A Local Characterization of Unions of Demazure Crystals

TL;DR

The paper provides a local, crystal-theoretic criterion for when a subset of a highest weight crystal is a Demazure crystal, characterizing such subsets as extremal, ideal, and principal. It introduces ideal subsets indexed by lower Bruhat ideals and proves these coincide with unions of Demazure crystals, further yielding disjoint decompositions into Demazure atoms and a link to dual Polo modules and Schubert filtrations. It shows that Demazure crystals are exactly the ideal subsets that are principal, establishing a precise, local criterion for Demazure structure and enabling a crystal-theoretic approach to relative Schubert filtrations. Overall, the work unifies Demazure crystals, extremal subsets, and Polo-module filtrations through a local, combinatorial framework applicable to symmetrizable Kac-Moody algebras.

Abstract

We characterize subsets of highest weight $\mathfrak{g}$-crystals that arise as unions of Demazure crystals, for any symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$. We provide a local characterization for these subsets and prove they admit disjoint decompositions into Demazure atoms. As a consequence, we give a new characterization for when a subset of a highest weight crystal is a Demazure crystal as well as a crystal-theoretic proof that any Polo module admits a relative Schubert filtration.

Figures (1)

• Figure 1: An illustration of the ideal condition: if $x,y \in X$ with $y = f_{i_m}^* \cdots f_{i_1}^* f_{j}^*(x)$, then $f_{i_m}^* \cdots f_{i_1}^*(x)\in X$.

Theorems & Definitions (41)

• Theorem I
• Theorem II
• Theorem III
• Definition A
• Definition B: Kas93
• Theorem 3.1: Kas93
• Lemma 3.2: BB05
• Lemma 3.3: ADG
• Definition C
• Remark 4.1