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\title{Cyclic sieving for reduced reflection factorizations of the Coxeter element}

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\author{Theo
Douvropoulos\thanks{\href{mailto:douvr001@irif.fr}{douvr001@irif.fr}. Supported by the European Research Council, grant ERC-2016- STG 716083 ``CombiTop''.}\addressmark{1}}

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\address{\addressmark{1}IRIF, UMR CNRS 8243, Universit\'e Paris Diderot, Paris 7, France}

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\abstract{In a seminal work, Bessis gave a geometric interpretation of the noncrossing lattice $NC(W)$, associated to a well-generated complex reflection group $W$. We use this framework to prove, in a unified way, various instances of the cyclic sieving phenomenon on the set of reduced reflection factorizations of the Coxeter element. These include in particular the conjectures of N.~Williams on the actions $\mathfrak{Pro}$ and $\mathfrak{Twist}$.} 


%Dans un travail pionnier, Bessis a donné une interprétation géométrique du treillis NC ( W ) des partitions non-croisées associé à un groupe de réflexion complexe W irréductible et bien engendré. Nous utilisons ce cadre pour démontrer, de manière unifiée, diverses instances du phénomène du crible cyclique sur l’ensemble des factorisations réduites de l’élément Coxeter en réflexions. Ces résultats incluent notamment les conjectures de N. Williams sur les actions Pro et Twist.

%% put your French abstract here, or comment this out if you don't have one
\resume{Dans un travail pionnier, Bessis a donn\'e une interpr\'etation g\'eom\'etrique du treillis $NC(W)$ des partitions non-crois\'ees, associ\'e \`a un groupe de r\'eflexion complexe $W$ irr\'eductible et bien engendr\'e. Nous utilisons ce cadre pour d\'emontrer, de mani\`ere unifi\'ee, diverses instances du ph\'enom\`ene du crible cyclique sur l'ensemble des factorisations r\'eduites de l'\'el\'ement de Coxeter en r\'eflexions. Ces r\'esultats incluent notamment les conjectures de N.~Williams sur les actions $\mathfrak{Pro}$ et $\mathfrak{Twist}$.}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Cyclic sieving phenomenon, Coxeter element, reflection
factorizations, Hurwitz action, Lyashko--Looijenga morphism}

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\section{Introduction}

Already in 1892, Hurwitz knew that there exist $n^{n-2}$ minimal length factorizations of the long cycle $(12\cdots n)\in S_n$ into transpositions. This same number counts maximal chains of the noncrossing lattice $NC(n)$, labeled trees on $n$ vertices, topologically distinct branched coverings of the sphere by itself, and\footnote{Although these last two objects are naturally overcounted by a factor of $n$.} classes of degree $n$ polynomials with prescribed critical values.

It often happens that the most exciting phenomena first observed in the symmetric group $S_n$, have suitable analogs for other reflection groups $W$. Our main object of study in this note is the set $\op{Red}_W(c)$ of minimal length reflection factorizations of a Coxeter element $c$ of $W$. It is enumerated by the Hurwitz number $$|\op{Red}_W(c)|=\frac{h^nn!}{|W|},$$ where $h$ is the order of the Coxeter element.
 
Bessis used the noncrossing lattice $NC(W)$ as a combinatorial recipe to build up the universal covering spaces of reflection arrangement complements. In the most difficult part of his work, he had to refine this construction to work with centralizer subgroups of $W$. The insight for that came from a topological understanding of his and Reiner's work \cite{bessis-reiner} on the CSP:
 
 
\subsubsection*{The cyclic sieving phenomenon (CSP)}
In 2004, the cyclic sieving phenomenon was first observed in Minnesota \cite{reiner-stanton-white}. It occurs when a polynomial $X(q)$ carries orbital information about the action of a cyclic group $C$ on a space $X$. More precisely, and if $C$ is generated by an element $c$ of order $n$, we say that the triple $(X,X(q),C)$ {\it exhibits the cyclic sieving phenomenon (CSP)} if for all integers $d$, \begin{equation}\label{Defn: CSP}|X^{c^d}|=X(\zeta^d),\end{equation} where $\zeta=e^{2\pi i/n}$ and $X^{c^d}$ denotes the set of elements of $X$ fixed by $c^d$.

Already in his thesis Drew Armstrong had conjectured cyclic sieving phenomena for the poset $NC^{(k)}(W)$ of generalized noncrossing partitions. Bessis and Reiner however, were the first \cite{bessis-reiner} to prove a CSP for $NC(W)$. Their work was later used by Krattenthaler and M\"uller to prove Armstrong's conjectures, albeit with an approach relying on case by case calculations. 

Below we recall the cyclic actions $\Phi_{\op{cyc}}$ and $\Psi_{\op{cyc}}$ that have been considered originally in the context of $NC^{(k)}(W)$, and the action $\Twist$ on factorizations of the Coxeter element:


\begin{defn}[{\cites{bessis-reiner}{nathan-williams-thesis}}]\label{Defn: cyclic actions}
For a well-generated group $W$ (see Section \ref{Section:
Sec2-background}), we consider the following cyclic actions on the set
$\op{Red}_W(c)$ of reduced reflection factorizations of the Coxeter
element $c$ of $W$:
\begin{alignat}{2}
&\Phi_{\op{cyc}}:(t_1,\cdots t_n)\rightarrow \big( ^{(^ct_n)}t_1,^c\!t_n,t_2,\cdots,t_{n-1}\big) &&\, \Psi_{\op{cyc}}:(t_1,\cdots, t_n)\rightarrow (^ct_n,t_1,\cdots,t_{n-1}) \nonumber\\
&\quad\quad\quad\quad\quad\quad\quad \Twist:(t_1,\cdots, t_n)\rightarrow \Big(\  ^{(t_1\cdots t_{n-1})}t_n,&&^{(t_1\cdots t_{n-2})}t_{n-1},\cdots,^{t_1}t_2,t_1\Big), 
\end{alignat}
where $^wt:=wtw^{-1}$ stands for conjugation.
\end{defn}

