Homework Set #5

1. Let S be a set of real numbers and X_n,S the subset of Zⁿ consisting of vectors v such that v⋅v∈S. Show that X_n,{1,2} is a root system for all n≥1 and that X_4,{2,4} is a root system.
2. Show that if S is a set of real numbers and Φ is a root system, then as long as Φ_S = {v∈Φ | v⋅v∈S} is non-empty, it is a root system. Prove that if Φ is irreducible, then the rank of Φ_S equals the rank of Φ.
3. Prove that a root system Φ⊂Rⁿ is irreducible if and only if it is not contained in any union of two hyperplanes in Rⁿ.
4. Prove that the Weyl group of any root system contains a subgroup of index 2.

Optional Bonus Problem: Show that if a root system Φ contains a subsystem Ψ of type G₂, then every root in Φ which does not belong to Ψ is perpendicular to every root in Ψ.

Created September 24, 2007.