A shifted Berenstein-Kirillov group and the cactus group

TL;DR

This paper introduces a shifted version of the Berenstein-Kirillov group using shifted tableau switching, demonstrating its isomorphism to a quotient of the cactus group and providing new presentations and proofs related to crystal actions.

Contribution

It defines a shifted Berenstein-Kirillov group, shows its relation to the cactus group, and offers an alternative presentation and proof methods for their actions on shifted tableau crystals.

Findings

Shifted Berenstein-Kirillov group acts on shifted tableau crystals.

It is isomorphic to a quotient of the cactus group.

Provides an alternative presentation for the cactus group.

Abstract

The Bender-Knuth involutions on semistandard Young tableaux are known to coincide with the tableau switching on horizontal border strips of two adjacent letters, together with the swapping of those letters. Motivated by this coincidence and using the shifted tableau switching due to Choi, Nam and Oh (2019), we consider a shifted version of the Bender-Knuth involutions and define a shifted version of the Berenstein-Kirillov group (1995). Similarly to the classical case, the shifted version of the Berenstein-Kirillov group also acts on the straight-shaped shifted tableau crystals introduced by Gillespie, Levinson and Purbhoo (2020), via partial Sch\"utzenberger involutions, thus coinciding with the action of the cactus group on the same crystal, due to the author. Following the works of Halacheva (2016, 2020), and Chmutov, Glick and Pylyavskyy (2020), on the relation between the actions of the Berenstein-Kirillov group and the cactus group on a crystal of straight-shaped Young tableaux, we also show that the shifted Berenstein-Kirillov group is isomorphic to a quotient of the cactus group. Not all the known relations that hold in the classic Berenstein-Kirillov group need to be satisfied by the shifted Bender-Knuth involutions, but the ones implying the relations of the cactus group are verified. Hence, we have an alternative presentation for the cactus group in terms of the shifted Bender-Knuth involutions. We also use the shifted growth diagrams due to Thomas and Yong (2016) to provide an alternative proof concerning the mentioned cactus group action.