Corollary 2

Tag 0ATT, permanent. Cite it as [Bourbaki, Tag 0ATT].

Lie Groups and Lie Algebras, chapter V, § 6. The Coxeter transformation

If c is a Coxeter transformation of W,

\exp \left( \frac{2i\pi}{h} \right) \quad \text{and} \quad \exp \left( \frac{-2i\pi}{h} \right)

are eigenvalues of c of multiplicity 1.

Otherwise, there would exist two non-proportional homogeneous invariants of degree 2 in S, and hence two non-proportional quadratic forms on V* invariant under W, contrary to § 2, no. 1, Prop. 1.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Lemma 4, § 1

Cites

§ 2, no. 1, Prop. 1