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

TL;DR

This paper provides an explicit combinatorial description of how Kashiwara-Nakashima tableaux in types B and C embed into Lusztig data, using elementary algorithms like jeu de taquin and RSK, with a spinor model as an intermediary.

Contribution

It introduces a simple, elementary combinatorial method to describe the embedding of Kashiwara-Nakashima tableaux into Lusztig data for types B and C, utilizing the spinor model.

Findings

Explicit combinatorial embedding described

Uses jeu de taquin and RSK algorithms

Connects tableaux with Lusztig data via spinor model

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.