A note on Weyl groups and root lattices

Abstract

We follow the dual approach to Coxeter systems and show for Weyl groups that a set of reflections generates the group if and only if the related sets of roots and coroots generate the root and the coroot lattices, respectively. Previously, we have proven if (W, S) is a Coxeter system of finite rank n with set of reflections T and if $t_1, \ldots t_n \in T$ are reflections in W that generate W, then $P:= \langle t_1, \ldots t_{n-1}\rangle$ is a parabolic subgroup of (W, S) of rank $n-1$ (Baumeister et al. in J Group Theory 20:103–131, 2017, Theorem 1.5). Here we show if (W, S) is crystallographic as well, then all the reflections $t \in T$ such that $\langle P, t\rangle = W$ form a single orbit under conjugation by P.