Abstract. We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element, that is, if and only if it has a reduced decomposition into a product of reflections that generate a parabolic subgroup.

Abstract. In this note, we provide a short and self-contained proof that the braid group on n strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank n into products of reflections. We moreover use the same argument to also show that all factorizations of an element in a parabolic subgroup of W also lie in this parabolic subgroup.

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 t1, . . . tn ∈ T are reflections in W that generate W then P := t1, . . . tn−1 is a parabolic subgroup of (W, S) of rank n − 1 [BGRW17, Theorem 1.5]. Here we show if (W, S) is crystallographic as well then all the reflections t ∈ T such that P, t = W form a single orbit under conjugation by P .

We call an element of a Coxeter group a parabolic quasi-Coxeter element if it has a reduced decomposition into a product of reflections that generate a parabolic subgroup. We show that for a parabolic quasi-Coxeter element in an affine Coxeter group the Hurwitz action on its set of reduced decompositions into a product of reflections is transitive.

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