Symplectic cacti, virtualization and Berenstein–Kirillov groups

Abstract

We explicitly realize an internal action of the symplectic cactus group, recently defined by Halacheva for any complex, reductive, finite-dimensional Lie algebra, on crystals of Kashiwara–Nakashima tableaux. Our methods include a symplectic version of jeu de taquin due to Sheats and Lecouvey, symplectic reversal, and virtualization due to Baker. As an application, we define and study a symplectic version of the Berenstein–Kirillov group and show that it is a quotient of the symplectic cactus group. In addition two relations for symplectic Berenstein–Kirillov group are given that do not follow from the defining relations of the symplectic cactus group.