Projection of root systems and the generalized injectivity conjecture for exceptional groups

TL;DR

This paper studies projections of root systems in Euclidean spaces, identifying conditions under which these projections contain root systems of a given rank, and applies this to prove a conjecture for most exceptional groups.

Contribution

It provides criteria for when projected root systems contain sub-root systems of a specific rank and verifies the generalized injectivity conjecture for most exceptional groups.

Findings

Identified conditions for root system projections to contain sub-root systems of a given rank.

Listed all exceptional root systems that can appear in projected root systems.

Proved the generalized injectivity conjecture for most cases involving exceptional groups.

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 $d$. When considering the case of $\Sigma$ of type exceptional, we give a list of all exceptional root systems that can occur in $\Sigma_\Theta$ and use it to prove the generalized injectivity conjecture in most exceptional groups cases.