A result on generating root systems by linearly independent subsets

Abstract

Let Φ be a root system in a finite dimensional Euclidean space F and S be a subset of Φ. Let the smallest in the collection of all root systems in F which contain S—i.e., the intersection of all such root systems—be denoted by R(S). It can be easily shown that Φ has linearly independent subsets X such that R(X)=Φ—e.g., for any base Δ of Φ, R(Δ)=Φ. We prove a result that generalizes the preceding fact: If Ψ is any subset of Φ, then there exists a linearly independent subset S of Ψ such that R(S)⊇Ψ. In the process of deriving the above one, we find a sufficient condition for a root system to be isomorphic to one of the root systems in {An,Dn+3:n∈N} and obtain a simple proof of the following known result on exceptional root systems: Let k,ℓ be integers such that 6⩽k⩽ℓ⩽8; if X is a subset of the exceptional root system E8 such that R(X) is isomorphic to Eℓ, then for some linearly independent subset Y of X, R(Y) is isomorphic to Ek.