Corollary

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

Lie Groups and Lie Algebras, chapter V, § 4. The geometric representation of a Coxeter group

Let $X$ be a subset of $S$ and $x^*$ an element of $C_X$ . The stabiliser of $x^*$ in $W$ is $W_X$ .

Now let $U$ be the union of the $w(\overline{C})$ for $w \in W$ , and let $\mathscr{F}$ be the set of subsets of $U$ of the form $w(C_X)$ , with $X \subset S$ and $w \in W$ . By the above, $\mathscr{F}$ is a partition of $U$ .

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

Cited by

Theorem 2, § 4

Cites

Nothing.