A Local Characterization of Unions of Demazure Crystals

TL;DR

This paper provides a local characterization of unions of Demazure crystals within highest weight crystals for symmetrizable Kac-Moody algebras, revealing their structure and decompositions.

Contribution

It introduces a new local criterion for identifying unions of Demazure crystals and proves their disjoint decomposition into Demazure atoms.

Findings

Characterization of unions of Demazure crystals as subsets of highest weight crystals.

Disjoint decomposition of these subsets into Demazure atoms.

A crystal-theoretic proof that Polo modules admit a relative Schubert filtration.

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.