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\title[The Hurwitz action in complex reflection groups]{The Hurwitz action in complex reflection groups}

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\author[Joel Brewster Lewis and Jiayuan Wang]{Joel Brewster Lewis\thanks{\href{mailto:jblewis@gwu.edu}{jblewis@gwu.edu}  Partially supported by an ORAU Powe Award, a Simons Collaboration Grant (634530), and the GWU UFF.}\addressmark{1} \and Jiayuan Wang\thanks{\href{mailto:j453w588@gwu.edu}{j453w588@gwu.edu}}\addressmark{1}}

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\address{\addressmark{1}Department of Mathematics, George Washington University, Washington, DC, USA 20052 }

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\abstract{We study the reflection factorizations of an arbitrary element in the complex reflection groups $G(m,p,n)$ under the Hurwitz action. Using combinatorial and graph theoretical techniques, we present an if-and-only-if statement of when two factorizations are Hurwitz equivalent. Then we prove a formula that counts the number of Hurwitz orbits of an arbitrary element.}

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\keywords{reflection factorizations, complex reflection groups, Hurwitz action}

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\section{Introduction}

Given a group $G$ with a generating set $T$ closed under conjugacy, the \emph{Hurwitz action} is a natural action of the $n$-strand braid group $\mathcal{B}_n$ on $T^n$.  This action was introduced in the late 19th century by Hurwitz in the case that $G = \Symm_n$ is the symmetric group and $T$ is the set of transpositions in $G$, as part of his study~\cite{Hurwitz} of covering surfaces of the sphere with given monodromy.  More recently, the Hurwitz action played an important role in Bessis' proof~\cite{Bessis} of the $K(\pi, 1)$ property for complements of complex reflection arrangements, this time in the case that $G$ is a complex reflection group and $T$ is its subset of reflections; in this setting, the transitivity of the Hurwitz action on certain tuples of reflections is a key part of the \emph{Coxeter--Catalan combinatorics} associated to the reflection group $G$ (see, e.g., \cite{cataland}).

In the recent paper \cite{BGRW}, the authors characterized those elements $w$ in a real reflection group $W$ (i.e., a finite Coxeter group) for which the Hurwitz action is transitive when restricted to minimum-length reflection factorizations of $w$.  In particular, they showed that these elements are precisely the \emph{quasi-Coxeter elements} for certain special subgroups, that is, the elements whose %minimum-length 
factorizations all generate the subgroup in question.

In the present paper, we extend the work of \cite{BGRW} to the complex realm.  Our main result (Theorem~\ref{main theorem}) gives an explicit formula for the number of Hurwitz orbits of minimum-length factorizations of an arbitrary element $g$ in the complex reflection group $G = G(m, p, n)$.  As a consequence, we characterize the elements for which the action is transitive (Cor.~\ref{cor transitive}).  As a second consequence, we give an elegant ``inverse'' result: we show that two minimum-length reflection factorizations of an element $g \in G$ belong to the same Hurwitz orbit if and only if they generate the same subgroup of $G$ (Cor.~\ref{cor:invariants}).

Our approach is combinatorial, taking advantage of the natural graph-theoretic structure of reflections in $G(m, p, n)$, and building on earlier work of Kluitmann \cite{Kluitmann} and Ben-Itzhak--Teicher \cite{BIT} in the symmetric group.

% The remainder of this abstract is organized as follows: in Section~\ref{sec:background}, we establish notational conventions and provide the necessary background on complex reflection groups and the Hurwitz action needed to understand the present work.  In Section~\ref{sec:main}, we introduce our main result and its corollaries, and provide a detailed sketch of their proofs.
%%%%
%%%% this paper is short and almost self-explanatory, so maybe we don't need preceding paragraph
%%%%

\acknowledgements

We are grateful to Alejandro Morales, Vic Reiner, and especially to Theo Douvropoulos, for their comments and suggestions at various stages of this work.  We also thank two anonymous referees for many constructive comments.

\section{Background and notation}
\label{sec:background}
\subsection{Conventions}
\label{sec:conventions}
Throughout this abstract, $m$, $p$, and $n$ will represent positive integers with $p \mid m$. Since $p$ divides $m$, the cyclic group $\ZZ/m\ZZ$, whose elements are equivalence classes of integers modulo $m$, has a unique subgroup $p\ZZ / m\ZZ \cong \ZZ / (m/p)\ZZ$ of order $m/p$: it consists of those equivalence classes whose elements are divisible by $p$.  We may write $k \equiv 0 \pmod{p}$ to indicate that an element $k$ in $\ZZ/m\ZZ$ belongs to this subgroup, and $k \equiv 0 \pmod{m}$ to indicate that $k$ is the identity in $\ZZ/m\ZZ$.  As in the previous sentence, we do not distinguish notationally between the integer $k$ and its equivalence class modulo $m$; this should cause no confusion in practice.

Given a collection of $k_1, k_2, \ldots, k_n$ of elements of $\ZZ/m\ZZ$, there is some minimal subgroup $k\ZZ/m\ZZ$ that contains all of them.  If we take $k_1, \ldots, k_n$ to be any representatives of their equivalence classes, then the smallest positive representative $k$ of $k\ZZ$ is $k = \gcd(m, k_1, \ldots, k_n)$.  All greatest common divisors that appear in this paper (particularly, as in Definition~\ref{def r star}) will be meant in this sense.

%Again throughout the paper, $\omega = \exp(2\pi i / m)$ will denote a fixed primitive $m$th root of unity.  The collection $\{ \omega^i \colon i \in \ZZ\}$ of all $m$th roots of unity is in natural bijection with $\ZZ/m\ZZ$, and the $(m/p)$th roots of unity correspond exactly to the elements $k \equiv 0 \pmod{p}$.



\subsection{Complex reflection groups}

%\jbl{Something here is necessary! At a minimum, we have to establish notations.}

Given a finite-dimensional complex vector space $V$, a  \emph{complex reflection} is a linear transformation $t: V \rightarrow V$ whose fixed space $\operatorname{ker}(t-1)$ is a hyperplane (i.e., has codimension $1$), and a finite subgroup $G$ of $GL(V)$ is called a \emph{complex reflection group} if $G$ is generated by its subset $R$ of complex reflections.
%\footnote{In the literature, it is sometimes required \emph{a priori} that reflections have finite order or that complex reflection groups live in the unitary group on $V$.  The former condition is implied by the fact that $G$ is finite, and the second may be recovered by a standard averaging trick \cite[Lem.~1.3]{LehrerTaylor}.} 
Complex reflection groups were classified by Shephard and Todd~\cite{ST54}: every complex reflection group is a direct product of irreducibles, and every irreducible is either of the form
\[
G(m, p, n) = \left\{\begin{array}{l} n \times n \text{ monomial matrices whose nonzero entries are}\\m\text{th} \text{ roots of unity with product a } \frac{m}{p}\text{th} \text{ root of unity}\end{array} \right\}
\]
for positive integers $m$, $p$, $n$ with $p \mid m$ or is one of $34$ exceptional examples.  

