The canonical reflection homomorphism, roots, reflections, and the positive cone

Let $S$, $m$, $V$, $B$ be as in The real Coxeter form, its radical, reflections, and form-preserving maps, let $W$ be the presented Coxeter group of Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups with its universal property, and put $r_s:=r_{e_s}$ for $s\in S$ (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order).

(1) Write $\mathrm{G}\mathrm{L}(V)$ for the group of invertible linear maps $V\to V$, the unit group of the monoid of linear endomorphisms under composition (Invertible linear maps, linear isomorphisms, and inverse linear maps, The space $\mathcal{L}(V,W)$ of linear maps with pointwise addition and scalar multiplication, Identity maps and composites of linear maps are linear, The invertible elements of a monoid form a group under the restricted operation). The canonical reflection homomorphism is the group homomorphism $\rho :W\to \mathrm{G}\mathrm{L}(V)$ (Monoid homomorphism and group homomorphism) that satisfies $\rho (s)=r_s$ for every $s\in S$; its existence and uniqueness are established by Descent of the reflection representation, unit root norms, and conjugation of reflections ↗, which is this definition's recorded justifier. For $w\in W$ and $v\in V$ write $\rho (w)v$ for the image.

(2) The root system of the pair is $\mathrm{\Phi }:=\{\rho (w)e_s:w\in W,s\in S\}\subset V$; its elements are the roots. The set of reflections of $W$ is $T:=\{ws{w}^{-1}:w\in W,s\in S\}\subset W$; that $\rho (ws{w}^{-1})$ is the reflection $r_{\rho (w)e_s}$ and that every root is $B$-non-isotropic are proved in Descent of the reflection representation, unit root norms, and conjugation of reflections ↗.

(3) The positive cone is $V_{+}:=\{{\sum }_{s\in S}{\lambda }_s{e}_s:{\lambda }_s\in \mathbb{R},{\lambda }_s\ge 0\}$ and the negative cone is $-V_{+}$ (Linear combination of a finite list, and the span $\mathrm{span}(S)$ as the smallest linear subspace containing $S$).

No positivity of $\rho (w)e_s$ for fixed $w$, no faithfulness or injectivity of $\rho$, no discreteness and no nondegeneracy of $B$ is asserted by this definition.