Related papers: Hurwitz equivalence of reflection factorizations i…
We show that in the complex reflection group $G_6$, reflection factorizations of a Coxeter element that have the same length and multiset of conjugacy classes are in the same Hurwitz orbit. This confirms one case of a conjecture of Lewis…
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…
We extend a result of Lewis and Reiner from finite Coxeter groups to all Coxeter groups by showing that two reflection factorizations of a Coxeter element lie in the same Hurwitz orbit if and only if they share the same multiset of…
We prove that the Hurwitz action on reflection factorizations of Coxeter elements is transitive up to certain natural constraints in the complex reflection groups G4 and G5. This affirms a more general conjecture by Lewis and Reiner in…
For a positively presented group G, we provide a criterion for two tuples of positive group generators of G to be Hurwitz equivalent or Hurwitz-conjugation equivalent. We also present an algorithmic approach to solve the Hurwitz equivalence…
We provide a variety of cases in which two factorizations have Hurwitz orbits of the same size. We begin with prototypical results about factorizations of length two, and show that cycling elements or flipping and inverting elements in any…
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 classical Hurwitz numbers count the fixed-length transitive transposition factorizations of a permutation, with a remarkable product formula for the case of minimum length (genus $0$). We study the analogue of these numbers for…
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 $G(d, d, n)$.
We consider the Hurwitz action on quasipositive factorizations of a 3-braid. In a previous paper, for any given 3-braid we described a certain finite set which contains at least one representative of each orbit. Here we give an algorithm to…
We investigate the so-called dual Matsumoto property or Hurwitz action in finite, affine and arbitrary Coxeter groups. In particular, we want to investigate how to reduce reflection factorizations and how two reflection factorizations of…
We enumerate Hurwitz orbits of shortest reflection factorizations of an arbitrary element in the infinite family $G(m, p, n)$ of complex reflection groups. As a consequence, we characterize the elements for which the action is transitive…
We give uniform formulas for the number of full reflection factorizations of a parabolic quasi-Coxeter element in a Weyl group or complex reflection group, generalizing the formula for the genus-0 Hurwitz numbers. This paper is the…
A necessary condition for uniqueness of factorizations of elements of a finite group $G$ with factors belonging to a union of some conjugacy classes of $G$ is given. This condition is sufficient if the number of factors belonging to each…
A Hurwitz generating triple for a group $G$ is an ordered triple of elements $(x,y,z) \in G^3$ where $x^2=y^3=z^7=xyz=1$ and $\langle x,y,z \rangle = G$. For the finite quasisimple exceptional groups of types $F_4$, $E_6$, $^2E_6$, $E_7$…
In this paper we prove certain Hurwitz equivalence properties of $B_n$. In particular we prove that for $n=3$ every two Artin's factorizations of $\Delta _3 ^2$ of the form $H_{i_1} ... H_{i_6}, \quad F_{j_1} ... F_{j_6}$ (with $i_k, j_k…
We define parabolic quasi-Coxeter elements in well generated complex reflection groups. We characterize them in multiple natural ways, and we study two combinatorial objects associated with them: the collections $\operatorname{Red}_W(g)$ of…
If $G$ is a finite primitive complex reflection group, all reflection subgroups of $G$ and their inclusions are determined up to conjugacy. As a consequence, it is shown that if the rank of $G$ is $n$ and if $G$ can be generated by $n$…
We prove that the exceptional group $E_6(2)$ is not a Hurwitz group. In the course of proving this, we complete the classification up to conjugacy of all Hurwitz subgroups of $E_6(2)$, in particular, those isomorphic to $L_2(8)$ and…
In this paper we prove certain Hurwitz equivalence properties in $B_n$. Our main result is that every two Artin's factorizations of $\Delta_n ^2$ of the form $H_{i_1} ... H_{i_{n(n-1)}}, \quad F_{j_1} ... F_{j_{n(n-1)}}$ (with $i_k, j_k \in…