Descent of the reflection representation, unit root norms, and conjugation of reflections

For every $s\in S$ the map $r_s=r_{e_s}$ is linear, $B$-preserving and satisfies $r_s^2={\mathrm{i}\mathrm{d}}_V$: these are the identities of [F1] for $a=e_s$, whose hypothesis $B(e_s,e_s)\mathrm{\ne }0$ holds by [F2]. Since $r_s^2={\mathrm{i}\mathrm{d}}_V$, the map $r_s$ is invertible with $r_s^{-1}=r_s$, so $r_s\in \mathrm{G}\mathrm{L}(V)$.

For all distinct $s,t\in S$ with $m(s,t)<\mathrm{\infty }$ one has $(r_s{r}_t{)}^{m(s,t)}={\mathrm{i}\mathrm{d}}_V$, directly from the exact order statement of [F1] applied to the plane spanned by $e_s$ and $e_t$: the product $r_s{r}_t$ acts there with exact order $m(s,t)$, so its $m(s,t)$-th power is the identity on $V$.

Conjugation formula. Let $g\in \mathrm{G}\mathrm{L}(V)$ be $B$-preserving and let $a\in V$ with $B(a,a)\mathrm{\ne }0$. Then $B(ga,ga)=B(a,a)\mathrm{\ne }0$, and for every $v\in V$ the $B$-preservation of $g$ gives $B(g^{-1}v,a)=B(g(g^{-1}v),ga)=B(v,ga)$. Substituting into the definition of $r_{ga}$ and writing $v=g(g^{-1}v)$, $r_{ga}(v)=g(g^{-1}v)-\frac{2B(g^{-1}v,a)}{B(a,a)}ga=g(g^{-1}v-\frac{2B(g^{-1}v,a)}{B(a,a)}a)=g{r}_a{g}^{-1}(v),$ so $g{r}_a{g}^{-1}=r_{ga}$.

Descent. The assignment $s\mapsto r_s$ sends the relator $s^2$ to $r_s^2={\mathrm{i}\mathrm{d}}_V$ by 1.1 and the relator $(st{)}^{m(s,t)}$ to $(r_s{r}_t{)}^{m(s,t)}={\mathrm{i}\mathrm{d}}_V$ by 1.2; these are exactly the relators of the presented Coxeter group $W$ of Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, and the images $r_s$ lie in the group $\mathrm{G}\mathrm{L}(V)$. So the universal property of that presentation gives a unique group homomorphism $\rho :W\to \mathrm{G}\mathrm{L}(V)$ with $\rho (s)=r_s$ for every $s\in S$.

$\rho$ preserves the form, and roots have unit norm. Let $H:=\{w\in W:B(\rho (w)u,\rho (w)w^{\mathrm{\prime }})=B(u,w^{\mathrm{\prime }})\text{ for all }u,w^{\mathrm{\prime }}\in V\}$. Every $s\in S$ lies in $H$ by 1.1, since $\rho (s)=r_s$. If $g,h\in H$ then $B(\rho (gh)u,\rho (gh)w^{\mathrm{\prime }})=B(\rho (g)\rho (h)u,\rho (g)\rho (h)w^{\mathrm{\prime }})=B(\rho (h)u,\rho (h)w^{\mathrm{\prime }})=B(u,w^{\mathrm{\prime }})$, using the homomorphism property of 2.1 and the $B$-preservation of $\rho (g)$ and then of $\rho (h)$, so $gh\in H$; and if $g\in H$, then $B(\rho (g^{-1})u,\rho (g^{-1})w^{\mathrm{\prime }})=B(\rho (g)\rho (g^{-1})u,\rho (g)\rho (g^{-1})w^{\mathrm{\prime }})=B(u,w^{\mathrm{\prime }})$, so $g^{-1}\in H$. Thus $H$ is a subgroup containing $S$, and since every element of $W$ is the value of a finite word in $S$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), so that the images of the elements of $S$ generate $W$, [F3] gives $H=W$. In particular $B(\rho (w)e_s,\rho (w)e_s)=B(e_s,e_s)=1$ for all $w\in W$ and $s\in S$ by [F2], so every root $\alpha \in \mathrm{\Phi }$ has $B(\alpha ,\alpha )=1\mathrm{\ne }0$ and the reflection $r_{\alpha }$ is defined.

Conjugation of reflections by $\rho$. Let $w\in W$ and $s\in S$. By 3.1 the map $\rho (w)$ is $B$-preserving, and $r_s=\rho (s)$ by 2.1; applying the conjugation formula 1.3 with $g=\rho (w)$ and $a=e_s$ therefore gives $\rho (w)r_s\rho (w{)}^{-1}=r_{\rho (w)e_s}$. Since $\rho$ is a homomorphism, $\rho (w{)}^{-1}=\rho (w^{-1})$ and $\rho (w)\rho (s)\rho (w{)}^{-1}=\rho (ws{w}^{-1})$, so $\rho (ws{w}^{-1})=r_{\rho (w)e_s}\mathrm{.}$ Hence for every $t=ws{w}^{-1}\in T$ the element $\rho (t)$ is the reflection in the root $\rho (w)e_s\in \mathrm{\Phi }$, as asserted.