Symplectic cacti, virtualization and Berenstein-Kirillov groups

TL;DR

This paper constructs an explicit action of the symplectic cactus group on Kashiwara-Nakashima tableaux crystals, introduces a symplectic Berenstein-Kirillov group, and explores their algebraic relations and quotients.

Contribution

It provides a new realization of the symplectic cactus group action and defines a symplectic Berenstein-Kirillov group with novel relations and quotient structure.

Findings

Explicit realization of symplectic cactus group action on crystals

Definition of a symplectic Berenstein-Kirillov group as a quotient

Identification of new relations not derived from cactus group relations

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.