In fact, Williams \cite{nathan-williams-thesis} was the first to conjecture CSP's for the set $\op{Red}_W(c)$. He considered the action $\mathfrak{Pro}$, which is the inverse of $\Psi_{\op{cyc}}$ above, and the action $\Twist$. We are now ready to state our main theorem:


\begin{thm}\label{Thm: main-thm}
For an irreducible, well-generated, complex reflection group $W$, with
degrees $d_1,\cdots,d_n$ (see Section~\ref{Section: Sec2-background}), Coxeter
element $c$ and Coxeter number $h=d_n$, the triple $$\Big(\
\op{Red}_W(c),\ \prod_{i=1}^n\dfrac{[ih]_q}{[d_i]_q},\  C\ \Big)$$
exhibits the cyclic sieving phenomenon, where $C$ may be generated by any
of the actions $\Phi_{\op{cyc}},\Psi_{\op{cyc}}$, or $\Twist$ given in
Definition~\ref{Defn: cyclic actions} and where $[n]_q:=\dfrac{1-q^n}{1-q}$.
\end{thm}

We prove this theorem (see Section \ref{Section last} after
Lemma~\ref{Lemma-Vic}) by interpreting the three combinatorial cyclic
actions on factorizations, as Galois actions in a topological covering
induced by the Lyashko--Looijenga morphism $LL$ (see Section \ref{Section:
The LL map} and Section \ref{Section: Cyclic actions via labeling maps}).
These, it turns out, correspond to scalar actions on certain symmetric
fibers of the $LL$ map. Finally a geometric Lemma~\ref{Lemma-Vic}
provides a CSP on the latter ones with respect to the Hilbert series of
the special fiber $LL^{-1}({\bf 0})$, which is exactly the polynomial
that appears in Theorem~\ref{Thm: main-thm}.  


\section{Background on reflection groups and their complex geometry}
\label{Section: Sec2-background}

Given a complex vector space $V\cong \CC^n$, we say that a {\it finite} subgroup $W$ of $\op{GL}(V)\cong \op{GL}_n(\CC)$ is a {\it complex reflection group} if it is generated by pseudo-reflections. Those are $\CC$-linear maps $t$ whose fixed spaces $V^t:=\op{ker}(t-{\bf \op{id}})$ are hyperplanes (i.e. $\op{codim}(V^t)=1$).

Complex reflection groups act naturally on the polynomial algebra
$\CC[V]:=\op{Sym}(V^*)$ via precomposition (that is,
$(w*f)(v):=f(w^{-1}\cdot v)$). The Shephard--Todd--Chevalley theorem states that their invariant algebra $\CC[V]^W$, is itself a polynomial algebra. It is generated by $n$ algebraically independent polynomials $(f_1,\cdots,f_n)$, which can be chosen to be homogeneous. Even though there is no  canonical selection for them, their degrees $d_i:=\op{deg}(f_i)$ are uniquely determined.

One of the most important applications of the Shephard--Todd--Chevalley theorem is on the {\it geometric invariant theory} (GIT) of $W$. It implies that the orbit space $W\setminus V$ can be identified with an $n$-dimensional complex space, and that we can think of the {\it fundamental invariants} $f_i$ as its coordinates (i.e. $\op{Spec}(\CC[f_1,\cdots,f_n])=W\setminus V$). Furthermore it gives an algebraic description on the abstract quotient map $\rho:V\rightarrow W\setminus V$:
\begin{equation}\label{quotient map}
\CC^n\cong V\ni{\bf x}:=(x_1,\cdots,x_n)\xrightarrow{\ \displaystyle\rho\ } {\bf f}({\bf x}):=\big(f_1({\bf x}),\cdots,f_n({\bf x})\big)\in W\backslash V\cong \CC^n
\end{equation}

In this note, we will be interested in the subclass of complex reflection groups that are generated by $n=\op{dim}(V)$-many quasi-reflections. Those are called {\it \color{blue} well-generated} and they possess good analogs of the Coxeter elements of real reflection groups. For us, and if $h:=d_n$ and $\zeta=e^{2\pi i/h}$, a {\it \color{blue} Coxeter element} will be a Springer $\zeta$-regular element. That is, an element $c\in W$ with a $\zeta$-eigenvector $v$ that is regular (i.e. lies in a free  orbit of $W$).

\subsection{Braid groups and discriminant hypersurfaces}

Each complex reflection group $W$ comes with an associated arrangement $\mathcal{A}_W:=\bigcup_{H\in R} H$ of its reflection hyperplanes $H\subset V$. It is a theorem of Steinberg that the restriction of the quotient map $\rho: V\rightarrow W\setminus V$ on the complement $\reg{V}:=V\setminus\mathcal{A}_W$ is a Galois covering. We define the {\it \color{blue} braid group} $B(W)$ to be the fundamental group of the base of this covering and use the following s.e.s. to produce a surjection $B(W)\xtwoheadrightarrow{\pi} W$.

\begin{equation}\label{1PBW1}
1\hookrightarrow\underset{\overset{\rotatebox{90}{\,:=}}{\displaystyle P(W)}}{\pi_1(\reg{V})}
\xrightarrow{\ \displaystyle \rho_*\ }\underset{\overset{\rotatebox{90}{\,:=}}{\displaystyle B(W)}}{\pi_1(W\backslash\reg{V})}
\xtwoheadrightarrow{\ \displaystyle \pi} W\rightarrow 1
\end{equation}

Given a choice of a basepoint $v\in\reg{V}$, a loop $b\in B(W)$ lifts to a path that connects $v$ to $b_*(v)$ (we call this the {\it Galois action} of $b$). Then, we define $w:=\pi(b)$ to be the {\it unique} element $w\in W$ such that $w\cdot v=b_*(v)$. We make now such a choice of a basepoint $v$ and in what follows we consider the surjection $\pi$ fixed.

