Exercise 7, § 4

Assume that (W, S) is irreducible and that B_M is positive.

a) Show that, for any subset T of S distinct from S, the group W_T is finite (use Th. 2 as well as Lemma 4 of § 3, no. 5).

b) Show that, if Card(S) \geq 3, the m(s, s') are all finite.

c) Assume that W is infinite. Show that, if T \subset S, T \neq S, the group \sigma(W_T) leaves stable a lattice in \mathbf{R}^T. Deduce that, if Card(S) \geq 3, the m(s, s'), s \neq s', all belong to the set {2, 3, 4, 6} (use Exerc. 6).