Polytopes for Crystallized Demazure Modules and Exremal Vectors

TL;DR

This paper introduces a geometric approach to understanding Demazure modules by parametrizing their crystal bases with convex polytopes and explicitly characterizing extremal vectors through linear systems.

Contribution

It provides a new polytope-based parametrization of Demazure modules' crystal bases and explicit descriptions of extremal vectors, advancing the combinatorial understanding of these modules.

Findings

Crystal bases parametrized by lattice points in convex polytopes

Explicit description of extremal vectors via linear equations

Enhanced combinatorial understanding of Demazure modules

Abstract

We give a parametrization for crystal bases of Demazure modules as a set of lattice points in some convex polytope and we also describe explicitly the extremal vectors as solutions of some system of linear equations.