%The significance of \eqref{1PBW1} lies in that it gives a topological interpretation of $W$ as the group of deck transformations of a covering map $\rho:\reg{V}\rightarrow W\backslash \reg{V}$, for which we have an explicit algebraic formula \eqref{quotient map}.

It is well understood that much of the combinatorics of real reflection groups $W$ is determined by the associated arrangement $\mathcal{A}_W$. As we transition to the complex case, $\mathcal{A}_W$ is replaced by its image $W\setminus \mathcal{A}_W$ in the quotient space. It is (yet another) consequence of the GIT of $W$ that this image is an algebraic variety in $W\setminus V\cong \CC^n$ which we call the {\it \color{blue} discriminant hypersurface} of $W$ and denote by $\mathcal{H}$.

Furthermore, $\mathcal{H}$ is cut out by a single polynomial in the $f_i$'s, which we also call the {\it discriminant} of $W$, and which we denote by $\Delta(W,{\bf f})$. When $W$ is a well-generated group, we can always choose a system of fundamental invariants $f_i$, such that the discriminant takes the form
\begin{equation}\label{Eqn:disc}\Delta(W,{\bf f})=f_n^n+\alpha_2 f_n^{n-2}+\cdots+\alpha_n,\end{equation}
where $\alpha_i\in\CC[f_1,\cdots,f_{n-1}]$ are quasi-homogeneous polynomials of weighted degree $h i$. 

\section{Geometric factorizations of a Coxeter element}
\label{Section: Geometric factorizations of a Coxeter element}

Recall that we have identified (see \eqref{quotient map}) the orbit space $W\setminus V$ with an $n$-dimensional complex vector space whose coordinates are the fundamental invariants $f_i$. Let us now define the {\it \color{blue} base space} $Y:=\op{Spec}\CC[f_1,\cdots,f_{n-1}]$, so that $W\setminus V\cong Y\times \CC$ with coordinates written $(y,x)$ or sometimes $(y,f_n)$.

The special slice $L_0:={\bf 0}\times\CC$, given by $f_1=\cdots
=f_{n-1}=0$ and $f_n$ arbitrary, is independent of the choice of $f_i$'s
(see \cite[Remark~25]{theo-thesis}). Because of the monicity of the
discriminant \eqref{Eqn:disc} in $f_n$, $L_0$ intersects the hypersurface
$\mathcal{H}$ only at the origin $({\bf 0},0)$. In particular, the loop
inside $L_0$ given by $f_n(t)=e^{2\pi i t}$, $t\in [0,1]$ (and the rest
of the $f_i$'s equal to $0$), is an element of the braid group
$B(W)=\pi_1(W\setminus V-\mathcal{H})$. We denote it by $\delta$ (see
Figure~\ref{Fig:label map}).

We can map $\delta$ to an element of $W$ via the fixed surjection $\pi:B(W)\twoheadrightarrow W$ from \eqref{1PBW1}. Because of our explicit knowledge of the quotient map $\rho$, it is easy to see that the Galois action of $\delta$ on the basepoint $v\in \reg{V}$ is multiplication by $e^{2\pi i/h}$:

\begin{prop}[{\cite[Lemma~6.13]{bes-kpi1}}]
The element $c:=\pi(\delta)$ is a Coxeter element for $W$.
\end{prop} 

As a loop around the unique point of contact $\{({\bf 0},0)\}=\mathcal{H}\bigcap L_0$ gives rise to a {\it geometrically constructed} Coxeter element, we might consider nearby ``vertical'' slices $L_y:=y\times\CC\subset W\setminus V\cong Y\times\CC$. In that case, the intersection $L_y\bigcap\mathcal{H}$ consists of $n$ points (counted with multiplicity), which are the solutions (see \eqref{Eqn:disc}) of the equation
\begin{equation} \big(\Delta(W,{\bf f});(y,t)\big):=t^n+\alpha_2(y)t^{n-2}+\cdots+\alpha_n(y)=0.\label{Eq: discriminant with y and t}\end{equation}
Here $y\in Y\cong\CC^{n-1}=\op{Spec}(\CC[f_1,\cdots,f_{n-1}])$ is fixed and $t$ is the parameter, and we write $(y,x_i)$ for the solutions.

Bessis describes a way of drawing loops around these points $(y,x_i)$ and shows that, via the fixed surjection $\pi$, they map to factorizations of the Coxeter element. This process of assigning factorizations to points $y$ of the base space will be called a {\it \color{blue} labeling map}. The construction depends nontrivially (and this is a subtlety that is important for us) on a choice of ordering of the complex numbers. In what follows, we will consider the following two:
\begin{defn}\label{Defn: orders}\
\begin{enumerate}
\item {\it \color{blue} Complex lexicographic order (clo)}: We order the points $x_i\in\CC$ by increasing real part first and we cut ties by increasing imaginary part.
\item {\it \color{blue} Polar lexicographic order (plo)}: We order the points $x_i\in\CC$ via their polar coordinates $(r,\theta)=(|x|,\op{arg}(x))$, where $r\geq 0$ and $\theta\in[-3\pi/2,\pi/2]$. We order them by increasing length $r$ first, and by {\bf decreasing} argument $\theta$ to cut ties (where $\theta$ starts at the 12:00 position).
\end{enumerate}
\end{defn}



After we have decided on such an ordering, we proceed by picking a path
$\theta$ in $Y$ that connects ${\bf 0}$ to $y$. We lift it to a path
$\beta_{\theta}$ in $Y\times\CC$, starting at $({\bf 0},1)$, which always
stays ``above'' (i.e. has bigger imaginary part than) all the points in
the intersections $L_{y'}\bigcap\mathcal{H}$ (see Figure~\ref{Fig:label map}). We call the endpoint of this path $(y,x_{\infty})$ to indicate that it lies in the slice $L_y$ and above all points $(y,x_i)$.