For every $m$, $p$, $n$, there is a natural projection map 
$
\pi: G(m, p, n) \twoheadrightarrow G(1, 1, n) = \Symm_n
$
defined as follows: for $g \in G(m, p, n)$, the matrix $\pi(g)$ is the result of replacing every root of unity in the matrix of $g$ with $1$.  The resulting permutation is called the \emph{underlying permutation} of $g$.  It will be convenient to use the following, more compact, notation for elements of $G(m, p, n)$: one writes $g = [u; (a_1, \ldots, a_n)]$ where $u = \pi(g)$ and $a_j \in \ZZ/m\ZZ$ is the exponent of $\exp(2\pi i/m)$ in the nonzero entry of the $j$th column of $g$. This notation reveals that $G(m, 1, n)$ has the structure of a \emph{wreath product} $G(m, 1, n) \cong (\ZZ/m\ZZ) \wr \Symm_n $, with multiplication given by
$
[u; (a_1, \ldots, a_n)] \cdot [v; (b_1, \ldots, b_n)] = \left[uv; (a_{v(1)} + b_1, \ldots, a_{v(n)} + b_n)\right]
$.

Given an element $g = [u; (a_1, \ldots, a_n)]$ of $G(m, p, n)$, the value $a_j$ is called the \emph{weight} of $j$.  Further, for any subset $S \subset \{1, \ldots, n\}$, we define $\sum_{j \in S} a_j$ to be the \emph{weight} of $S$.  This notion will be particularly relevant when $S$ is the set of entries of a cycle of $g$, or when $S = \{1, \ldots, n\}$ and $a_1 + \ldots + a_n$ is the weight of $g$.  In this language, an element of $G(m,1,n)$ belongs to $G(m,p,n)$ if and only if its weight is a multiple of $p$.

When $p < m$, the group $G(m, p, n)$ contains two types of complex reflections: for $a \in \{0, 1, \ldots, m - 1\}$, the \emph{transposition-like reflections}
\begin{equation}
\label{eq:transposition-like reflection}
[(i\, j); a] \defeq [(i\, j); (0, \ldots, 0, a, 0, \ldots, 0, -a, 0, \ldots, 0)]
\end{equation}
(with $i$ having weight $a$ and $j$ having weight $-a$)
of weight $0$ and order $2$, and for $b \in \{p, 2p, \ldots, m - p\}$ the \emph{diagonal reflections} 
$
[\id$; $(0, \ldots, 0, b, 0, \ldots, 0)]
$
of weight $b$ and various orders; the group $G(m, m, n)$ contains only the transposition-like reflections.  %In the particular case of transposition-like reflections, we abbreviate the notation even further, writing $[(i\, j); a]$ for the reflection in Equation~\ref{eq:transposition-like reflection}.


\subsection{Shi's formula for reflection length}
Fix a complex reflection group $G$ with reflections $R$.  Since $G$ is a reflection group, every element $g$ of $G$ can be written as a product of reflections.  If $f = (t_1, \ldots, t_\ell)$ is a tuple of reflections such that $g = t_1 \cdots t_\ell$, we say that $f$ is a \emph{(reflection) factorization} of $g$.  We say that a reflection factorization $F$ of $g$ is \emph{shortest}, \emph{minimum}, or \emph{of minimum length} if there is no reflection factorization of $g$ using fewer reflections, and we define the \emph{reflection length} $\lR(g)$ of $g$ to be the length
\[
\lR(g)=\operatorname{min}\{\ell: g=t_{1}t_{1}\cdots t_{\ell} \text{ for some } t_{i}\in R\}
\]
of the shortest factorizations. Throughout the paper, we use the word ``factorizations'' as a shorthand for ``reflection factorizations of minimum length''.

%When $G$ is a finite real reflection group, the reflection length of an element can be interpreted geometrically: one has $\lR(g) = \codim_V(\fix(g))$, where $\fix(g) = \{v \in V \colon g(v) = v\}$ is the \emph{fixed space} of $g$ \cite{Carter}. The same is true in $G(m, 1, n)$, but \emph{not} in any of the other irreducible complex reflection groups \cite{Shi, FG}.  

In \cite{Shi}, Shi gave a combinatorial formula for reflection length in the group $G(m, p, n)$ that we now describe. For an element $g\in G(m, p, n)$, let $\cyc(g)$ be the number of cycles in $\pi(g)$.  A \emph{cycle partition} $\Pi$ of $g$ is a set partition of the set $\{ C_{1}, C_{2}, \ldots, C_{\cyc(g)} \}$ of cycles of $g$ such that for every part in $\Pi$, the corresponding cycle weights sum to $0\pmod{p}$.  (Such partitions always exist because the weight of $g$ is $0 \pmod{p}$.) For example, the element $g= [\id; (2, 2, 2, 2)] \in G(4,4,4)$ has four cycle partitions:
\[
\Par(g) = \Big\{\pt{(1)(2) \mid (3)(4)}, \pt{(1)(3) \mid (2)(4)}, \pt{(1)(4) \mid (2)(3)}, \pt{(1)(2)(3)(4)} \Big\}.
\]
Observe that the set of cycle partitions depends on the choice of the group containing $g$: if we view this element $g$ as an element of $G(4, 2, 4)$ then $\Par(g)$ consists of all fifteen set partitions of the four cycles.

Given a partition $\Pi$ of an element $g \in G(m, p, n)$, let $|\Pi|$ denote the number of parts of $\Pi$ and let $v_m(\Pi)$ denote the number of parts of $\Pi$ of weight $0 \pmod{m}$ (not just $0 \pmod{p}$). Shi defines the \emph{value} $v(\Pi)$ of a cycle partition $\Pi$ to be
$
v(\Pi) = {|\Pi|} + v_m(\Pi)
$.
A partition is \emph{maximum} if its value is the largest among the values of all possible cycle partitions of $g$ (relative to the given $m, p$), and we denote by $\Parmax(g)$ the set of maximum cycle partitions of $g$.  For example, with $g = [\id; (2, 2, 2, 2)] \in G(4, 4, 4)$ as above, the three partitions into two parts have value $4$, while the partition in one part has value $2$, and so 
$
\Parmax(g)=\Big\{\pt{(1)(2) \mid (3)(4)}, \pt{(1)(3) \mid (2)(4)}, \pt{(1)(4) \mid (2)(3)}\Big\}.
$ 


\begin{thm}[{Shi~\cite{Shi}}]\label{shi}
Given an element $g \in G(m, p, n)$ with reflections $R$, we have 
\[
\lR(g)= n + \cyc(g)-v(\Pi)
\] 
where $\cyc(g)$ is the number of cycles of $g$ and $\Pi\in \Parmax(g)$ is a maximum cycle partition of $g$.
\end{thm}
Using Shi's formula, we see that the element $g = [\id; (2, 2, 2, 2)] \in G(4, 4, 4)$ has reflection length $\lR(g)=4+4-(2+2)=4$.  In general, computing $\Parmax(g)$ is computationally challenging: there is a standard reduction from the {\sf SubsetSum} problem, so it is NP-hard.

