In which it is proven that, for each parabolic quasi-Coxeter element in a finite real reflection group, the orbits of the Hurwitz action on its reflection factorizations are distinguished by the two obvious invariants

Abstract

We prove that two reflection factorizations of a parabolic quasi-Coxeter element in a finite Coxeter group belong to the same Hurwitz orbit if and only if they generate the same subgroup and have the same multiset of conjugacy classes. As a lemma, we classify the finite Coxeter groups for which every reflection generating set that is minimal under inclusion is also of minimum size.

10 pages, 1 figure, comments very much welcome! v2: minor corrections; simplification of one argument in Section 2 (and consequent renumbering)

References in corpus (5)

• On cycle decompositions in Coxeter groups
• The Hurwitz action in complex reflection groups
• Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements
• Hurwitz numbers for reflection groups I: Generatingfunctionology
• Hurwitz equivalence of reflection factorizations in $G_7$