2021/11/22 by Azenhas, Olga, Santos, João Miguel
#05E05 #05E10 #14M15 #17B37 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2111.11160
We attach a \mathfraksl2 crystal, called cocrystal, to a symplectic Kashiwara-Nakashima (KN) tableau, whose vertices are skew KN tableaux connected via the Lecouvey-Sheats symplectic 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 jeu de taquin. On the other hand, we prove that Baker virtualization by folding A2n-1 into Cn commutes with dilatation of crystals. Thus we may alternatively utilize this Baker virtualization to embed a type Cn Demazure crystal, its opposite and atoms into A2n-1 ones. The right, respectively left keys of a KN tableau are thereby computed as A2n-1 semistandard tableaux and returned back via reverse Baker embedding to the Cn crystal as its right respectively left symplectic keys. In particular, Baker embedding also virtualizes the crystal of Lakshmibai-Seshadri paths as Bn-paths into the crystal of Lakshmibai-Seshadri paths as \mathfrakS2n-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,ℂ) case, left and right key maps as a tool to test whether a symplectic KN tableau is standard on a Schubert or Richardson variety in the flag variety Sp(2n,ℂ)/B, with B a Borel subgroup.