Remark

Suppose that (W, S) is a Coxeter system. There exists a homomorphism $\varepsilon$ from W to the group {1, -1} characterized by $\varepsilon(s) = -1$ for all $s \in S$. We call $\varepsilon(w)$ the signature of w; it is equal to $(-1)^{l(w)}$. The formula $\varepsilon(ww') = \varepsilon(w)\varepsilon(w')$ thus translates into $l(ww') \equiv l(w) + l(w') \mod 2$.

¹ This means that (W, S) has the following universal property: given a group G and a map f from S to G such that $(f(s)f(s'))^{m(s,s')} = 1$ for (s, s') in I, there exists a homomorphism g from W to G extending f. This homomorphism is unique because S generates W. An equivalent form of this definition is the following. Let $\overline{W}$ be a group, f a homomorphism from $\overline{W}$ to W and h a map from S to $\overline{W}$ such that $f(h(s)) = s$ and $(h(s)h(s'))^{m(s,s')} = 1$ for (s, s') in S and such that the h(s) (for $s \in S$) generate $\overline{W}$. Then f is injective (and hence is an isomorphism from $\overline{W}$ to W).