Proposition 8

Assume that the group $W$ is essential, irreducible and infinite. Then:

(i) No point of $E$ is invariant under $W$. (ii) We have $\operatorname{Card} \mathscr{M} = \dim T + 1$, and there exist real numbers $c_H > 0$ such that $\sum_{H \in \mathscr{M}} c_H e_H = 0$. If the real numbers $c'_H$ are such that $\sum_{H \in \mathscr{M}} c'_H e_H = 0$, there exists a real number $\xi$ such that $c'_H = \xi c_H$ for all $H$ in $\mathscr{M}$. (iii) The chamber $C$ is an open simplex.

Assertion (i) follows from Prop. 4. On the other hand, since $W$ is essential, the vectors $(e_H)_{H \in \mathscr{M}}$ generate $T$. We have $(e_H|e_{H'}) \leq 0$ for $H, H' \in \mathscr{M}$ and $H \neq H'$ (Prop. 3) and, since $W$ is irreducible, there does not exist a partition of $\mathscr{M}$ into two disjoint subsets $\mathscr{M}'$ and $\mathscr{M}''$ such that $H' \in \mathscr{M}'$ and $H'' \in \mathscr{M}''$ imply that $(e_{H'}|e_{H''}) = 0$. We can thus apply Lemma 5 of no. 5, and case 1) of that lemma is excluded; indeed, the $e_H$ are not linearly independent, since $W$ has no fixed point. Assertion (ii) follows.

We now prove (iii). Number the walls of $C$ as $H_0, H_1, \ldots, H_d$ and put $t_m = e_{H_m}$. By (ii), the vectors $t_1, \ldots, t_d$ form a basis of $T$, so the hyperplanes $H_1, \ldots, H_d$ have a point $a_0$ in common, and there exists a basis $(t'_1, \ldots, t'_d)$ of $T$ such that $(t_m|t'_n) = \delta_{mn}$; moreover, again by (ii) there exist real numbers $c_1 > 0, \ldots, c_d > 0$ such that

Since the vector $t_0$ is orthogonal to the hyperplane $H_0$, there exists a real number $c$ such that $H_0$ is the set of points $x = t + a_0$ of $E$ with $(t|t_0) = -c$.

Every point of $E$ can be written uniquely in the form $x = t + a_0$ with $t = \xi_1 t'_1 + \cdots + \xi_d t'_d$ and $\xi_1, \ldots, \xi_d$ real. The point $x$ belongs to $C$ if and only if it is on the same side of $H_m$ as $t_m$ for $0 \leq m \leq d$; this translates into the inequalities $(t|t_1) > 0, \ldots, (t|t_d) > 0$, and $(t|t_0) > -c$, or equivalently by $\xi_1 > 0, \ldots, \xi_d > 0, c_1 \xi_1 + \cdots + c_d \xi_d < c$. Since $C$ is non-empty, $c > 0$. Put $a_m = a_0 + \frac{c}{c_m} t'_m$ for $1 \leq m \leq d$; then the chamber $C$ consists of the points of $E$ of the form $a_0 + \sum_{m=1}^d \lambda_m (a_m - a_0)$ with $\lambda_1 > 0, \ldots, \lambda_d > 0$ and $\lambda_1 + \cdots + \lambda_d < 1$, so $C$ is the open simplex with vertices $a_0, \ldots, a_d$. Q.E.D.