From $x_{\infty}$ we now construct paths $\beta_i$ in $L_y$ to the (ordered) points $x_i$ such that they never cross each other (or themselves), and their order as they leave $x_{\infty}$ is given by their indices (i.e. $\beta_1$ is the leftmost one).

Given this information, we can now easily construct elements
$b_{(y,x_i)}$ of $B(W)$: First, we follow the path $\beta_{\theta}$ from
the basepoint $({\bf 0},1)$ to $(y,x_{\infty})$, then we go down
$\beta_i$ but before we reach its end, we trace a small counterclockwise
circle around $x_i$, and finally we return by the same route (see Figure~\ref{Fig:label map}). 


\begin{wrapfigure}{r}{0.43\textwidth}
  \begin{center}
	\includegraphics[trim={12cm 2.5cm 12cm 3.5cm},clip,width=0.43\textwidth]{figures/label-map.png}
  \end{center}
\caption{\footnotesize Factorizations via the slice $L_y$.}\label{Fig:label map}
\end{wrapfigure}
The product $\delta_y=b_{(y,x_1)}\cdots b_{(y,x_k)}$ (where $k$ is the number of {\it geometrically distinct} points $x_i$) surrounds $\mathcal{H}$ and is, of course, homotopic to our loop $\delta$. We have finally made it; from each slice $L_y$, we have constructed a geometric factorization of $\delta$. The fixed surjection $\pi: B(W)\twoheadrightarrow W$ allows us to turn this into a factorization of the Coxeter element $c$:

\begin{defn}\label{Defn: labeling map}
There is a collection of labeling maps which, to any point $y\in Y$
assign a factorization $c_1\cdots c_k=c$, where the factors
$c_i:=\pi(b_{(y,x_i)})$ are defined as above. Such labeling maps are
differentiated by the choice  of ordering of the complex numbers as
discussed before Definition~\ref{Defn: orders}. 

We denote the labeling maps corresponding to the orders (plo) and (clo)
as $\op{clbl}$ and $\op{rlbl}$ respectively (referring to ``cyclic''
labels vs ``reduced''\footnote{This term is not related to some length
function; it is only here to allow the exposition be consistent with
\cite[Definition~7.14]{bes-kpi1}.} labels).
\end{defn}

\section{The Lyashko--Looijenga morphism ($LL$) and trivialization theorems}
\label{Section: The LL map}

In the previous section we described a way to produce factorizations of
the Coxeter element, by intersecting the discriminant hypersurface
$\mathcal{H}$ with the slices $L_y$. The most striking fact of this
theory though, is that this geometric construction is sufficient to
produce {\it all} reduced block factorizations of the Coxeter element
$c$. In fact, if we additionally keep track of the intersection
$L_y\bigcap \mathcal{H}$, each such factorization is attained exactly
once (Theorem~\ref{Thm-triv} records this for reflection factorizations, which is sufficient for us).

The geometric object that keeps track of the point configurations
$L_y\bigcap \mathcal{H}$ for the various $y\in Y$ is the Lyashko--Looijenga morphism. We use the notation $E_n$ for the space of {\it centered configurations} of $n$ points (that need not be distinct):

\begin{defn}\label{Defn: LL map}
For an irreducible well-generated complex reflection group $W$, we define
the {\it \color{blue} Lyashko--Looijenga} map by:
\begin{alignat*}{3}
& \ Y &&\xrightarrow{\quad LL \quad}\ \ & &E_n\\
 y=(f_1,&\cdots,f_{n-1}) &&\xrightarrow{\quad \quad\quad}\ \ & \text{ multiset of ro}&\text{ots of } \big( \Delta(W,{\bf f});(y,t)\big)=0
\end{alignat*}
and denote it by $LL$. Notice that there is a simple description of $LL$ as an algebraic morphism. Indeed the (multiset of) roots of a polynomial is completely determined by its coefficients, therefore we can express $LL$ as the map:
\begin{alignat*}{3}
Y&\cong \CC^{n-1} &&\xrightarrow{\quad LL\quad } &\quad E_n &\cong\CC^{n-1}\\
y=(f_1,&\cdots,f_{n-1}) &&\xrightarrow{\quad\quad\quad} &\quad \big( \alpha_2(f_1,\cdots,f_{n-1}),&\cdots,\alpha_n(f_1,\cdots,f_{n-1})\big)
\end{alignat*}
where the $\alpha_i$'s are as in \eqref{Eq: discriminant with y and t}.
\end{defn}

The $LL$ map is a proper, finite morphism and its ramification and
monodromy are intimately related to the combinatorics of the noncrossing
lattice (see \cite[Chapter~7]{theo-thesis} and \cite[Theorem~7.20]{bes-kpi1}). However, in what follows we will mainly take advantage of just two statements about $LL$.

To set them up, we introduce the notation $\reg{E_n}$ for the subset of $E_n$ that contains those centered configurations that have $n$ {\bf distinct} points. Furthermore, we let $\reg{Y}$ be the preimage $LL^{-1}(\reg{E_n})$ (that is, the set of those points $y\in Y$ for which the slice $L_y$ intersects the discriminant $\mathcal{H}$ in $n$-many distinct points).

\begin{prop}[{\cite[Lemma~5.7]{bes-kpi1}}]\label{Prop: LL is a covering map on Y-K}
The restriction $LL:\reg{Y}\rightarrow \reg{E_n}$ is a topological covering map. In particular, any path in $\reg{E_n}$ can be lifted to a path in $\reg{Y}$.
\end{prop}

\begin{thm}[{\cite[Theorem~7.5]{bes-kpi1}}]\label{Thm-triv}
The map $LL\times\op{rlbl}:\reg{Y}\rightarrow \reg{E_n}\times\op{Red}_W(c)$ is a bijection. The same is true for the map $\op{clbl}$.
\end{thm} 

