Homework Set #6

  1. Let Φ be a root system and let Ψ = {(α⋅α)-1α | α∈Φ}. Prove that Ψ is a root system.
  2. Let Φ denote the subset of Z8 consisting of vectors of length 2√2 such that all coordinates have the same parity and the sum of coordinates is divisible by 4. Prove Φ is a root system.
  3. Prove that every inclusion of Dynkin diagrams induces an inclusion of the corresponding root systems.
  4. Let Γ be a connected graph with vertices {1,2,...,n}. If the quadratic form
    Q(x1,x2,...,xn) = x12+...+xn2 - Σxixj ≥ 0
    for all (x1,x2,...,xn) &isinRn, where the sum is taken over edges (i,j)∈Γ, i<j, prove that
    Q(x1,x2,...,xn) > 0
    when all of the coordinates are ≥ 0, at least one is 0, and at least one is strictly positive.
  5. Use the previous result to classify Γ such that
    Q(x1,x2,...,xn) ≥ 0
    for all (x1,x2,...,xn) &isinRn.
Revised October 2, 2007.