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\title{On enumerating factorizations in reflection groups}

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

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

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\abstract{We describe an approach, via Malle's permutation $\Psi$ on the set of irreducible characters $\op{Irr}(W)$, that gives a uniform derivation of the Chapuy-Stump formula for the enumeration of reflection factorizations of the Coxeter element. It also recovers its weighted generalization by delMas, Reiner, and Hameister, and further produces structural results for factorization formulas of arbitrary regular elements.}

\resumetitle{R\'esum\'e}
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\resume{Nous d{\'e}crivons une approche, via la permutation de Malle $\Psi$ sur l'ensemble des caract{\`e}res irr{\'e}ductibles $\op{Irr}(W)$, qui donne une d{\'e}rivation uniforme de la formule de Chapuy-Stump pour l'{\'e}num{\'e}ration des factorisations de l'{\'e}l{\'e}ment Coxeter en r{\'e}flexions. Il r{\'e}cup{\`e}re {\'e}galement sa g{\'e}n{\'e}ralisation pond{\'e}r{\'e}e par delMas, Reiner et Hameister et produit en outre des r{\'e}sultats structurels pour des formules de factorisation d'{\'e}l{\'e}ments r{\'e}guliers arbitraires.}

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\keywords{Frobenius lemma, Hecke algebras, Malle's permutation}

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\section{Introduction}

A famous theorem of Cayley states that there are $n^{n-2}$ vertex-labeled trees on $n$ vertices. The same number, as Hurwitz knew already by the end of the $19^{\text{th}}$ century, enumerates the set of shortest length factorizations $t_1\cdots t_{n-1}=(12\cdots n)\in S_n$ of the long cycle into transpositions $t_i$. A natural generalization of this problem, is to enumerate such factorizations of arbitrary length.

It took almost a hundred years for the community to return to this question, but by the end of the 80's Jackson \cite[Corol.~4.2]{jackson-first-proof-of-arb-length} had computed an explicit answer. If $\op{FAC}_{S_n}(t)$ denotes the exponential generating function for the number of arbitrary length factorizations of the long cycle in transpositions (see \eqref{Eq: Frob formula no Cox}), then Jackson's result can be reinterpreted as follows: \begin{equation}
\op{FAC}_{S_n}(t)=\dfrac{e^{t\binom{n}{2}}}{n!}\big( 1-e^{-tn}\big)^{n-1}.\label{EQ: Jackson's formula}
\end{equation}

As it often happens with some of the most fascinating properties of the symmetric group, the previous statements are special cases of more general theorems that hold for a bigger class of reflection groups $W$. A natural analog of the long cycle is the Coxeter element $c\in W$, while transpositions are replaced by reflections. Then, if $W$ is well-generated and of rank $n$, $\mathcal{R}$ denotes its set of reflections, and $h$ is the order of $c$, Bessis \cite[Prop.~7.6]{bes-kpi1} proved the following enumeration: \begin{equation}
\#\Big\{ (t_1,\cdots,t_n)\in \mathcal{R}^n\ |\ t_1\cdots t_n=c\Big\}\ =\ \dfrac{h^nn!}{|W|}.\label{EQ: Hurwitz number for W}
\end{equation}

The $W$-analog of Jackson's formula \eqref{EQ: Jackson's formula} regarding arbitrary length factorizations was discovered (and proved) by Chapuy and Stump \cite{chapuy-stump} soon after. If $\op{FAC}_W(t)$ denotes the corresponding exponential generating function, they showed that \begin{equation}
\op{FAC}_W(t)=\dfrac{e^{t|\mathcal{R}|}}{|W|}\big( 1-e^{-th}\big)^n.\label{EQ: Chapuy-Stump formula}
\end{equation}

The reduced case \eqref{EQ: Hurwitz number for W}, which can easily be derived by calculating the leading term of $\op{FAC}_W(t)$, has a long history and appears in connection to many a mathematical endeavour. It originated in singularity theory, in combinatorics it appeared as the number of maximal chains in the noncrossing lattice $NC(W)$, and more importantly it was essential in Bessis' proof of the $K(\pi,1)$-conjecture \cite{bes-kpi1}.