\subsection{Interpretation of the Hurwitz action via the labeling maps}

The driving (combinatorial) force behind Theorem~\ref{Thm-triv} is the transitivity of the Hurwitz action on the set $\op{Red}_W(c)$. Its connection to such a geometric statement comes from the labeling map. Let us, however, first recall the definition:

\begin{defn}\label{Defn: The (algebraic) Hurwitz action}
For {\it any} group $G$, there is a natural action of the (usual) braid group $B_n$ on the set of $n$-tuples of elements $t_i$ of $G$. The generator $s_i$ acts via:
\begin{equation}\label{Eqn: Si bradi action}
s_i*(t_1,\cdots,t_i,t_{i+1},\cdots,t_n)=(t_1,\cdots,t_{i-1},t_it_{i+1}t_i^{-1},t_i,t_{i+2},\cdots,t_n).\end{equation}
We call this the (left) {\it \color{blue} Hurwitz action} of $B_n$ on $G_n$. It is clear that the Hurwitz action respects the product of the $t_i$'s. Therefore, we may restrict it on tuples that encode fixed length factorizations of elements of $G$. In our context, we consider the Hurwitz action on the set of reduced reflection factorizations $\op{Red}_W(c)$.
\end{defn}

The following Lemma~\ref{Lemma:Hurwit-Galois equiv}, which describes how the label $\op{rlbl}(y)$ is affected as $y$ varies in the base space $Y$, is the main technical support of this note. It is based on the observation, already by Fox and Neuwirth \cite{fox-neuwirth-braids}, that we may identify braids $g\in B_n$ with loops $\gamma$ in $\reg{E_n}$. This of course demands that we first choose a basepoint ${\bf e}\in\reg{E_n}$ {\it and} order its elements (${\bf e}$ is a point configuration), just so that we may even define the isomorphism. 

On the other hand, these loops $\gamma$ can be lifted to paths in
$\reg{Y}$ via the covering map $LL:\reg{Y}\rightarrow \reg{E_n}$ (see
Proposition~\ref{Prop: LL is a covering map on Y-K}). If $y$ is a lift of the basepoint ${\bf e}$, we write $\gamma\cdot y$ for the Galois action (that is $\gamma\cdot y$ is the endpoint when we lift the loop $\gamma$ starting at $y$). Then we have:


\begin{lem}[{\cite[Corollary~6.20]{bes-kpi1}}]\label{Lemma:Hurwit-Galois equiv}
The labeling maps are equivariant with respect to the Hurwitz action and the Galois action. That is,
$$\op{rlbl}(\gamma\cdot y)=g*\op{rlbl}(y),$$ where $\gamma$ and $g$ are as above. The same is true for $\op{clbl}$.
\end{lem}
\begin{proof}
Any loop $\gamma\in\reg{E_n}$ can be decomposed as a sequence of moves that transpose two neighboring points (we call these {\it Hurwitz moves}). In this case, the corresponding braid $g\in B_n$ is just one of the canonical generators $s_i$ whose action is given by \eqref{Eqn: Si bradi action}.

The following Figure~\ref{fig-hurw-action} describes the effect of such a
Hurwitz move on the labeling maps. The loops $\beta_i$ in
Figure~\ref{fig-hurw-act-a} are the ones we used in Section~\ref{Section: Geometric factorizations of a Coxeter element} to define the labels. We write $rlbl(y)=(\beta_1,\beta_2)$ forgetting the surjection $\pi:B(W)\twoheadrightarrow W$.

The next two Figures~\ref{fig-hurw-act-b} and~\ref{fig-hurw-act-c} show
the slice $L_{\gamma\cdot y}$ and on it are drawn two pairs of loops. The
blue loops $(\beta_1,\beta_2)$ are homotopic (notice that the homotopy
must happen inside $B(W)=\pi_1(W\setminus V-\mathcal{H})$) to those in
Figure~\ref{fig-hurw-act-a}. The red ones $(\beta_1',\beta_2')$, on the other hand, are those assigned by the labeling map. As we can see, we have $$\op{rlbl}(\gamma\cdot y)=(\beta_1',\beta_2')=(\beta_1\beta_2\beta_1^{-1},\beta_1)=s_1*(\beta_1,\beta_2)=s_1*\op{rlbl}(y).$$

\begin{figure}[h]
\centering
\begin{subfigure}[t]{0.3\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{figures/pic1.png}
\caption{\footnotesize As the two points in $LL(y)$ move around each other...}
\label{fig-hurw-act-a}
\end{subfigure}\quad
\begin{subfigure}[t]{0.3\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{figures/pic2.png}
\caption{\footnotesize ...the loop $\beta_2$ stretches to avoid $\beta_1$.}
\label{fig-hurw-act-b}
\end{subfigure}\quad
\begin{subfigure}[t]{0.3\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{figures/pic3.png}
\caption{\footnotesize Our previous loop $\beta_2$ is now homotopic to $\beta_2'^{-1}\beta_1'\beta_2'$.}
\label{fig-hurw-act-c}
\end{subfigure}
\caption{The Hurwitz action.}
\label{fig-hurw-action}
\end{figure}
\end{proof}

\begin{remark}\label{Remark: different theorem}
This is indeed a different theorem for the two labeling maps. The reason
is that the way we associate the braid $g$ to the loop $\gamma$ depends
on a choice of ordering of the points in the configuration ${\bf e}$.
This is different for the various labeling maps as we saw in
Definition~\ref{Defn: orders}. This nuance will become clearer in
Section~\ref{Section: Cyclic actions via labeling maps}, where in
Propositions~\ref{Prop. equivariance clbl} and~\ref{Prop: rlbl equiv.} we apply this Lemma to both maps $\op{clbl}$ and $\op{rlbl}$. 
\end{remark}

\section{Cyclic actions via the labeling maps}
\label{Section: Cyclic actions via labeling maps}

