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

TL;DR

It is proved that Baker virtualization by folding $A_{2n-1}$ into $C_n$ commutes with dilatation of crystals, and that Baker embedding virtualizes the crystal of Lakshmibai-Seshadri paths as $B_n$-paths into the crystal of Lakshmibai-Seshadri paths as $\mathfrak{S}_{2n}$-paths.

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.

Figures (6)

• Figure 1: The type $C_2$ crystal graph $\mathcal{KN}((2,1),2)$ containing the $A_1$ crystal $\mathcal{SSYT}((2,1),2)$, consisting of the two top left most tableaux, as a subcrystal. The type $C_2$ lowering crystal operators are $f_1$, $\color{blue}{\rightarrow}$, and $~~~~~~~~f_2$, $~~\color{red}{\rightarrow}$.
• Figure 2: The dilatation of the crystal $\textsf{KN}((2,1),2)$ in Figure \ref{['cristal21only']}, by $m=6$, the least common multiple of the maximal $i$-string lengths, inside $\textsf{KN}((12,6),2)\simeq \mathfrak{B}(K(2,1)^{\otimes 6},2)$, exhibiting the right and left keys of each vertex of $\textsf{KN}((2,1),2)$ as the leftmost respectively rightmost factor in each $6$-fold tensor product of keys in $O_{B_2}(2,1)$. The blue (resp. red) arrow means $f_1^6$ (resp. $f_2^6$).
• Figure 3: The crystal $\mathbf{B}(2,1)$ of L-S paths of shape $\lambda=\Lambda_2+\Lambda_1=(2,1)$ obtained from the dilatation of the $C_2$ crystal $\mathfrak{B}(2,1)$. The $C_2$ Weyl group $B_2=<s_1,s_2|s_1^2=s_2^2=1,\,(s_1s_2)^4=1>$ with long element $\textbf{w}_0=s_2s_1s_2s_1$.
• Figure 4: The partition of the $C_2$ crystal graph $\mathfrak{B}{(2,1)}$ into $\left | B_2(2,1)\right|=8$ Demazure atom crystals .
• Figure 5: The $C_2$ crystal graph $\mathfrak{B}(2,1)$ split into opposite Demazure crystal atoms.

Theorems & Definitions (43)

• Definition 2.1: 1CC
• Example 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6
• Example 2.7
• Example 2.8
• Example 2.9
• Remark 2.10