Shifted Bender-Knuth moves and a shifted Berenstein-Kirillov group

TL;DR

This paper introduces a shifted version of the Bender-Knuth operators and the Berenstein-Kirillov group, establishing their relation to the cactus group and providing an alternative presentation for the cactus group using shifted involutions.

Contribution

It defines shifted Bender-Knuth operators, constructs a shifted Berenstein-Kirillov group, and proves its isomorphism to a quotient of the cactus group, extending known relations to the shifted setting.

Findings

The shifted Berenstein-Kirillov group is isomorphic to a quotient of the cactus group.

Shifted Bender-Knuth involutions satisfy relations of the cactus group.

An alternative presentation of the cactus group via shifted involutions is provided.

Abstract

The Bender-Knuth involutions on Young tableaux are known to coincide with the tableau switching on two adjacent letters, together with a swapping of those letters. Using the shifted tableau switching due to Choi, Nam and Oh (2019), we introduce a shifted version of the Bender-Knuth operators and define a shifted version of the Berenstein-Kirillov group. The actions of the cactus group, due to the author, and of the shifted Berenstein-Kirillov group on the Gillespie-Levinson-Purbhoo straight-shaped shifted tableau crystal (2017, 2020) coincide. Following the works of Halacheva (2016, 2020), and Chmutov, Glick and Pylyavskyy (2016, 2020), on the relation between the actions of the Berenstein-Kirillov group and the cactus group on the crystal of straight-shaped Young tableaux, we 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 via the shifted Bender-Knuth involutions.