A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements

Abstract

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.

On the Hurwitz action in finite Coxeter groups

“…These results are needed later in Section 6. It is known for the Coxeter groups of type A n that all the elements are parabolic Coxeter elements in the sense of [BDSW14]. For types B n and I 2 (m), the sets of parabolic Coxeter elements and parabolic quasi-Coxeter elements coincide as it is shown in Section 6.…”

“…We now give the main result of [BDSW14] Theorem 2.8. Let (W, T ) be a dual Coxeter system of finite rank n and let c = s 1 · · · s m be a parabolic Coxeter element in W .…”

“…We now define (parabolic) Coxeter elements and introduce (parabolic) quasi-Coxeter elements. The second item of the definition below is borrowed from [BDSW14].…”

“…For types B n and I 2 (m), the sets of parabolic Coxeter elements and parabolic quasi-Coxeter elements coincide as it is shown in Section 6. In particular, Theorem 1.1 is true for A n , B n and I 2 (m) as a consequence of [BDSW14]. For the other types it is in general false that parabolic quasiCoxeter elements coincide with parabolic Coxeter elements.…”

“…For more on the topic we refer to [BDSW14], and the references therein, where a simple proof of the transitivity of the Hurwitz action was shown for (suitably defined) parabolic Coxeter elements in a (not necessarily finite) Coxeter group. See also [IS10].…”