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The Bender-Knuth involutions on Young tableaux are known to coincide with thetableau switching on two adjacent letters, together with a swapping of thoseletters. Using the shifted tableau switching due to Choi, Nam and Oh (2019), weintroduce a shifted version of the Bender-Knuth operators and define a shiftedversion of the Berenstein-Kirillov group. The actions of the cactus group, dueto the author, and of the shifted Berenstein-Kirillov group on theGillespie-Levinson-Purbhoo straight-shaped shifted tableau crystal (2017, 2020)coincide. Following the works of Halacheva (2016, 2020), and Chmutov, Glick andPylyavskyy (2016, 2020), on the relation between the actions of theBerenstein-Kirillov group and the cactus group on the crystal ofstraight-shaped Young tableaux, we show that the shifted Berenstein-Kirillovgroup is isomorphic to a quotient of the cactus group. Not all the knownrelations that hold in the classic Berenstein-Kirillov group need to besatisfied by the shifted Bender-Knuth involutions, but the ones implying therelations of the cactus group are verified. Hence we have an alternativepresentation for the cactus group via the shifted Bender-Knuth involutions.

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