Berenstein and Kirillov have introduced and studied the group generated by the Bender-Knuth involutions modulo the relations they do satisfy on semi-standard tableaux of straight shape. Halacheva has generalized the Henriques-Kamnitzer cactus group to Dynkin diagrams of classical Cartan types and defined an internal action of those cacti on the crystals of the corresponding Cartan types.
Halacheva and Chmutov, Glick, Pylyavskyy have shown that the known relations for the Berenstein-Kirillov group reduce to the Cartan type A cactus relations plus one coming from the braid relations of the symmetric group.We introduce the analogue symplectic Berenstein-Kyrillov group using the Kashiwara-Nakashima tableaux. We study it by virtual analysis and provide a set of relations that reduce to those of the sympletic cactus group plus two relations coming from the braid relations of the group of signed permutations. As in the previous case it is not known whether these relations form a complete set. This is a joint work with Mojdeh Tarighat Feller and Jacinta Torres.
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