PBW bases and marginally large tableaux in type D

TL;DR

This work addresses two realizations of the crystal $B(\infty)$ in type $D$: marginally large tableaux and a Kostant-partition/PBW model. It provides an explicit crystal isomorphism $\Psi$ between $\mathcal{T}(\infty)$ and $\mathrm{Kp}(\infty)$, describing the map row-by-row via the roots $\beta_{i,k}$ and $\gamma_{j,\ell}$ and proving $f_i^{\mathcal{T}}\Psi(T)=\Psi(f_i^{\mathrm{Kp}}T)$. Unlike the type $A$ case, the type $D$ map is not purely local and requires coordinated changes across multiple boxes; this is made concrete through a rowwise, diagrammatic analysis and the introduced stack notation. The results extend the type $A$ correspondence to type $D$ and provide practical, diagrammatic tools for understanding the crystal structure of $B(\infty)$ in this setting.

Abstract

We give an explicit description of the unique crystal isomorphism between two realizations of $B(\infty)$ in type $D$: that using marginally large tableaux and that using PBW monomials with respect to one particularly nice reduced expression of the longest word.

Figures (1)

• Figure 2.1: The fundamental crystal of type $D_n$.

Theorems & Definitions (21)

• Definition 2.1
• Example 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Example 2.6
• Proposition 2.7
• proof
• Remark 2.9
• Example 2.10