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

TL;DR

This paper introduces a new combinatorial framework using cocrystals and virtualization techniques to efficiently compute symplectic keys of Kashiwara-Nakashima tableaux, with applications to Schubert calculus.

Contribution

It develops a novel cocrystal structure for symplectic tableaux and demonstrates virtualization methods to compute keys without jeu de taquin, advancing combinatorial representation theory.

Findings

Cocrystals encode key information for symplectic tableaux.

Virtualization via Baker embedding simplifies key computation.

Applications include testing standardness of tableaux in flag varieties.

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.