Characters of Feigin-Stoyanovsky subspaces and Brion's theorem

TL;DR

This paper provides an alternative proof for the character formula of Feigin-Stoyanovsky subspaces using Brion's theorem on convex polyhedra, connecting geometric and algebraic methods in representation theory.

Contribution

It introduces a novel geometric approach to compute characters of subspaces in affine Lie algebra representations, leveraging polyhedral combinatorics.

Findings

Character formula derived via polyhedral geometry

Application of Brion's theorem to representation theory

New perspective on monomial bases in affine Lie algebra modules

Abstract

We give an alternative proof of the main result of the paper http://arxiv.org/abs/math/0112104, the proof relies on Brion's theorem about convex polyhedra. The result itself can be viewed as a formula for the character of the Feigin-Stoyanovsky subspace of an integrable irreducible representation of the affine Lie algebra $\widehat{\mathfrak{sl}_n}(\mathbb{C})$. Our approach is to assign integer points of a certain polytope to the vectors comprising a monomial basis of the subspace and then compute the character via (a variation of) Brion's theorem.