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 i-Lusztig data associated to a family of reduced expressions i of the longest element w0. 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.