In this section we give an interpretation of (all three) cyclic actions $\Phi,\Psi$, and $\Twist$, from the introduction, as Hurwitz actions. They are in fact induced by particular loops in the configuration space $\reg{E_n}$ via Lemma~\ref{Lemma:Hurwit-Galois equiv}. We begin by introducing the latter ones:

\begin{defn}
We consider the following centered configurations of $\reg{E_n}$:
$${\bf e}_{(n)}:=\{e^{\epsilon +2\pi i k/n}\ |\ k=1,\cdots n\}\quad\text{and}\quad {\bf e}_{(n-1,1)}:=\{0\}\cup\{e^{\epsilon +2\pi i k/(n-1)}\ |\ k=1,\cdots,n-1\},$$
for small $\epsilon$, and the following loops (also in $\reg{E_n}$) based on them: 
$$\gamma_{(n)}(t):=e^{2\pi i t/n}\cdot {\bf e}_{(n)},\ t\in [0,1],\quad\text{ and }\quad \gamma_{(n-1,1)}(t):=e^{2\pi i t/(n-1)}\cdot {\bf e}_{(n-1,1)},\ t\in [0,1].$$
Notice that the configurations are cyclically symmetric and the loops $\gamma$ correspond to the minimal rotations that respect them. We will further consider two {\it twisting} loops that rotate the two configuration by $180^{\circ}$:
$$\gamma^{\mathfrak{T}}_{(n)}(t):=e^{\pi it}\cdot {\bf e}_{(n)},\ t\in [0,1],\quad\text{ and }\quad \gamma^{
\mathfrak{T}}_{(n-1,1)}(t):=e^{\pi i t}\cdot {\bf e}_{(n-1,1)},\ t\in [0,1],$$ defined when $n$ is even or odd respectively. %Notice the immediate relations $\gamma_{(n)}^{n/2}=\gamma_{\mathfrak{T},(n)}$ and $\gamma_{(n-1,1)}^{(n-1)/2}=\gamma_{\mathfrak{T},(n-1,1)}$.
\end{defn}


\begin{prop}\label{Prop. equivariance clbl}
The labeling map $\op{clbl}:\reg{Y}\rightarrow \op{Red}_W(c)$ is equivariant with respect to the Galois actions of $\gamma_{(n)}$ and $\gamma_{(n-1,1)}$, and the actions of $\Phi_{\op{cyc}}$ and $\Psi_{\op{cyc}}$, respectively. That is,
$$\Phi_{\op{cyc}}*\op{clbl}(y_1)=\op{clbl}(\gamma_{(n)}\cdot y_1)\quad\text{ and }\quad \Psi_{\op{cyc}}*\op{clbl}(y_2)=\op{clbl}(\gamma_{(n-1,1)}\cdot y_2),$$
for any $y_1\in LL^{-1}({\bf e}_{(n)})$ and any $y_2\in LL^{-1}({\bf e}_{(n-1,1)})$.
\end{prop}
\begin{proof}
The proof is encoded in Figure~\ref{fig-clbl-phi-psi}. We write
$\op{clbl}(y_1)=(\beta_1,\cdots,\beta_n)$ and similarly
$\op{clbl}(\gamma_{(n)}\cdot y_1)=(\beta_1',\cdots,\beta_n')$ forgetting
the fixed surjection $\pi:B(W)\twoheadrightarrow W$. As the Galois action
of $\gamma_{(n)}$ rotates the configuration in
Figure~\ref{fig-map-phi-1}, the blue loop $\beta_i$ ends up encircling
the $(i+1)^{\op{th}}$ point in Figure~\ref{fig-map-phi-2}. That is, it is homotopic to its label there ($\beta_i=\beta_{i+1}'$).

The  situation is different only for the last loop $\beta_n$. As we can
see in Figure~\ref{fig-map-phi-2}, after the rotation $\beta_n$ is no longer a label. To express it in terms of the $\beta_i'$'s, notice that it surrounds the configuration clockwise, then follows $\beta_1'$, and then it unwinds again. That is $\beta_n=c^{-1}\beta_1'c$, which we reverse to obtain $\beta_1'=c\beta_nc^{-1}$ and complete the proof for $\Phi_{\op{cyc}}$.

The proof is similar for $\Psi_{\op{cyc}}$. As before, we have $\beta_i=\beta_{i+1}'$, $i=2\cdots n-1$. But now, the last loop $\beta_n$ ends up encircling the {\it second} point (after the rotation). That is, $\beta_2'=c\beta_nc^{-1}$. Moreover, the loop $\beta_1$ is not a label anymore as it is now on the right of $\beta_2'$. We compute the Hurwitz move $\beta_1'=\beta_2'^{-1}\beta_1\beta_2'$, which after replacing $\beta_2'$ with $c\beta_nc^{-1}$ gives us precisely $\Psi_{\op{cyc}}$.
\begin{figure}[h]
\centering
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[trim={2cm 0 2cm 0},clip,width=1.1\linewidth]{figures/mypic1.png}
\caption{\footnotesize As the configuration rotates, the loop}
\label{fig-map-phi-1}
\end{subfigure}\ 
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[trim={2cm 0 2cm 0},clip,width=1.1\linewidth]{figures/mypic2.png}
\caption{\footnotesize ... $\beta_5$ can no longer be part of the label.}
\label{fig-map-phi-2}
\end{subfigure}\ 
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[trim={2cm 0 2cm 0},clip,width=1.1\linewidth]{figures/mypic3.png}
\caption{\footnotesize As ${\bf 0}$ is labeled first (Definition~\ref{Defn: orders}) in (plo),}
\label{fig-map-psi-1}
\end{subfigure}\
\begin{subfigure}[t]{0.24\textwidth}
\centering
\includegraphics[trim={2cm 0 2cm 0},clip,width=1.1\linewidth]{figures/mypic4.png}
\caption{\footnotesize ... in this case, $\beta_1$ cannot be a label either.}
\label{fig-map-psi-2}
\end{subfigure}
\caption{A Hurwitz interpretation of $\Phi_{\op{cyc}}$ and $\Psi_{\op{cyc}}$.}
\label{fig-clbl-phi-psi}
\end{figure}
\end{proof}

