Crystals for Demazure Modules of Classical Affine Lie Algebras

TL;DR

This paper investigates the structure of crystals for Demazure modules in affine Lie algebras, revealing simplified structures for certain highest weights using path realizations and affine Weyl group elements.

Contribution

It introduces a specific sequence of affine Weyl group elements that simplifies the Demazure crystal structure for selected perfect crystals.

Findings

Demazure crystals have a simple structure for highest weight $l\\La_0$.

A special sequence of affine Weyl group elements is identified.

The study covers multiple types of affine Lie algebras.

Abstract

We study, in the path realization, crystals for Demazure modules of affine Lie algebras of types $A^{(1)}_n,B^{(1)}_n,C^{(1)}_n,D^{(1)}_n, A^{(2)}_{2n-1},A^{(2)}_{2n}, and D^{(2)}_{n+1}$. We find a special sequence of affine Weyl group elements for the selected perfect crystal, and show if the highest weight is $l\La_0$, the Demazure crystal has a remarkably simple structure.