Corollary

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

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

Let $R_0$ be the set of unit vectors of $V$ orthogonal to an element of $\mathfrak{H}$ . If $W$ is irreducible,

\sum_{u \in R_0} (x|u)^2 = h(x|x)

for all $x \in V$ .

Put $f(x) = \sum_{u \in R_0} (x|u)^2$ . It is clear that $f$ is a positive quadratic form invariant under $W$ , and non-degenerate since the $e_i$ form a basis of $V$ . Since

W is irreducible, there exists a constant $\beta$ such that $f(x) = \beta(x|x)$ (§ 2, no. 1, Prop. 1). If $(x_i)_{1 \leq i \leq l}$ is an orthonormal basis of V for the scalar product $(x|y)$ , then

\beta l = \sum_{i=1}^l \beta(x_i|x_i) = \sum_{i=1}^l f(x_i) = \sum_{i=1}^l \sum_{u \in R_0} (x_i|u)^2 \\ = \sum_{u \in R_0} 1 = \mathrm{Card}(R_0) = 2\mathrm{Card}(\mathfrak{H}) = hl.

Thus

\beta = h

, which proves (4).

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

Cited by

Cites

§ 2, no. 1, Prop. 1