifferential geometrygeometric topologyoperator algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

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