Cocrystals of symplectic Kashiwara-Nakashima tableaux, symplectic Willis like direct way, virtual keys and applications

Abstract

We attach a $\mathfrak{sl}_2$ crystal, called cocrystal, to a symplectic Kashiwara-Nakashima (KN) tableau, whose vertices are skew KN tableaux connected via the Lecouvey-Sheats symplectic \emph{jeu de taquin}. These cocrystals contain all the needed information to compute right and left keys of a symplectic KN tableau. Motivated by Willis' direct way of computing type $A$ right and left keys, we also give a way of computing symplectic, right and left, keys without the use of the symplectic \emph{jeu de taquin}. On the other hand, we prove that Baker virtualization by folding $A_{2n-1}$ into $C_n$ commutes with dilatation of crystals. Thus we may alternatively utilize this Baker virtualization to embed a type $C_n$ Demazure crystal, its opposite and atoms into $A_{2n-1}$ ones. The right, respectively left keys of a KN tableau are thereby computed as $A_{2n-1}$ semistandard tableaux and returned back via reverse Baker embedding to the $C_n$ crystal as its right respectively left symplectic keys. In particular, Baker embedding also virtualizes the crystal of Lakshmibai-Seshadri paths as $B_n$-paths into the crystal of Lakshmibai-Seshadri paths as $\mathfrak{S}_{2n}$-paths. Lastly, as an application of our explicit symplectic right and left key maps, thanks to the isomorphism between Lakshmibai-Seshadri path and Kashiwara crystals we use, similarly to the ${{Gl}(n,\mathbb{C})}$ case, left and right key maps as a tool to test whether a symplectic KN tableau is \emph{standard} on a Schubert or Richardson variety in the flag variety $Sp(2n,\mathbb{C})/B$, with $B$ a Borel subgroup.