A Note on the Hurwitz Action on Reflection Factorizations of Coxeter Elements in Complex Reflection Groups

We show that the Hurwitz action is "as transitive as possible" on reflection factorizations of Coxeter elements in the well-generated complex reflection groups $G(d,1,n)$ (the group of $d$-colored permutations) and $G(d,d,n)$.

On the Hurwitz action in finite Coxeter groups

We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element. We call an element of the Coxeter group parabolic quasi-Coxeter element if it has a factorization into a product of reflections that generate a parabolic subgroup. We give an unusual definition of a parabolic subgroup that we show to be equivalent to the classical one for finite Coxeter groups.

Hurwitz Equivalence in Tuples of Dihedral Groups, Dicyclic Groups, and Semidihedral Groups

Let $D_{2N}$ be the dihedral group of order $2N$, ${\it Dic}_{4M}$ the dicyclic group of order $4M$, $SD_{2^m}$ the semidihedral group of order $2^m$, and $M_{2^m}$ the group of order $2^m$ with presentation $$M_{2^m} = \langle \alpha, \beta \mid \alpha^{2^{m-1}} = \beta^2 = 1,\ \beta\alpha\beta^{-1} = \alpha^{2^{m-2}+1} \rangle.$$ We classify the orbits in $D_{2N}^n$, ${\it Dic}_{4M}^n$, $SD_{2^m}^n$, and $M_{2^m}^n$ under the Hurwitz action.

ISOTROPY SUBGROUP OF HURWITZ ACTION OF THE 3-BRAID GROUP ON THE BRAID SYSTEMS

The Hurwitz action of the n-braid group Bn on the n-fold direct product Bmn of the m-braid group Bm is studied. We show that the isotropy subgroup of the Hurwitz group action of B3 at the triple of the standard generators of B4 has index 16, by explicitly describing a complete system of coset representatives. An application to the braided surfaces in 4-space is also given.

Hurwitz Equivalence in Tuples of Generalized Quaternion Groups and Dihedral Groups

Let $Q_{2^m}$ be the generalized quaternion group of order $2^m$ and $D_N$ the dihedral group of order $2N$. We classify the orbits in $Q_{2^m}^n$ and $D_{p^m}^n$ ($p$ prime) under the Hurwitz action.