Lusztig data of Kashiwara-Nakashima tableaux in type D

TL;DR

This work provides an explicit crystal embedding of Kashiwara-Nakashima tableaux of type $D$ into Lusztig data for a family of reduced expressions compatible with a maximal Levi subalgebra of type $A$. The construction routes the KN crystal ${\bf KN}_{\lambda}$ through a parabolic Verma crystal ${\bf V}_{\lambda}$ via a separation procedure based on Schützenberger slides, and then into the PBW/Lusztig data crystal ${\bf B}_{\boldsymbol{\rm i}}$ through a Burge-type RSK correspondence for type $D_n$. The main results include a detailed separation algorithm (including proofs of its crystal-compatibility) and the explicit embedding into Lusztig data, with the KN–spinor isomorphism playing a crucial role. These results connect KN tableaux in type $D$ to type $A$-combinatorics and to PBW bases, enabling a concrete description of the crystal embedding and applications to branching rules and Kirillov–Reshetikhin crystals. The approach yields a combinatorial framework that unifies KN tableaux, spinor models, parabolic Verma modules, and Lusztig data through explicit, algorithmic maps.

Abstract

We describe the embedding from the crystal of Kashiwara-Nakashima tableaux in type $D$ of an arbitrary shape into that of $\mathbf{i}$-Lusztig data associated to a family of reduced expressions $\mathbf{i}$ which are compatible with the maximal Levi subalgebra of type $A$. The embedding is described explicitly in terms of well-known combinatorics of type $A$ including the Schützenberger's jeu de taquin and an analog of RSK algorithm.