Proposition 21

*Let C (resp. $C'$) be a chamber of R, $\Sigma$ (resp. $\Sigma'$) a subset of $B(C)$ (resp. $B(C')$), Q (resp. $Q'$) the set of linear combinations of elements of $\Sigma$ (resp. $\Sigma'$) with negative integer coefficients, and $P = Q \cup R_+(C)$ (resp. $P' = Q' \cup R_+(C')$). If there exists an element of the Weyl group transforming P to $P'$, then there exists an element of the Weyl group transforming C to $C'$ and $\Sigma$ to $\Sigma'$.

We reduce immediately to the case $P = P'$. Let $V_1$ be the vector subspace of V generated by $P \cap (-P)$. Then $\Sigma$ and $\Sigma'$ are bases of the root system $R_1 = P \cap (-P)$ in $V_1$ (Cor. 6 of Prop. 20). Hence there exists $g_1 \in W(R_1)$ such that $g_1(\Sigma) = \Sigma'$. It is clear that $g_1$ is induced by an element $g$ of $W(R)$ that is a product of the symmetries $s_\sigma$ with $\sigma \in \Sigma$. Let $\gamma = \sum_{\beta \in B(C)} c_\beta \beta$ be an element of $P - R_1$. Then $c_\beta > 0$ for at least one $\beta \in B(C) - \Sigma$. Moreover, if $\sigma \in \Sigma$, then $s_\sigma(\gamma) - \gamma \in V_1$, so $s_\sigma(\gamma)$ has at least one coordinate $> 0$ with respect to $B(C)$ (no. 1, formula (5)), hence $s_\sigma(\gamma) \in R_+(C)$ and finally $s_\sigma(\gamma) \in P - R_1$. It follows that $P - R_1$ is stable under the $s_\sigma, \sigma \in \Sigma$, and hence under $g$, so $g(P) = P$. We are thus reduced to proving the proposition when $P = P'$ and $\Sigma = \Sigma'$. In this case, $Q = Q'$, so $R_+(C) = P - Q = P - Q' = R_+(C')$, and hence $C = C'$ (Cor. 1 of Prop. 20).