Positive-root criterion for Coxeter length
= Positive-root criterion for Coxeter length
In the geometric representation of a Coxeter system,
$$
\ell(ws_i)>\ell(w)\iff w(e_i)\text{ is a positive root},
$$
and
$$
\ell(s_iw)>\ell(w)\iff w^{-1}(e_i)\text{ is a positive root}.
$$
Replacing “positive” by “negative” reverses either length inequality.