English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We attach a $\mathfrak{sl}_2$ crystal, called cocrystal, to a symplecticKashiwara-Nakashima (KN) tableau, whose vertices are skew KN tableaux connectedvia the Lecouvey-Sheats symplectic \emph{jeu de taquin}. These cocrystalscontain all the needed information to compute right and left keys of asymplectic KN tableau. Motivated by Willis' direct way of computing type $A$right and left keys, we also give a way of computing symplectic, right andleft, keys without the use of the symplectic \emph{jeu de taquin}. On the otherhand, we prove that Baker virtualization by folding $A_{2n-1}$ into $C_n$commutes with dilatation of crystals. Thus we may alternatively utilize thisBaker virtualization to embed a type $C_n$ Demazure crystal, its opposite andatoms into $A_{2n-1}$ ones. The right, respectively left keys of a KN tableauare thereby computed as $A_{2n-1}$ semistandard tableaux and returned back viareverse Baker embedding to the $C_n$ crystal as its right respectively leftsymplectic keys. In particular, Baker embedding also virtualizes the crystal ofLakshmibai-Seshadri paths as $B_n$-paths into the crystal ofLakshmibai-Seshadri paths as $\mathfrak{S}_{2n}$-paths. Lastly, as anapplication of our explicit symplectic right and left key maps, thanks to theisomorphism between Lakshmibai-Seshadri path and Kashiwara crystals we use,similarly to the ${{Gl}(n,\mathbb{C})}$ case, left and right key maps as a toolto test whether a symplectic KN tableau is \emph{standard} on a Schubert orRichardson variety in the flag variety $Sp(2n,\mathbb{C})/B$, with $B$ a Borelsubgroup.

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