Proposition 9

With the preceding notations, $W$ is the subgroup of $\mathbf{O}(E)$ generated by the reflections with respect to the hyperplanes of $\mathfrak{H}$. It is an essential group ($§ 3$, no. 7) and $C_0$ is a chamber of $E$ relative to $\mathfrak{H}$.

The first assertion is trivial. On the other hand, if $x \in E$ is invariant under $W$, it is orthogonal to all the $e_s$, and hence is zero; this shows that $W$ is essential. Finally, the isomorphism $E \to E^*$ defined by $B_M$ transforms $C_0$ to the set $C$ of no. 4; the property $(P_n)$ proved there shows that, for all

$^2$ The roots of the equation $z^3 - 1 = 0$ are 1 and $\frac{-1 \pm i \sqrt{3}}{2}$. Thus $\cos \frac{2\pi}{3} = -\frac{1}{2}$ and consequently $\cos \frac{\pi}{3} = \frac{1}{2}$. Note also that $\sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2}$, hence

Similarly, the roots of the equation $z^2 - i = 0$ are $\pm \frac{1+i}{\sqrt{2}}$, so

$w \in W$ and all $s \in S$, $w(C_0)$ does not meet $H_s$. It follows that $C_0$ is contained in the complement $U$ of the union of the hyperplanes of $\mathfrak{H}$, and since $C_0$ is connected, open and closed in $U$, it is a chamber of $E$ relative to $\mathfrak{H}$. Q.E.D.

We can apply to $W$ and $C_0$ all the properties proved in § 3. In particular, $\overline{C_0}$ is a fundamental domain for the action of $W$ on $E$ (in other words, the cone $U$ defined in no. 6 is equal to the whole of $E$).

Conversely, let $E$ be a finite dimensional real vector space, provided with a scalar product $(x|y)$ and let $W$ be an essential finite group of displacements of $E$ leaving 0 fixed; assume that $W$ is generated by reflections. Let $C_0$ be a chamber of $E$ with respect to $W$ (cf. § 3), and let $S$ be the set of orthogonal reflections relative to the walls of $C_0$. Then $(W, S)$ is a finite Coxeter system ($§ 3$, no. 2, Th. 1). Moreover, if $s \in S$, denote by $H_s$ the wall of $C_0$ corresponding to $s$, and denote by $e_s$ the unit vector orthogonal to $H_s$ and on the same side of $H_s$ as $C_0$. If $(m(s, s'))$ denotes the Coxeter matrix of $(W, S)$, Props. 3 and 7 of $§ 3$ show that

and that the $e_s$ form a basis of $E$. The natural representation of $W$ on $E$ can thus be identified with the representation $\sigma$ of no. 3.