\subsection{The Hurwitz action}

The \emph{Hurwitz move} $\sigma_i$ on a tuple $(t_1, \ldots, t_n)$ of elements of a group $G$ is the operation $\sigma_i( \ldots, t_i, t_{i+1}, \ldots)=(\ldots, t_{i+1}, t_{i+1}^{-1}t_{i}t_{i+1}, \ldots)$.  It is easy to check that $\sigma_1, \ldots, \sigma_{n - 1}$ satisfy the braid relations, and consequently give rise to the \emph{Hurwitz action} of the braid group on $n$ strands.  This action was first studied by Hurwitz~\cite{Hurwitz} in the context of the symmetric group $\Symm_{n}$.  The question of its orbit structure on reflection factorizations in Coxeter groups and complex reflection groups has been an area of substantial interest in the past two decades, especially in cases when the action is transitive, i.e., when there is a single Hurwitz orbit (see, e.g., \cite{BDSW, BGRW, BIT, Bessis, IS10, LR, WY19}). The following result of Kluitmann~\cite{Kluitmann} on the symmetric group will be of particular use for us. 

\begin{thm}[{Kluitmann~\cite{Kluitmann}}]\label{Kluitmann}
The set of all transposition factorizations $(t_{1}, t_{2}, \ldots, t_{k})$ of an element $w\in \Symm_n$ such that $\langle t_{1}, t_{2},\ldots, t_{k}\rangle = \Symm_{n}$ forms a single orbit under the Hurwitz action.
\end{thm}

% Bessis~\cite{Bessis} proved that, for a well generated complex reflection group, the shorstest factorizations of a Coxeter element form a single Hurwitz orbit. The corresponding result for arbitrary Coxeter groups was proved by Igusa and Schiffler~\cite{IS10}, with a short and self-contained proof in~\cite{BDSW}. In finite Coxeter groups, i.e., finite real reflection groups, the elements whose shortest factorizations form a single Hurwitz orbit are completely classified in~\cite{BGRW}. Moreover, Lewis and Reiner~\cite{LR} showed that in a finite Coxeter group, the reflection factorizations (of any length) of a Coxeter element are Hurwitz equivalent if and only if they share the same multiset of conjugacy classes. This result was extended to arbitrary Coxeter group by Wegener and Yahiatene~\cite{WY19}. 

% In~\cite{LR}, Lewis and Reiner showed that, in a finite real reflection group, every factorization is Hurwitz equivalent to a factorization of a particular form. This is important when we later define a standard form.
% 
% \begin{cor}[{Lewis--Reiner~\cite{LR}}]\label{cor}
% Given a finite real reflection group $G$ and set $R$ of generators for $G$, if $\lR(g) = \ell$, then every factorization of $g\in G$ into $k$ reflections lies in the Hurwitz orbit of some $\mathbf{t}=(t_{1}, \ldots, t_{k})$ such that 
% \[ \begin{array}{c}
% t_{1}=t_{2},\\
% t_{3}=t_{4},\\
% \vdots\\
% t_{k-\ell-1} = t_{m-\ell}.
% \end{array}\]
% and $(t_{k-\ell+1}, \ldots, t_{k})$ is a shortest reflection factorization of $g$.
% \end{cor}

\section{The Main Theorem}
\label{sec:main}
To state our main results, we need one further piece of terminology.  (The reader may wish to recall our convention for greatest common divisors from Section~\ref{sec:conventions}.)

\begin{definition}\label{def r star}
Let $g\in G(m, p, n)$, let $\Pi$ be a cycle partition of $g$, and let $B$ be a block in $\Pi$. Suppose that the weights of the cycles in $B$ are $(k_{1}, k_{2}, \ldots, k_{|B|})$. Define \[r(B)=\gcd(m, k_{1},k_{2}, \ldots, k_{|B| - 1}, k_{|B|}).\]
\end{definition}

\begin{thm}\label{main theorem}
Given an element $g \in G(m, p, n)$, the number of Hurwitz orbits of its shortest factorizations is given by 
\[
\sum_{\Pi\in\Parmax(g)} \prod_{B\in \Pi} (r(B))^{|B|-1}.
\]
\end{thm}


For example, the element $g = [\id; (2, 2, 2, 2)] \in G(4, 4, 4)$ discussed above has $12$ Hurwitz orbits of shortest reflection factorizations, including $4 = 2^{2 - 1} \times 2^{2 - 1}$ from each of its three maximum cycle partitions.


\begin{cor}\label{cor transitive}
Let $g\in G(m,p,n)$. The shortest factorizations of $g$ form a single orbit under the Hurwitz action if and only if ${|\Parmax(g)|} = 1$ and either ${|B|} = 1$ or $r(B)=1$ for every block $B$ in $\Pi\in \Parmax(g)$. In particular, if $g$ has a single cycle, then $g$ is Hurwitz transitive. Further, 
when $p=1$, $g$ is Hurwitz transitive if and only if no two cycles of $g$ have nonzero weights that sum to $0\pmod{m}$.
\end{cor}

\begin{cor}
\label{cor:invariants}
Two shortest factorizations $f_1$ and $f_2$ of an element $g$ in $G = G(m, p, n)$ are Hurwitz-equivalent if and only if they generate the same subgroup of $G$.
\end{cor}

The remainder of this extended abstract is devoted to the proof of these results. We begin by defining a standard form for factorizations, and show that every factorization is Hurwitz-equivalent to a factorization in standard form.%; next, we present a sufficient and necessary condition for two standard form factorizations to be Hurwitz equivalent, with an explicit way to construct a Hurwitz path; and lastly, we conclude with the proof for Theorem~\ref{main theorem}.

\subsection{Standard forms}
In this section, we define a standard form for factorizations in $G(m, p, n)$ and show that every shortest factorization is Hurwitz-equivalent to a factorization in standard form.  It is easiest to describe the standard forms in terms of a graph object associated to a factorization.

\begin{definition}\label{def graph}
Given a factorization $f = (t_{1}, t_{2}, \ldots,  t_{\ell})$ of an element $g = t_1 \cdots t_\ell \in G(m,p,n)$, the \emph{factorization graph} of $f$ is the graph $\Gamma_f = (V,E)$ on (labeled) vertex set $V=\{1, \dots, n\}$ with (labeled) edges $E = \{e_1, \ldots, e_\ell\}$ defined as follows: if $t_k$ has underlying permutation $(i\, j)$ then $e_k$ joins vertices $i$ and $j$, while if $t_k$ is diagonal with nonzero weight in position $i$ then $e_k$ is a loop at vertex $i$.
\end{definition}

Fix an element $g$ in $G(m, p, n)$ and a factorization $f = (t_1, \ldots, t_\ell)$ of $g$.  The connected components of the graph $\Gamma_f$ form a set partition of $\{1, \ldots, n\}$.  In fact, this set partition corresponds to a cycle partition of $g$%: the action of each transposition $\pi(t_i)$ respects the set partition, so $\pi(g)$ must as well, and consequently all elements of a single cycle belong to the same part
.  We denote by $\Pi_f$ the cycle partition induced in this way by the factorization $f$.  If $B$ is a part in $\Pi_f$, we denote by $|B|$ the number of cycles in $B$ and by $\Gamma_f|_B$ the connected component of $\Gamma_f$ that corresponds to $B$.

