Abstract:
In  this  talk,  we  begin  by  presenting  the  crystal  structure  of  finite-dimensional  irreducible  representations  of  the  Lie algebras $\mathfrak{sl}_n$ in terms of Gelfand-Tsetlin patterns.  We then define a crystal structure using the set of symplectic Zhelobenko patterns,  parametrizing bases for finite-dimensional irreducible representations of $\mathfrak{sp}_4$.  This is obtained by a bijection with Kashiwara-Nakashima tableaux and the symplectic jeu de taquin of Sheats and Lecouvey.  We offer some conjectures on the generalization of this structure to rank $n$.