[Paper Review] PBW bases and marginally large tableaux in type D

Ben Salisbury, Adam Schultze+1 more|arXiv (Cornell University)|Jun 8, 2016

Algebraic structures and combinatorial models9 references3 citations

TL;DR

This paper establishes an explicit crystal isomorphism between two realizations of $B(\infty)$ in type $D_n$: marginally large tableaux and PBW monomials derived from a specific reduced expression of the longest word $w_0$. The isomorphism is non-local, requiring simultaneous consideration of multiple boxes in the tableau to map to Kostant partitions, and is fully described via diagrammatic rules that generalize type $A$ constructions.

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.

Motivation & Objective

• To establish a canonical crystal isomorphism between two combinatorial realizations of $B(\infty)$ in type $D_n$.
• To resolve the challenge that the isomorphism is non-local in type $D$, unlike in type $A$, where individual boxes map directly to roots.
• To provide a diagrammatic interpretation of Kostant partitions and crystal operators, mirroring the multisegment picture used in type $A$.
• To offer a concrete, algorithmic description of the map from marginally large tableaux to PBW monomials using bracketing rules on Kostant partitions.
• To extend the type $A$ framework of [3] to type $D$, accounting for the structural differences in root systems and crystal operators.

Proposed method

• Construct $B(\infty)$ using marginally large tableaux as defined in [6], with explicit filling rules and weight conditions.
• Realize $B(\infty)$ via PBW monomials associated with a specific reduced expression of the longest word $w_0$ in type $D_n$, using bracketing rules on Kostant partitions.
• Define a crystal isomorphism by mapping each tableau to a Kostant partition through a non-local rule involving multiple boxes and their root contributions.
• Introduce a diagrammatic model for Kostant partitions and crystal operators, using labeled nodes and edges to represent root additions and removals.
• Verify the isomorphism by checking that the map preserves the crystal structure: it commutes with Kashiwara operators $e_i$ and $f_i$.
• Use computational tools (Sage) to verify and motivate the construction, particularly for small $n$.

Experimental results

Research questions

• RQ1 How can the crystal isomorphism between marginally large tableaux and PBW monomials in type $D_n$ be explicitly described?
• RQ2 Why is the isomorphism in type $D$ non-local, unlike in type $A$, and how can this be systematically captured?
• RQ3 Can a diagrammatic model for Kostant partitions and crystal operators be constructed that mirrors the multisegment picture in type $A$?
• RQ4 What is the precise rule that maps a marginally large tableau to a Kostant partition via PBW monomials?
• RQ5 How do the Kashiwara operators $e_i$ and $f_i$ act on the PBW monomial realization, and how do they correspond to actions on tableaux?

Key findings

• The unique crystal isomorphism between marginally large tableaux and PBW monomials in type $D_n$ is explicitly described, with a non-local rule that maps multiple boxes simultaneously to a Kostant partition.
• The isomorphism is not local: unlike in type $A$, where each box maps independently to a root, in type $D$ the mapping depends on the collective configuration of boxes.
• The paper provides a diagrammatic model for Kostant partitions and crystal operators, using labeled nodes and edges to represent root additions and removals, generalizing the multisegment picture from type $A$.
• The map preserves the crystal structure: it commutes with all Kashiwara operators $e_i$ and $f_i$, confirming it is a well-defined crystal isomorphism.
• The construction is validated through computational checks using Sage, particularly for small $n$, and is consistent with the known PBW and tableau realizations of $B(\infty)$.
• The result extends the type $A$ framework of [3] to type $D$, resolving the structural differences in root systems and crystal actions.