We are now prepared for the key definition of this section.

\begin{definition}\label{def std form}
Given a shortest factorization $f$ of an element $g\in G(m,p,n)$, with factorization graph $\Gamma_f$ and cycle partition $\Pi_f$.  Let $B_1$, $B_2$, \ldots, $B_{|\Pi_f|}$ be the parts of $\Pi_f$, and for each part $B_i \in \Pi_f$, say that $C_{i, 1}$, \ldots, $C_{i, |B_i|}$ are the cycles of $g$ contained in $B_i$, and that for each $j$, $v_{i,j}$ is the smallest element of $\{1, \ldots, n \}$ permuted by $C_{i,j}$.  Without loss of generality, assume that the indices of the $B_i$ and $C_{i, j}$ have been chosen so that $v_{i,1} < v_{i,2} <\ldots < v_{i,|B_i|}$ and $v_{1, 1} < v_{2, 1} < \ldots < v_{|\Pi_f|, 1}$.  We say that $f$ is in \emph{standard form} if the following conditions are met:
(1) if $i' < i$ then every edge in $\Gamma_f|_{B_{i'}}$ has smaller label than all edges in $\Gamma_f|_{B_i}$; 
(2) for each $i$, the first $2|B_i|-2$ factors $t_1, \ldots, t_{2|B_i|-2}$ in $f$ that correspond to edges in $\Gamma_f|_{B_i}$ have underlying transpositions
% \begin{align*}
% \pi(t_{1}) & = \pi(t_{2}) = (v_{i,1} \, v_{i,2}),\\
% \pi(t_{3}) & = \pi(t_{4}) = (v_{i,2} \, v_{i,3}),\\
% & \vdots\\
% \pi(t_{2|B_i|-3}) & = \pi(t_{2|B_i|-2}) = (v_{i,|B_i| - 1} \, v_{i,|B_i|});
% \end{align*}
$\pi(t_{1}) = \pi(t_{2}) = (v_{i,1} \, v_{i,2})$,
$\pi(t_{3}) = \pi(t_{4}) = (v_{i,2} \, v_{i,3})$, 
\ldots,
$\pi(t_{2|B_i|-3}) = \pi(t_{2|B_i|-2}) = (v_{i,|B_i| - 1} v_{i,|B_i|})$;
and
(3) for each $i$, if there is a loop in $\Gamma_f|_{B_i}$ then it is the edge labeled $2|B_i| - 1$ and acts on vertex $v_{i, |B_i|}$.
\end{definition}
For example, the factorization $[(13); 0] \cdot [(13); 1] \cdot [(35); 1]\cdot [(35); 3]\cdot [(12); 0]\cdot [(46); 0]\cdot [(46); 4]\cdot [\id; (0,0,0,0,0,3,0,0,0)]\cdot [(49); 1]\cdot [(67); 2]\cdot [(89); 3]$ of the element \[[(12)(3)(498)(5)(67); (0, 1, 1, 1, 7, 2, 6, 6, 6)] \in G(9, 3, 9)\] is in standard form, with factorization graph
\begin{center}
\vspace{-.1in}
 \begin{tikzpicture}[-,>=stealth',auto,node distance=1.2cm,
  thick, main node/.style={circle,draw, font=\sffamily\bfseries\small}]

  \node[main node] (1) {$1$};
  \node[main node] (2) [left of=1] {$2$};
  \node[main node] (3) [right of=1] {$3$};
  \node[main node] (5) [right of=3] {$5$};
%  \node[main node] (5) [right of=4] {$5$};
  \node[main node] (6) [right of=5] {$8$};
  \node[main node] (7) [right of=6] {$9$};
  \node[main node] (8) [right of=7] {$4$};
  \node[main node] (9) [right of=8] {$6$};
  \node[main node] (10) [right of=9] {$7$};
  
  \path[every node/.style={font=\sffamily\small}]
    (1) edge node [above] {5} (2)
    (1) edge[bend left] node [above] {1} (3)
    (1) edge[bend right] node [below] {2} (3)
    (3) edge[bend left] node [above] {3} (5)
    (3) edge[bend right] node [below] {4} (5)
%    (4) edge node [above] {\tiny 5} (5)
    (8) edge[bend left] node [above] {6} (9)
    (8) edge[bend right] node [below] { 7} (9)
    (6) edge node [above] { 11} (7)
    (7) edge node [above] { 9} (8)
    (9) edge node [above] { 10} (10) 
    (9) edge [out=500, in=60, looseness=5] node [above] { 8} (9);
 \end{tikzpicture}
 \;
 \raisebox{.1in}{.}
\end{center}

\begin{lem}\label{lemma any in p}
For any $g \in G(m, p, n)$ and any minimum-length factorization $f$ of $g$, there is a standard form factorization of $g$ that is Hurwitz-equivalent to $f$.
\end{lem}
\begin{proof}[Proof sketch]
The factorization $f$ induces a cycle partition $\Pi_f$.  It is straightforward to write down a standard form factorization $f'$ of $g$ such that $\Pi_{f'} = \Pi_f$.  One can check using Shi's formula (Theorem~\ref{shi}) that every connected component in $\Gamma_f$ must have the same number of edges as the corresponding component in $\Gamma_{f'}$.  By Kluitmann's theorem (Theorem~\ref{Kluitmann}), the projected factorizations $\pi(f)$ and $\pi(f')$ are Hurwitz-equivalent, i.e., there is some braid $\beta$ such that $\beta(\pi(f)) = \pi(f')$.  Then $\beta(f)$ is the desired %standard form 
factorization% of $g$
.
\end{proof}


\subsection{Hurwitz paths between standard form factorizations}
\label{sec:building paths}

In this section, we construct explicit sequences of Hurwitz moves connecting certain standard form factorizations to each other. The main result of the section is Lemma~\ref{lemma h path}.

\begin{definition}\label{def doubled path}
For $n \geq 1$, we say that a factorization in $G(m, p, n)$ is a \emph{doubled path} if it is of the form
\[
\Big([(12);a_{1}], \; [(12);b_{1}], \; [(23); a_{2}], \; [(23); b_{2}], \ldots, [(n - 1 \; n); a_{n - 1}], \; [(n - 1 \; n); b_{n - 1}] \Big)
\] 
or
\[
\Big([(12);a_{1}], \; [(12);b_{1}], \ldots, [(n - 1 \; n); a_{n - 1}], \; [(n - 1 \; n); b_{n - 1}], \; [\id; (0,\ldots,0,d)] \Big).
\]
\end{definition}

By multiplying out, it's easy to see that every doubled path in $G(m, p, n)$ is a factorization of a diagonal element $g = [\id; (k_1, \ldots, k_n)] \in G(m,p, n)$ of weight~$0$ (if no diagonal element is present) or~$d$, and that for each $i$ one has $b_i = a_i + k_1 + k_2 + \ldots + k_i$.  Consequently, given the product $g$, the $a_i$ determine the entire factorization.  This suggests the following definition.

\begin{definition}\label{def pair weight and constant}
Let 
$
f= \Big([(12);a_{1}], [(12); b_{1}], [(23); a_{2}], [(23); b_{2}], \ldots \Big)
$
be a doubled path, factoring an element $g = [\id; (k_1, \ldots, k_n)] \in G(m,p, n)$. Define the \emph{pair weight} of the $i$-th pair of factors (i.e., with underlying transposition $(i\, i+1)$) to be $a_{i}$ and the corresponding \emph{pair difference} to be the difference $d_i \defeq b_i - a_i = k_{1}+\cdots+k_{i}$.
\end{definition}

The next result gives a sufficient condition for two doubled paths to belong to the same Hurwitz orbit.


