It is about an exercise in Humphrey's Reflection groups and Coxeter groups exercise 1 section 5.6. Let (W,S) be a Coxeter system. It is assumed throughout the chapter that S is finite. Let $\sigma : W \rightarrow GL(V) $ be the geometric representation of W. The exercise is
If W is infinite then the length function takes arbitrarily large values and so the root system $\Phi$ is infinite. I was able to prove this result.
Ans: Since the generating set S is finite there are only finitely many words of a particular length and so W is infinite will occur only when the length function takes arbitrarily large values. Since $\Phi=\{w(\alpha_s): w \in W, s \in S\}$ it follows that $\Phi$ is infinite.
My doubt is from this how it follows that $-1 \in GL(V)$ does not lie in $\sigma(W)$. Please help me to figure out this.