Circuits and Hurwitz action in finite root systems

Abstract

In a finite real reflection group, two factorizations of a Coxeter element into an arbitrary number of reflections are shown to lie in the same orbit under the Hurwitz action if and only if they use the same multiset of conjugacy classes. The proof makes use of a surprising lemma, derived from a classification of the minimal linear dependences (matroid circuits) in finite root systems: any set of roots forming a minimal linear dependence with positive coefficients has a disconnected graph of pairwise acuteness.

18 pages, one attached figure and six auxiliary data files. v2: minor changes

Cited by in corpus (4)

• Hurwitz Actions on Reflection Factorizations in Complex Reflection Group $G_6$
• Hurwitz equivalence of reflection factorizations in $G_7$
• Hurwitz Transitivity of Longer Reflection Factorizations in G4 and G5
• Extended Weyl groups, Hurwitz transitivity and weighted projective lines I: Generalities and the tubular case