% \begin{prop}\label{prop h path in m}
% Suppose 
% $
% f_{1}=
% \Big(
% [(12); a_{1}], [(12); b_1], \dots 
% \Big)
% $
% and
% $
% f_{2}=\Big(
% [(12); a'_{1}], [(12); b'_1], \dots%,  [(n-1\, n); a'_{n-1}], [(n-1\, n); b'_{n-1}]
% \Big)
% $
% are two doubled paths factoring the same element $g$ of weight $d$ in $G(m, p, n)$.  If there exists an $(n-1)\times n$ $\ZZ$-matrix $M = (m_{ij})$ such that $m_{ij} = m_{ji}$ for $i, j \leq n - 1$ and
% \begin{equation}
% \label{modular matrix relation}
% \begin{pmatrix} a_{1} \\ \vdots \\ a_{n-1}\end{pmatrix}
% +
% M\cdot \begin{pmatrix} d_1 \\  \vdots\\ 
% d_{n - 1} \\ d
% \end{pmatrix} \equiv
% \begin{pmatrix}a'_{1}\\  \vdots\\ a'_{n-1}\end{pmatrix} \pmod{m},
% \end{equation}
% then there exists a Hurwitz path from $f_{1}$ to $f_{2}$.
% \end{prop}

\begin{prop}\label{prop h path in m}
Suppose 
$
f_{1}=
\Big(
[(12); a_{1}], [(12); b_1], \dots 
\Big)
$
and
$
f_{2}=\Big(
[(12); a'_{1}], [(12); b'_1], \dots%,  [(n-1\, n); a'_{n-1}], [(n-1\, n); b'_{n-1}]
\Big)
$
are two doubled paths factoring the same element $g$ of weight $d$ in $G(m, p, n)$.  If there exists an $n \times (n - 1)$ $\ZZ$-matrix $M = (m_{ij})$ such that $m_{ij} = m_{ji}$ for $i, j \leq n - 1$ and
\begin{equation}
\label{modular matrix relation}
\begin{pmatrix} a_{1} & \cdots & a_{n-1}\end{pmatrix}
+
\begin{pmatrix} d_1 &  \cdots & 
d_{n - 1} & d
\end{pmatrix} \cdot M \equiv
\begin{pmatrix}a'_{1} &  \cdots & a'_{n-1}\end{pmatrix} \pmod{m}
\end{equation}
(with congruence taken coordinate-wise), then there exists a Hurwitz path from $f_{1}$ to $f_{2}$.
\end{prop}

\begin{figure}
\begin{center}
\raisebox{.4cm}{\rotatebox{90}{\begin{tikzpicture}
\pic[braid/.cd, number of strands=6, 
line width=2pt , width = .75cm, height=.65cm
,gap =0.15,
, name prefix=braid,]{braid={a_4 a_5-a_3^{-1} a_4^{-1}-a_2^{-1} a_1^{-1}-a_3^{-1} a_2^{-1} a_2^{-1} a_1^{-1}-a_3^{-1} a_2^{-1}-a_4 a_3-a_5^{-1} a_4^{-1}}};
\end{tikzpicture}}}%
\quad
\rotatebox{90}{\begin{tikzpicture}
\pic[braid/.cd, number of strands=7, 
line width=2pt 
,gap =0.15, width = .75cm, height=.65cm,
, name prefix=braid,]{braid={a_6 a_5^{-1} a_4 a_3^{-1} a_2 a_1 a_1 a_2 a_3 a_4^{-1} a_5 a_6^{-1}}};
\end{tikzpicture}}
\end{center}
\caption{The braids corresponding to the operations $\tau_{1, 3}$ and $\gamma_{1}$ in the case $n=4$.}
\label{fig:braids}
\end{figure}

\begin{proof}[Proof sketch]
We define two families of operations that can be applied to doubled paths: 
For any pair of indices $i, j$ with $1 \leq i < j \leq n-1$, 
define $\tau_{i,\,j}$ to be the following sequence of Hurwitz moves: 
\begin{multline*}
\tau_{i,\,j} \defeq \sigma_{2j-2} \sigma_{2j-1} \sigma_{2j-3}^{-1} \sigma_{2j-2}^{-1}\cdots  
\sigma_{2i+2} \sigma_{2i+3} \sigma_{2i+1}^{-1} \sigma_{2i+2}^{-1} \; \circ \;
 \sigma_{2i}^{-1}  \sigma_{2i+1}^{-1}  \sigma_{2i-1}^{-1}  \sigma_{2i}^{-1} \; \circ  \\ \sigma_{2i}^{-1}  \sigma_{2i+1}^{-1}  \sigma_{2i-1}^{-1}  \sigma_{2i}^{-1} \; \circ 
 \sigma_{2i+2} \sigma_{2i+1}  \sigma_{2i+3}^{-1}  \sigma_{2i+2}^{-1} \cdots  \sigma_{2j-2} \sigma_{2j-3} \sigma_{2j-1}^{-1} \sigma_{2j-2}^{-1},
\end{multline*}
and for any index $i \in \{1, \ldots, n - 1\}$, define $\gamma_{i}$ to be the following sequence of Hurwitz moves:
\begin{multline*}
\gamma_{i} \defeq \sigma_{2n-2} \sigma_{2n-3}^{-1} \sigma_{2n-4} \sigma_{2n-5}^{-1}  \cdots \sigma_{2i+2} \sigma_{2i+1}^{-1}  \; \circ \; \sigma_{2i} \sigma_{2i-1} \sigma_{2i-1} \sigma_{2i} \; \circ \\
\sigma_{2i+1} \sigma_{2i+2}^{-1}  \cdots  \sigma_{2n-5} \sigma_{2n-4}^{-1}  \sigma_{2n-3} \sigma_{2n-2}^{-1}.
\end{multline*}
The associated braids are illustrated in Figure~\ref{fig:braids}.

