-\Phi^+ = \Phi - \Phi^+$ is the set of [*negative pseudo-roots*]{}; see Definitions \[def\_pseudo\_root\_system\] and \[def\_positive\_pseudo\_roots\]. Then $\mathcal{H}=(H_\alpha)_{\alpha\in\Phi^+}$ is an essential hyperplane arrangement on $V$. The set of chambers ${\mathscr{T}}$ of this arrangement is in canonical bijection with $W$: the unit element $1\in W$ corresponds to the chamber $B=\bigcap_{\alpha\in\Phi^+}H_\alpha^+=\bigcap_{s\in S}H_{e_s}^+$, and an arbitrary element $w$ of $W$ corresponds to the chamber $w(B)$.

More generally, a *parabolic subgroup* of $W$ is a subgroup $W_I$ generated by a subset $I$ of $S$, and the left cosets of parabolic subgroups of $W$ are called *standard cosets*. The *Coxeter complex* $\Sigma(W)$ of $W$ is the set of standard cosets of $W$ ordered by reverse inclusion. It is a simplicial complex, and we have an isomorphism of posets from $\Sigma(W)$ to the face poset ${\mathscr{L}}$ of ${\mathcal{H}}$ sending a standard coset $wW_I$ to the cone $\{x\in V : \forall s\in I,\ (x,w(e_s))=0\text{ and }
\forall s\in S-I,\ (x,w(e_s))>0\}$. The fact that this is an isomorphism is proved in [@Bourbaki Chapitre V § 4  6 pp. 96–97], since the representation of $W$ on $V^\vee$ is isomorphic to its canonical representation on $V$ by Theorem \[thm\_W\_finite\]. The fact that $\Sigma(W)$ is a simplicial complex then follows from 