Root-sign and inversion-length facts

Finite subsystems. For finite $J\subseteq S$, let $W_J^{\ast }$ be the group presented by the restricted Coxeter matrix, and let $V_J$ be the span of its simple roots. The natural map $W_J^{\ast }\to W$ is injective: an equality of two finite $J$-words in the presentation of $W$ has a derivation using finitely many defining relators and letters, all contained in some finite $K\supseteq J$; [F3] then gives the equality already in $W_J^{\ast }$. Similarly, the intrinsic length of an element of $W_J^{\ast }$ equals its length in $W$: any shorter ambient word and a finite derivation of its equality lie in some finite $K$, contradicting [F3]. The action of $W_J^{\ast }$ on $V_J$ is its finite-rank geometric representation, because the defining reflection formulas restrict to those for $J$.

Root signs and inversion counts. Each root $u{\alpha }_i$ belongs to a finite subsystem containing the letters of a word for $u$ and $s_i$; [F1] therefore gives it exactly one sign in the full basis. Fix $w$, choose finite $J$ containing a word for it, and write $n=\mathrm{\ell }(w)$. By [F1] and 1.1 the subsystem on $J$ supplies $n$ distinct positive roots made negative by $w$. Conversely, any finite collection of such full-system roots, together with words representing them and $w$, is contained in a finite subsystem $K$. Their signs in that subsystem agree with their signs in the full basis, and [F1] and 1.1 show that at most $n$ of them occur. Hence the full inversion set has exactly $n$ elements. Reversing words in the involutory generators gives $\mathrm{\ell }(w^{-1})=\mathrm{\ell }(w)$.

Faithful dual reflection action. If $w\mathrm{\ne }1$, choose finite $J$ containing a word for $w$. By 1.1 its image in $W_J^{\ast }$ is nontrivial, so [F2] shows that $w$ acts nontrivially on $V_J$, and hence on $V$. The algebraic dual separates vectors: a nonzero finite linear combination of basis vectors is detected by a coordinate functional. Thus a nonidentity invertible operator on $V$ has nonidentity contragredient action on $V^{\ast }$. For each generator, the dual formula is $(s_{i}f)(v)=f(v)-2B(v,{\alpha }_i)f({\alpha }_i)$; it fixes pointwise the hyperplane $f({\alpha }_i)=0$ and negates the nonzero functional $B(-,{\alpha }_i)$, so it is a linear reflection. These generator actions faithfully realize $W$.

Descent. If $w^{-1}{\alpha }_i$ is negative, take a finite subsystem containing $s_i$ and a word for $w$. By [F1] its intrinsic lengths satisfy $\mathrm{\ell }(w^{-1}{s}_i)<\mathrm{\ell }(w^{-1})$; 1.1 transfers this inequality to $W$. Reversal of words gives $\mathrm{\ell }(w^{-1}{s}_i)=\mathrm{\ell }(s_{i}w)$ and $\mathrm{\ell }(w^{-1})=\mathrm{\ell }(w)$. Together with 2.1 and 2.2, this proves every assertion.