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

摘要

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