As we discussed in Remark~\ref{Remark: different theorem}, the labeling maps $\op{rlbl}$ and $\op{clbl}$ are affected in a different way as we follow the paths $\gamma$ in the configuration space $\reg{E_n}$.

\begin{prop}\label{Prop: rlbl equiv.}
The labeling map $\op{rlbl}:\reg{Y}\rightarrow \op{Red}_W(c)$ is equivariant with respect to the Galois actions of $\gamma^{\mathfrak{T}}_{(n)}(t)$ and $\gamma^{
\mathfrak{T}}_{(n-1,1)}(t)$, and the action of $\Twist$, when $n$ is even or odd respectively. That is,
$$\Twist*\op{rlbl}(y_1)=\op{rlbl}(\gamma^{\mathfrak{T}}_{(n)}(t)\cdot y_1)\quad\text{ and }\quad \Twist*\op{rlbl}(y_2)=\op{rlbl}(\gamma^{
\mathfrak{T}}_{(n-1,1)}(t)\cdot y_2),$$
for $n$ even and odd respectively, and for any $y_1\in LL^{-1}({\bf e}_{(n)})$ and any $y_2\in LL^{-1}({\bf e}_{(n-1,1)})$.
\end{prop}
\begin{proof}\let\qed\relax
\end{proof}\vspace{-0.83cm}

\begin{wrapfigure}{r}{0.24\textwidth}
  \begin{center}\vspace{-0.5cm}
	\includegraphics[trim={2cm 0.3cm 2cm 0},clip,width=0.25\textwidth]{figures/pic12.png}
  \end{center}\vspace{-0.5cm}
\caption{\footnotesize 12 angry points}\label{fig:12 pts}\vspace{-0.5cm}
\end{wrapfigure}

\quad\quad The same method applies here as in Proposition~\ref{Prop.
equivariance clbl}. The difference is that the labeling map $\op{rlbl}$
orders the points differently, as we can see in Figure~\ref{fig:12 pts}. We write $\op{rlbl}(y_1)=(t_1,\cdots,t_n)$ and $\op{rlbl}(\gamma_{(n)}^{\mathfrak{T}}\cdot y_1)=(t_1',\cdots,t_n')$. The Galois action is clockwise rotation by $180^{\circ}$.

According to Lemma~\ref{Lemma:Hurwit-Galois equiv} a label $t_i$ is conjugated by a label $t_j$ only when $t_j$ crosses $t_i$ from above and to the right. In particular, as the points move in the upper half circle, the labels are not affected. For instance, $t_1$ travels to the $n^{\op{th}}$ point giving us $t_n'=t_1$.

If we follow $t_2$ however, using Figure~\ref{fig:12 pts} to enhance our imagination, we see that $t_1$ crosses it to the right; this gives us $t_{n-1}'=\ ^{t_1}t_2$. We may proceed by induction: Pick an even number $2k$ (think of $8$ in the figure) and start rotating it. It will first be conjugated by the odd numbers $7,5,\cdots$, or really $2k-1,2k-3,\cdots$ (in that order), and then by the labels of the even numbers too. Those however will have already reached their final version (since they are now on the upper half circle). That is, the total action on $t_{2k}$ until it becomes $t_{n+1-2k}'$ will be: $$t_{n+1-2k}'=wt_{2k}w^{-1}\quad \text{ for }w=\ ^{(t_1\cdots t_{2k-3})}t_{2k-2}\cdots\ ^{(t_1t_2t_3)}t_4\cdot\ ^{t_1}t_2\cdot t_1\cdot t_3\cdots t_{2k-1},$$
and direct calculations shows that $w=t_1t_2\cdots t_{2k-1}$. The proof is similar for an odd number $2k+1$ and similar for the Galois action of $\gamma_{(n-1,1)}^{\mathfrak{T}}$.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\qedsymbol

The following Lemma identifies the previous Galois actions on the symmetric configurations ${\bf e}$, essentially as scalar multiplication. Note that since the coordinates $f_i$ of the space $Y$ have weights $d_i$, the correct way to define a scalar action on it, called {\it\color{blue} weighted multiplication}, is via:
\begin{equation}
\lambda\in \CC^{*}, y=(f_1,\cdots,f_{n-1})\ \Rightarrow \lambda*y:=(\lambda^{d_1}f_1,\cdots,\lambda^{d_{n-1}}f_{n-1}).
\end{equation} 

\begin{lem}\label{Lem: Galois action is weighted multiplication}
The Galois actions of $\gamma_{(n)}$ on $LL^{-1}({\bf e}_{(n)})$, and of $\gamma_{(n-1,1)}$ on $LL^{-1}({\bf e}_{(n-1,1)})$, are identical to {\bf weighted} multiplication by $e^{-2\pi i/nh}$ and $e^{-2\pi i/(n-1)h}$ respectively. Similarly, both twisting loops $\gamma_{(n)}^{\mathfrak{T}}$ and $\gamma_{(n-1,1)}^{\mathfrak{T}}$ act on the corresponding fibers as {\bf weighted} multiplication by $e^{-\pi i/h}$. 
\end{lem}
\begin{proof}
The loops $\gamma$ simply rotate the roots of the discriminant polynomial \eqref{Eq: discriminant with y and t} by some angle $\theta$. To do this is to scale its coefficients $\alpha_k(y)$ by $(e^{i\theta})^k$. Since the $\alpha_k$ are of weighted degree $kh$, we see that this translates to weighted multiplication on $y$ by $e^{i\theta/h}$.
\end{proof}

\begin{defn}\label{Defn: Cyclic subgroups}
We record here the cyclic subgroups of the weighted $\CC^{*}$-action considered above. We will view the fibers $LL^{-1}({\bf e})$ as modules over them. We define:
$$C_{nh}:=\Big\langle e^{\textstyle \frac{-2\pi i}{nh}}\Big\rangle\quad\text{ and }\quad C_{(n-1)h}:=\Big\langle e^{\textstyle\frac{-2\pi i}{(n-1)h}}\Big\rangle\quad\text{ and }\quad C_{2h}:=\Big\langle e^{\textstyle\frac{-\pi i}{h}}\Big\rangle  .$$
\end{defn}


