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 Schützenberger'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.

arXiv comment41 pages, introduction is updated, the proof of Lemma 5.2 is revised with mored details, some typos in the proof of Theorem 5.7 are corrected