Homogeneous Bases for Demazure Modules

TL;DR

This paper constructs explicit bases for Demazure modules' associated graded spaces using polytopes, revealing their monomial annihilating ideals across various Lie types.

Contribution

It introduces a novel polytope-based basis construction for Demazure modules' graded spaces, establishing their monomial annihilating ideals in multiple Lie types.

Findings

Constructed normal polytopes labelling bases for Demazure modules.

Proved the annihilating ideals are monomial.

Extended results across types A, B, C, D.

Abstract

We study the PBW filtration on the Demazure modules $V_{r_{\gamma}}(\lambda)$ associated to reflections $r_{\gamma}$ at positive roots in type $A_n$, long roots in type $C_n$, short roots in type $B_n$ and positive roots not involving the simple root $\alpha_{n-1}$ in type $D_n$. In each type, we construct a normal polytope labelling a basis for the associated graded space $\operatorname{gr} V_{r_{\gamma}}(\lambda)$ and prove that the annihilating ideal of $\operatorname{gr} V_{r_{\gamma}}(\lambda)$ is a monomial ideal.