% \begin{figure}
% \begin{center}
% \rotatebox{90}{\begin{tikzpicture}
% \pic[braid/.cd, number of strands=7, 
% line width=2pt 
% ,gap =0.1, width = .75cm, %height=.85cm,
% , name prefix=braid,]{braid={a_6 a_5^{-1} a_4 a_3^{-1} a_2 a_1 a_1 a_2 a_3 a_4^{-1} a_5 a_6^{-1}}};
% \end{tikzpicture}}
% \end{center}
% \caption{The braid corresponding to the operation $\gamma_{1}$.} 
% \label{fig:braid 3}
% \end{figure}

By an inductive computation, one shows that if $f$ is a doubled path in $G(m, p, n)$ with pair weights $(a_1, \ldots, a_{n - 1})$ and pair differences $(d_1, \ldots, d_{n - 1})$, then $\tau_{i,j}(f)$ is a doubled path with the same pair weights and pair differences as $f$, except that the $i$th pair weight of $\tau_{i, j}(f)$ is $a_i + d_j$ and the $j$th pair weight of $\tau_{i, j}(f)$ is $a_j + d_i$.  
%
Similarly, if $f$ includes a diagonal factor of weight $d$, then $\gamma_{i}(f)$ is again a doubled path, and $\gamma_{i}(f)$ has the same pair weights and pair differences as $f$, except that the $i$-th pair weight of $\gamma_{i}(f)$ is $a_{i}+d$.
%
Finally, it is easy to check that (under the same hypotheses) $\sigma_{2i-1}(f)$ is a doubled path, with the same pair differences and pair weights as $f$,  except that the $i$-th pair weight of $\sigma_{2i - 1}(f)$ is $a_i + d_i$.

%%%%%%%%%%%%%%%
%%%% transposed
%%%%%%%%%%%%%%%

%Consequently, if \eqref{modular matrix relation} holds, applying the operations $\sigma_{2i - 1}^{m_{ii}}$ for $1 \leq i \leq n$, $\tau_{i, j}^{m_{ij}}$ for $1 \leq i < j \leq n - 1$, and $\gamma_i^{m_{in}}$ for $1 \leq i \leq n$ in any order to $f_1$ produces $f_2$, as needed.

Consequently, if \eqref{modular matrix relation} holds, applying the operations $\sigma_{2i - 1}^{m_{ii}}$ for $1 \leq i \leq n-1$, $\tau_{i, j}^{m_{ij}}$ for $1 \leq i < j \leq n - 1$, and $\gamma_i^{m_{nj}}$ for $1 \leq j \leq n-1$, to $f_1$ in any order produces $f_2$, as needed.
\end{proof}

%%%%%%%%%%%%%%%%%%%
%%%% I think this example would need to be expanded (writing down the factorizations explicitly, showing the effects fo the Hurwitz moves explicitly) to be really helpful for a reader, and we don't have the space to do that.
%%%%%%%%%%%%%%%%%%%

% \begin{ex}\label{ex M}
% Consider $g=[\id; (15, 15, 20, 10)]\in G(60,60,4)$. Then $g$ has a unique maximum cycle partition $\Pi=\pt{(1)(2)(3)(4)}$ and $\lR(g)=6$. Consider two doubled-paths $f_1$ with pair weights $\{0,0,0\}$ and $f_2$ with pair weights $\{10,15,25\}$. Then matrix $M=\begin{bmatrix} 2&1&5\\1&0&0\\5&0&5\end{bmatrix}$ satisfies the desired equivalence relations described in Proposition~\ref{modular matrix relation}. One can construct a Hurwitz path by reading off the entries in $M$. The diagonal entries $m_{ii}$ means applying $\sigma_{2i-1}$ to the first doubled-path $m_{ii}$ times. And $m_{i,j}$ means applying $\tau_{i,j}$ to the first doubled-path $m_{i,j}$ times. Then we have the Hurwitz path from $f_1$ to $f_2$: $f_2= (\sigma_5^{5}\circ \sigma_1^{2}\circ\tau_{1,2}\circ \tau_{1,3})(f_1)$.
% \end{ex}

By a moderately complicated argument in elementary number theory, it is possible to rephrase the previous proposition as follows.

\begin{cor}\label{prop m}
Follow the notation of Proposition~\ref{prop h path in m} and define $r(B)$ as in Definition~\ref{def r star}.  If $a_{i}\equiv a'_{i} \pmod{r(B)}$ for $i = 1, \ldots, n - 1$, then there exists a matrix $M$ as in Proposition~\ref{prop h path in m}.
\end{cor}
% \begin{hproof}
% If for every $i$, $a_i\equiv a'_i \pmod{r}$, then $a_i -a'_i \equiv 0\pmod{r}$. Recall that $r =\gcd(m,k_1, \ldots, k_n )$, this means that $r = \gcd(m, d_1, \ldots, d_{n-1})$. This implies that in modulo $r$, the difference $a_i - a'_i$ can be expressed as a sum of the $d_i$'s, which is done using Euclidean algorithm. Putting the coefficients for the $d_i$'s into a matrix form, we find an $M$ as desired.
% \end{hproof}

%In Example~\ref{ex M}, $r=5$. Since every pair weight in $f_1$ is equivalent to a pair weight in $f_2$ mod $5$, then we are guaranteed to find such a $M$. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%% odd -- incorporated above
%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
% Now we extend, we extend the discussion that began with Definition~\ref{def odd doubled path} to include the possibility that the factorization includes a diagonal reflection (in graph, a loop).
% \begin{definition}\label{def odd doubled path}
% For $n \geq 1$, we say that a factorization in $G(m, p, n)$ is an \emph{odd doubled path} if it is of the form
% \begin{multline*}
% \Big([(12);a_{1}], [(12);b_{1}], [(23); a_{2}], [(23); b_{2}], \ldots, \\ [(n - 1 \; n); a_{n - 1}], [(n - 1 \; n); b_{n - 1}], [\id; (0,\ldots,0,d)] \Big).
% \end{multline*} 
% The \emph{pair weights} and \emph{pair differences} of an odd doubled path are defined to be $(a_1, \ldots, a_{n - 1})$ and $(d_1, \ldots, d_{n - 1}) := (b_1 - a_1, \ldots, b_{n - 1} - a_{n - 1})$, respectively, as before.
% \end{definition}
% 
% We introduce a new operation to complement the $\sigma_{2i - 1}$ and $\tau_{i, j}$ in the context of odd doubled paths.
% 
% \begin{definition}\label{def operation four}
% Given $i$ where $1\leq i \leq n-1$, define $\gamma_{i}$ to be the following sequence of Hurwitz moves:
% \begin{multline*}
% \gamma_{i}:=\sigma_{2n-2}\circ\sigma_{2n-3}^{-1}\circ\sigma_{2n-4}\circ\sigma_{2n-5}^{-1} \circ\cdots\circ\sigma_{2i+2}\circ\sigma_{2i+1}^{-1}\circ {}\\
% \sigma_{2i}\circ\sigma_{2i-1}\circ\sigma_{2i-1}\circ\sigma_{2i}\circ {}\\
% \sigma_{2i+1}\circ\sigma_{2i+2}^{-1}\circ \cdots \circ\sigma_{2n-5}\circ\sigma_{2n-4}^{-1}\circ \sigma_{2n-3}\circ\sigma_{2n-2}^{-1}.
% \end{multline*}
% \end{definition}
% 
% The associated braid is illustrated in Figure~\ref{fig:braid 3}.
% 
% 
% 
% \begin{prop}\label{prop operation four}
% Let $f$ be an odd doubled path in $G(m, p, n)$ with pair weights $(a_1, \ldots, a_{n - 1})$ and pair differences $(d_1, \ldots, d_{n - 1})$, and with diagonal factor of weight $d$.  Then $\gamma_{i}(f)$ is again a doubled path. Moreover, $\gamma_{i}(f)$ has the same pair weights and pair differences as $f$, except that the $i$-th pair weight of $\gamma_{i}(f)$ is $a_{i}+d$.
% \end{prop}
% 
% Again, these braids do not commute in the braid group, but their actions commute when restricted to doubled paths.
% 
% Using the same number theoretic method, one can show that for two odd doubled-paths, if there is a $(n-1)\times n$ matrix $M$ that satisfies the appropriate system of equivalence relations, then the two doubled-paths are Hurwitz equivalent. Further, such a matrix exists when each respective pair weights are equivalent modulo $r$. 
% 
% We conclude our results in this section with the following lemma.

