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 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.
Keywords
Cite
@article{arxiv.2104.13679,
title = {Shifted Bender-Knuth moves and a shifted Berenstein-Kirillov group},
author = {Inês Rodrigues},
journal= {arXiv preprint arXiv:2104.13679},
year = {2021}
}
Comments
12 pages, 2 figures. This is an extended abstract of arXiv:2104.11799 for FPSAC 2021, to appear in S\'eminaire Lotharingien de Combinatoire. arXiv admin note: substantial text overlap with arXiv:2104.11799