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A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements. (English) Zbl 1343.20041
Summary: 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.

MSC:
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20F36 Braid groups; Artin groups
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