Bourbaki

Corollary 3

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

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

The homothety with ratio −1 of V belongs to W if and only if all the exponents of W are odd. In that case, h is even and ch/2=−1c^{h/2} = -1 for any Coxeter transformation c of W.

The first assertion follows from § 5, no. 3, Prop. 4. Assume that the exponents of W are odd. Then h is even by formula (2), and

(exp⁡2iπmjh)h/2=exp⁡(iπmj)=−1; \left( \exp \frac{2i\pi m_j}{h} \right)^{h/2} = \exp(i\pi m_j) = -1;

thus ch/2=−1c^{h/2} = -1 since c is a semi-simple automorphism of V (Algebra, Chap. IX, § 7, no. 3, Prop. 4).

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

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Transcribed from the printed volume. Not translated.