By definition, every connected component $\Gamma_f|_B$ in the factorization graph of a standard form factorization $f$ includes a doubled path of length $2|B| - 1$ or $2|B| - 2$.  This allows us to restate the previous result in terms of standard form factorizations.

\begin{lem}\label{lemma h path}
Suppose that $f_1$ and $f_2$ are two standard form factorizations of the element $g\in G(m, p, n)$ and that they induce the same cycle partition $\Pi$ of $g$.  Suppose furthermore that for each block $B$ of $\Pi$, the pair weights $(a_1, \ldots, a_{|B| - 1})$ and $(a'_1, \ldots, a'_{|B| - 1})$ in the doubled paths in $f_1|_B$ and $f_2|_B$ satisfy $a_i \equiv a'_i \pmod{r(B)}$ for all $i \in \{1, \ldots, |B| - 1\}$.  Then $f_1$ and $f_2$ are Hurwitz-equivalent.
\end{lem}
\begin{hproof}
We use Kluitmann's theorem (Theorem~\ref{Kluitmann}) to choose a braid $\beta_1$ such that $\Gamma_{\beta_1(f_1)} = \Gamma_{f_2}$ without changing the factors in the doubled paths.  For each connected component, Corollary~\ref{prop m} gives a second braid $\beta_2$ so that the weights of the doubled paths in $\beta_2\beta_1(f_1)|_B$ are the same as those in $f_2|_B$.  The weights of the other factors in the component are determined by $g$.  Applying this to each connected component gives a Hurwitz path from $f_1$ to $f_2$.
\end{hproof}



\subsection{When are two standard form factorizations not equivalent?}

%In this section, we answer the question stated in the title. Using Kluitmann's result~\ref{Kluitmann}, one can pick two standard form factorizations, of a fixed element in $G(m,p,n)$, that share the same underlying permutation. This means that the non-doubled-path portion of the two factorizations are identical. To determine whether these two factorizations are in the same orbit, it suffices to study whether the two doubled-paths are in the same orbit. To do that, we use an invariant of the Hurwitz action, i.e., the group generated by the factors. The main result of this section is Lemma~\ref{lemma subgroup}.

The main result of this section is Lemma~\ref{lemma subgroup}, which is the converse of Lemma~\ref{lemma h path}.  As our main tool, we construct an invariant that distinguishes different Hurwitz orbits.

\begin{definition}
Given a factorization $f$ of an element in $G(m, n, p)$, denote by $G_f$ the subgroup of $G(m, n, p)$ generated by the factors in $f$.
\end{definition}

It is easy to see that $G_f$ is preserved by Hurwitz moves.  Our first result is completely straightforward.

\begin{prop}
If $f$ is a shortest factorization with induced cycle partition $\Pi_f$, we have that $G_f$ is the direct product of the restrictions $G_{f|_B}$ to individual blocks $B$ of $\Pi_f$.
\end{prop}

Thus, in what follows, it suffices to consider factorizations $f$ for which the factorization graph $\Gamma_f$ is connected.  Also, by Lemma~\ref{lemma any in p}, it suffices to consider the case of factorizations in standard form.

\begin{prop}
Let $f$ be a connected standard form factorization of an element $g \in G(m, p, n)$, and let $d$ be the weight of $g$.  Then $G_f \cong G\left(\frac{m}{r}, \frac{\gcd(m, d)}{r}, n\right)$.  More concretely, there is a diagonal element $\delta \in G(m, 1, n)$ such that the conjugation map $\phi_\delta$ defined by $\phi_\delta(g) = \delta g \delta^{-1}$ restricts to an isomorphism $\phi_\delta: G_f \to G\left(\frac{m}{r}, \frac{\gcd(m, d)}{r}, n\right)$.
\end{prop}
\begin{hproof}
First, consider the following set of factors of $f$: for each pair of factors in the doubled path with same underlying transposition, include the first corresponding edge, and also include all edges not in the doubled path.  These edges form a spanning tree of $\Gamma_f$.  By iteratively choosing entries of $\delta$ to fix one edge of the tree at a time, we can choose $\delta$ that simultaneously conjugates each of these edges to a transposition (i.e., to a reflection $[(i \, j); 0]$ of weight $0$).  We claim this is the desired element.

It is straightforward to check that, after conjugation, every weight in every factor in $\delta f \delta^{-1}$ is a multiple of $r$, or equivalently every root of unity appearing in their matrices is an $\frac{m}{r}$-th root of unity.  Moreover, after conjugation, the diagonal factor will still have weight $d$.  It follows immediately that $\phi_\delta(G_f) \subseteq G\left(\frac{m}{r}, \frac{\gcd(m, d)}{r}, n\right)$.

For the reverse inclusion, we can use the maps $\tau_{i, j}$ and $\gamma_i$ from Section~\ref{sec:building paths} to explicitly construct a set of reflections in $\phi_\delta(G_f)$ that generate $G\left(\frac{m}{r}, \frac{\gcd(m, d)}{r}, n\right)$.
\end{hproof}

Since conjugation by a diagonal element fixes diagonal elements, the following consequence is immediate.

\begin{cor}
\label{cor:diagonal}
If $f_1$ and $f_2$ are two connected standard form factorizations of the same element $g$, then the diagonal subgroups of $G_{f_1}$ and $G_{f_2}$ are equal (not just isomorphic), and are equal to the diagonal subgroup of $G\left(\frac{m}{r}, \frac{\gcd(m, d)}{r}, n\right)$.
\end{cor}