\begin{cor}\label{Cor: Red_W(c)isom to fibers}
The following module isomorphisms hold:
\begin{alignat*}{4}
\op{Red}_W(c)&\cong_{C_{nh}}LL^{-1}({\bf e}_{(n)}),&&\quad \op{Red}_W(c)&&\cong_{C_{(n-1)h}}&&LL^{-1}({\bf e}_{(n-1,1)})\\
\op{Red}_W(c)&\cong_{C_{2h}}LL^{-1}({\bf e}_{(n)}),\ n\text{ even},&&\quad\op{Red}_W(c)&&\cong_{C_{2h}}&&LL^{-1}({\bf e}_{(n-1,1)}),\ n\text{ odd},
\end{alignat*}
where the cyclic actions on $\op{Red}_W(c)$ are those of
$\Phi_{\op{cyc}}$, $\Psi_{\op{cyc}}$, and $\Twist$ respectively, and the
actions on the fibers $LL^{-1}({\bf e})$ are via weighted multiplication
and according to Definition~\ref{Defn: Cyclic subgroups}.
\end{cor}
\begin{proof}
This is immediate after Propositions~\ref{Prop. equivariance clbl} and~\ref{Prop: rlbl equiv.}, and Lemma~\ref{Lem: Galois action is weighted multiplication}.
\end{proof}


\section{CSP's through finite quasi-homogeneous morphisms} 
\label{Section last}

\begin{lem}\label{Lemma-Vic}
Let ${\bm f}:\CC^n\rightarrow\CC^n$ be a polynomial morphism, finite and quasihomogeneous, which corresponds to the inclusion of algebras $\CC[x_1,\cdots,x_n]\hookleftarrow\CC[\theta_1,\cdots,\theta_n]$ over weighted parameters $x_i$ and $\theta_i$ of degrees $a_i$ and $b_i$ respectively. Consider further a point ${\bm\epsilon}=(\epsilon_1,\cdots,\epsilon_n )$ in $\CC^n$, such that the fiber ${\bm f}^{-1}({\bm \epsilon})$ is stable under {\bf weighted} multiplication by $c:=e^{-2\pi i/N}$ for some number $N$. Then, if $C_N:=\langle c\rangle$, we have the isomorphism of $C_N$-modules:
$${\bm f}^{-1}({\bm \epsilon})\cong_{C_N}\bigslant{\CC[x_1,\cdots,x_n]}{\big(\theta_1,\cdots,\theta_n\big)}=:\mathcal{R},$$
where the action on $\mathcal{R}$ is the {\it induced} action on functions. That is, $c*x_i=(e^{2\pi i/N})^{a_i}x_i$ (notice the change of sign). In particular, for $X:={\bm f}^{-1}({\bm \epsilon})$ and $X(q):=\op{Hilb}(\mathcal{R},q)=\prod_{i=1}^n\frac{[b_i]_q}{[a_i]_q}$, the triple $(X,X(q),C_N)$ exhibits the CSP.
\end{lem}
\begin{proof}[Sketch]
Since $\CC[{\bm x}]$ is Cohen-Macaulay and finite over $\CC[{\bm
\theta}]$, it is a free $\CC[{\bm \theta}]$-module. This allows us to
construct a $C_N$-stable filtration of the (otherwise ungraded)
coordinate ring $K[{\bm f}^{-1}({\bm \epsilon})]$ whose associated graded
ring is isomorphic to $\mathcal{R}$ as a $C_N$-module. The final
statement about CSP's is \cite[Proposition~2.1: $(i)\iff (iii)$]{reiner-stanton-white}.
\end{proof}


%Intuitively, it says that the scalar action that defines the weights of the morphism can be {\it transfered} to nearby fibers to the extent that their symmetry allows.
%Lemma~\ref{Lemma-Vic} is of a geometric nature and it takes advantage of the special role that is played by the origin ${\bf 0}$ for quasi-homogeneous morphisms. Intuitively, it says that the scalar action that defines the weights of the morphism can be {\it transfered} to nearby fibers to the extent.

\begin{proof}[Proof of Theorem~\ref{Thm: main-thm}]
Our main theorem is an immediate consequence of the previous lemma and
Corollary~\ref{Cor: Red_W(c)isom to fibers}. Indeed, the $LL$ map is
finite, quasi-homogeneous, and the Hilbert series of $LL^{-1}({\bf 0})$ is precisely \cite[Theorem~5.3]{bes-kpi1} the polynomial $\prod_{i=1}^n\frac{[ih]_q}{[d_i]_q}.$
\end{proof}
\begin{remark}
Lemma~\ref{Lemma-Vic} is of a geometric nature and it takes advantage of the special role that is played by the origin ${\bf 0}$ for quasi-homogeneous morphisms. Intuitively, it says that the scalar action that defines the weights of the morphism can be {\it transferred} to nearby fibers to the extent that their symmetry allows.

On the other hand, it gives a very pleasant elucidation of CSP's where the {\it same} polynomial encodes orbital information of different cyclic actions. In our case for instance, the different cyclic groups $C_{(n-1)h}$ and $C_{nh}$ appear as we deform the origin ${\bf 0}\in E_n$ to the configurations ${\bf e}_{(n-1,1)}$ and ${\bf e}_{(n)}$, which in turn have different cyclic symmetries.
\end{remark}

\acknowledgements{This work is inspired by Dennis Stanton's thought-provoking questions during our Thesis defense. We are immensely joyous for finally being able to provide an answer, and we hope he will not mind that it is not really what he asked for! We also thank Vic Reiner for fascinating discussions over this topic, and in particular for showing us Lemma~\ref{Lemma-Vic}.}

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