Let $A$ be a generalized Cartan matrix on the index set $I$. Define the Weyl group of $A$ as the Coxeter group on the basis $I$ with $m(i,j)=2,3,4,6,\infty$ according to whether $A_{ij} A_{ji}$ is $0,1,2,3$ or bigger. It acts on ${\mathbb Z}[I]$ by $s_j(i) := i - A_{ij} j$.
Question: Is this action faithful?
If $A$ is symmetrizable then the above representation is just a ${\mathbb Z}$-form of the geometric representation of $W$, defined for any Coxeter group. As the latter is always faithful, so is our representation of $W$ on ${\mathbb Z}[I]$.
What happens if $A$ is not symmetrizable?
The motivation for the question comes from Kac-Moody algebras: If $({\mathfrak h},{\mathfrak h}^{\ast},\{\alpha_i\},\{h_j\})$ is a minimal realization of $A$, then the above action of $W$ on ${\mathbb Z}[I]$ occurs as the restriction of the action of $W$ on ${\mathfrak h}^{\ast}$ to the lattice ${\mathbb Z}[\{\alpha_i\}_{i\in I}]\subseteq{\mathfrak h}^{\ast}$ generated by the roots $\alpha_i$. I would like to know if this action is faithful. Again, if $A$ is symmetrizable, then one can introduce a non-degenerate bilinear form on ${\mathfrak h}^{\ast}$ such that the $s_i$ act as orthogonal reflections associated to the (anisotropic) roots $\alpha_i$, and one can show that for any $w\in W\setminus\{e\}$ with reduced expression $s_{i_n}\cdots s_{i_1}$ we must have $w(\alpha_{i_1})<0$. In particular, $w$ does not act trivially on ${\mathbb Z}[\{\alpha_i\}]$ in this case. However, I didn't manage prove the exchange lemma involved here without the description of the $s_i$ as orthogonal reflections. Specifically, how do I know that if $w(\alpha_i)=\alpha_j$, then $s_j = w s_i w^{-1}$ (on ${\mathfrak h}^{\ast}$)?
Thank you!