% \begin{prop}\label{prop different}
% Still consider $f$ and $f'$. If there exists an $i$ such that $a_i\neq a'_i \pmod{r}$, then $G_f\neq G_f'$.
% \end{prop}
% \begin{hproof}
% Pick a certain $n$-cycle $w\in G_f$ and another $n$-cycle $w'\in G_{f'}$. Then $w^{-1}w'$ is a diagonal element that is not in $G(\frac{m}{r}, \frac{m}{r}, n)$ if $a_i\neq a'_i \pmod{r}$ for some $i$. 
% \end{hproof}


\begin{lem}\label{lemma subgroup}
Suppose that $f_1$ and $f_2$ are two Hurwitz-equivalent standard form factorizations of the element $g\in G(m, p, n)$.  Then $\Pi_{f_{1}}=\Pi_{f_2}$ and for every connected component $B$, the respective pair weights $(a_1, \ldots, a_{|B| - 1})$ and $(a'_1, \ldots, a'_{|B|-1})$ of the doubled paths in $f_{1}|_{B}$ and $f_{2}|_{B}$ satisfy the condition that $a_{i}\equiv a'_{i} \pmod{r(B)}$ for every $i\in [|B|-1]$.
\end{lem}

\begin{hproof}
We prove the contrapositive: suppose there is a block $B$ for which the pair weights in the two factorizations are not equivalent mod $r$.  Then $G_{f_1}$ contains the cycle $c_1 \defeq [(v_1 \, v_2); a_1] \cdots [(v_{|B| - 1} \, v_{|B|}); a_{|B| - 1}]$ (the product of half of the edges in the doubled path) and $G_{f_2}$ contains the cycle $c_2 \defeq [(v_1 \, v_2); a'_1] \cdots [(v_{|B| - 1} \, v_{|B|}); a'_{|B| - 1}]$.  By multiplying out, using the fact that $a_i \not \equiv a'_i \pmod{r}$ for some $i$, one sees that $c_1 c_2^{-1}$ is a diagonal element whose weights are not all multiples of $r$.  Such an element does not belong to $G\left(\frac{m}{r}, 1, n\right)$ and so by Corollary~\ref{cor:diagonal} does not belong to either $G_{f_1}$ or $G_{f_2}$.  Consequently $G_{f_1} \neq G_{f_2}$, and so $f_1$ and $f_2$ are not Hurwitz-equivalent.
\end{hproof}


\subsection{Proof of main results}

\begin{proof}[Proof of Theorem~\ref{main theorem}]
By Lemma~\ref{lemma any in p}, it suffices to study the standard form factorizations of $g$.  Every standard form factorization $f$ may be constructed by first choosing a maximum cycle partition $\Pi$, and then for every part $B\in\Pi$ choosing the pair weights $\{a_i\}_{i=1}^{|B|-1}$ of $f|_{B}$. By Lemmas~\ref{lemma h path} and~\ref{lemma subgroup}, choosing an orbit amounts to independently choosing for each $a_i$ one of the $r(B)$ equivalence classes modulo $r(B)$, and independently repeating this for each part $B$; this gives $\prod_{B \in \Pi} (r(B))^{|B|-1}$ orbits corresponding to the partition $\Pi$.  Summing over all possible maximum cycle partitions $\Pi\in\Parmax$ of $g$ gives the total number of Hurwitz orbits.
\end{proof}

Corollary~\ref{cor transitive} is an immediate consequence of Theorem~\ref{main theorem}.


%\subsection{Reformulation in terms of invariants}
%In this section, we present our second corollary with its proof. Since the Hurwitz action preserves the product of the tuple and the subgroup $H$ generated by its elements, and consequently two shortest factorizations that differ in either of these quantities belong to different Hurwitz orbits, our main result may be recast as saying that these are the only obstructions. We restate Corollary~\ref{cor:invariants} below. 

% \begin{corr}
% [\ref{cor:invariants}]
% Suppose that $f_1$ and $f_2$ are two shortest reflection factorizations of an element $g \in G = G(m, p, n)$. Then $f_1 \myeq f_2$ if and only if $f_1$ and $f_2$ generate the same subgroup of $G$.
% \end{corr}
\begin{proof}[Proof of Corollary~\ref{cor:invariants}]
By Lemma~\ref{lemma any in p}, it suffices to show that the result is true for standard form factorizations.  It follows from the proof of Lemma~\ref{lemma subgroup} that two standard form factorizations of $g$ generate the same subgroup if and only if they induce the same partition $\Pi$ and have the same pair weights modulo $r(B)$ for each block $B \in \Pi$.  By Lemmas~\ref{lemma h path} and~\ref{lemma subgroup}, this is equivalent to the condition that they generate the same subgroup of $G$.
\end{proof}


\begin{remark}
\label{rem:conj classes}
There is another natural invariant of the Hurwitz action, namely, the multiset of conjugacy classes (with respect to $G$, but also (stronger) with respect to the subgroup $H$ that they generate) of the factors.  Thus, a surprise consequence of Corollary~\ref{cor:invariants} is that among the shortest factorizations of an element $g$ in $G(m, p, n)$, those that generate the subgroup $H$ all have the same multiset of $H$-conjugacy classes of the factors.
\end{remark}

\begin{remark}
We are grateful to Theodosius Douvropoulos for the following example, which shows that Remark~\ref{rem:conj classes} does not hold in every complex reflection group.  The exceptional group $G = G_{16}$ is generated by two reflections $a, b$ subject to the relations $a^5 = b^5 = 1$, $aba = bab$.  Both generators have non-unit eigenvalue $\exp(2\pi i/5)$.  The element $g \defeq a^2 b^3$ is not a reflection (e.g., because it has determinant $1$) and so $f_1 = (a^2, b^3)$ is a shortest reflection factorization of $g$ that generates $G$.  It is slightly more work to check that $f_2 = (a^{-1}ba, b^{-2} a^{-1} b^2)$ is another shortest reflection factorization of $g$ that also generates $G$. However, the factorizations $f_1$ and $f_2$ cannot lie in the same Hurwitz orbit because their factors have different conjugacy classes: the reflections in $f_1$ have determinants $\exp(4\pi i/5)$ and $\exp(6\pi i/5)$, while those in $f_2$ have determinants $\exp(2\pi i/5)$ and $\exp(8\pi i/5)$.
\end{remark}


\begin{remark}
In~\cite{Berger11}, Berger showed that the three invariants
\[
\left(\text{product } g, \text{generated subgroup } H, \text{multiset of } H\text{-conjugacy classes}\right)
\]
distinguish Hurwitz orbits of reflection factorizations of arbitrary lengths in the dihedral group $G(m,m,2)$. That is, two tuples of reflections in a dihedral group belong to the same Hurwitz orbit if and only if they have the same product, generate the same subgroup, and have the same orbits under conjugacy by the subgroup.

The corresponding result can be shown in the symmetric group essentially by combining the work of Kluitmann~\cite{Kluitmann} and Ben-Itzhak and Teicher~\cite{BIT}.  The corresponding result has also been established in some of the small exceptional complex reflection groups ($G_4$, $G_5$, $G_6$, and $G_7$) in unpublished work of Minnick--Pirillo--Racile--Wang (private communication).

In \cite[\S5]{JBL} it was conjectured that the corresponding statement is valid in \emph{any} complex reflection group.  Corollary~\ref{cor:invariants} may be viewed as further evidence for this conjecture.
\end{remark}







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