Lusztig data of Kashiwara-Nakashima tableaux in types $B$ and $C$

Abstract

We provide an explicit combinatorial description of the embedding of the crystal of Kashiwara-Nakashima tableaux in types $B$ and $C$ into that of $\bi$-Lusztig data associated to a family of reduced expressions $\bi$ of the longest element $w_0$. The description of the embedding is simple and elementary using only the Sch\"{u}tzenberger's jeu de taquin and RSK algorithm. A spinor model for classical crystals plays an important role as an intermediate object connecting Kashiwara-Nakashima tableaux and Lusztig data.