[*left*]{}) [*weak Bruhat order*]{}, where the cover relation is $z {\prec}w$ if there exists $s\in S$ such that $z \cdot s= w$ (respectively $s \cdot z = w$) and $\ell(z) + 1 =\ell(w)$. The strong Bruhat order refines both the left and right weak Bruhat orders.

Let $V=\bigoplus_{s\in S}{{\mathbb{R}}}e_s$, with the symmetric bilinear form $(\cdot,\cdot)$ defined by $$(e_s,e_t)=-\cos\left({\pi}/{m_{s,t}}\right).$$ In particular, $(e_s,e_s) = 1$. The *canonical representation* of $(W,S)$ is the representation of $W$ on $V$ given by $$\begin{aligned}
\label{equation_canonical_representation}
s(v) & = v - 2 \cdot (e_s,v) \cdot e_s,\end{aligned}$$ for every $s\in S$ and every $v\in V$. Note that this formula defines an orthogonal isomorphism of $V$ for the symmetric bilinear form $(\cdot,\cdot)$. We refer the reader to [@Bourbaki Chapitre V, § 4,  8, Théorème 2 p. 98] for the next result.

\[thm\_W\_finite\] Equation  defines a faithful representation of $W$ on $V$, and the form $(\cdot,\cdot)$ is positive definite if and only if $W$ is finite.

From now on, we assume that $W$ is finite, and we write $\Phi=\{w(e_s) : w\in W,\ s\in S\}$ and $\Phi^+=\Phi\cap\sum_{s\in S}{{\mathbb{R}}}_{\geq 0} e_s$. The set $\Phi$ is a [*pseudo-root system*]{}, its subset $\Phi^+$ is the set of [*positive pseudo-roots*]{}, and the set $\Phi^- = 