Séminaire Lotharingien de Combinatoire, 85B.53 (2021), 12 pp.
Inês Rodrigues
Shifted Bender-Knuth Moves and a Shifted Berenstein-Kirillov Group
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-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-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.
Received: December 1, 2020.
Accepted: March 1, 2021.
Final version: April 29, 2021.
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