Projection of root systems

TL;DR

This paper investigates conditions under which the projection of a root system onto a subspace retains a root system structure of a specific rank, providing insights into the geometric and algebraic properties of root systems.

Contribution

It establishes criteria for when the projection of a root system contains a root system of a given rank, advancing understanding of root system projections in Lie theory.

Findings

Identifies conditions on subsets of simple roots for the projected set to form a root system.

Provides a characterization of the rank of the projected root system.

Enhances the understanding of the geometric structure of root systems under projection.

Abstract

Let $a$ be a real euclidean vector space of finite dimension and $\Sigma$ a root system in $a$ with a basis $\Delta$. Let $\Theta \subset \Delta$ and $M = M_{\Theta}$ be a standard Levi of a reductive group $G$ such that $a_\Theta = a_M / a_G$. Let us denote $d$ the dimension of $a_\Theta$, i.e the cardinal of $\Delta - \Theta$ and $\Sigma_{\Theta}$ the set of all non-trivial projections of roots in $\Sigma$. We obtain conditions on $\Theta$ such that $\Sigma_\Theta$ contains a root system of rank $d$.