Simplified presentations and embeddings of Demazure modules

TL;DR

This paper establishes an embedding of higher level Demazure modules into tensor products of lower level modules for affine Lie algebras, simplifying their presentation and providing crystal theoretic tools for classical decomposition analysis.

Contribution

It introduces a simplified presentation of Demazure modules and constructs an embedding into tensor products, extending previous results and connecting to crystal theory.

Findings

Embedding of higher level Demazure modules into tensor products

Simplified presentation of Demazure modules

Crystal-based method for classical decomposition

Abstract

For an untwisted affine Lie algebra we prove an embedding of any higher level Demazure module into a tensor product of lower level Demazure modules (e.g. level one in type A) which becomes in the limit (for anti-dominant weights) the well-known embedding of finite-dimensional irreducible modules of the underlying simple Lie algebra into the tensor product of fundamental modules. To achieve this goal, we first simplify the presentation of these modules extending the results of \cite{CV13} in the $\mathfrak{g}$-stable case. As an application, we propose a crystal theoretic way to find classical decompositions with respect to a maximal semi-simple Lie subalgebra by identifying the Demazure crystal as a connected component in the corresponding tensor product of crystals.