%\subsubsection*{A uniform argument}

Apart from the Weyl group case \cite{michel-deligne-lusztig-derivation}, neither \eqref{EQ: Chapuy-Stump formula} nor \eqref{EQ: Hurwitz number for W} were well understood. Although the statements are uniform for all well-generated groups, the proofs of Bessis and Chapuy-Stump have relied on the Shephard-Todd classification (a common misfortune for theorems regarding reflection groups, but also a driving reason for why the theory is as evolved as it is). Our main contribution  is a case-free proof (\cref{Cor: Chapuy-Stump for dn}) and generalization of them (\cref{Thm: Main not weighted,Thm: Main weighted}) which in fact, in terms of Bessis' work \cite{bes-kpi1}, completes the first uniform proof of the dual braid presentation of $B(W)$ for real $W$.% In particular, in terms of Bessis' pioneering work \cite{bes-kpi1}, we have completed a uniform proof of the "Trivialization theorem" \cite[Thm.~7.20]{bes-kpi1} for all real reflection groups $W$. In fact, the first case-free proof of the dual braid presentation of spherical Artin groups.

%For that matter, one of the main obstacles to producing a conceptual proof for \eqref{EQ: Chapuy-Stump formula} lies in that we have no nice, uniform construction of irreducible characters for complex reflection groups. Only for Weyl groups there is Springer's correspondence, which is however technically difficult for computations. 

%In this work we also start with the Frobenius Lemma, but instead of explicitly computing the characters $\chi\in\op{Irr}(W)$, we group them together with respect to an invariant called the Coxeter number $c_{\chi}$ (see Defn.~\ref{Defn: Coxeter numbers}). Then, Malle's cyclic action $\Psi$ on $\op{Irr}(W)$ allows us to cancel the contribution of those $\chi$ for which $c_{\chi}$ is not a multiple of $h$. The resulting expression is very rigid (Thm.~\ref{Thm: Main not weighted}) and the mere knowledge of bounds for the $c_{\chi}$ allows us to complete the proof.

%Ours is not the first approach towards a uniform proof of \eqref{EQ: Chapuy-Stump formula}. In \cite{michel-deligne-lusztig-derivation}, Michel also considers a grouping of the characters; the partition given by Lusztig's families. This is finer (and much more technologically advanced) and although the argument gives a very satisfying connection between \eqref{EQ: Jackson's formula} and \eqref{EQ: Chapuy-Stump formula}, it requires the existence of the elusive ``spets" when $W$ is not a Weyl group.

%Moreover, our strategy applies in further generality and produces structural results for any regular element $g\in W$ (which become explicit formulas for a larger class of groups than the well-generated ones, see Corol.~\ref{Cor: Chapuy-Stump for dn}). In addition, a refined version in Section~\ref{Section: The weighted enumeration} recovers (uniformly) and extends the main result of \cite{reiner-delMas-Hameister} on a weighted version of the Chapuy-Stump formula \eqref{EQ: Chapuy-Stump formula}.

%When $W$ is a real reflection group, all our theorems are completely case-free. In the complex case, although our approach is indeed uniform, it relies on the BMR-freeness theorem, a property of the Hecke algebra $\mathcal{H}(W)$ that is currently proven in a case-by-case way (see \S\ref{Section: On the uniformity of the proofs} for details).

\section{Complex reflection groups and regular elements}
\label{Section: Complex reflection groups and regular elements}


Given a complex vector space $V\cong \CC^n$, we call a {\it finite} subgroup $W\leq \op{GL}(V)$ a {\it\color{blue} complex reflection group} if it is generated by unitary reflections. These are $\CC$-linear maps $t$ whose fixed spaces $V^t:=\op{ker}(t-\op{id})$ are hyperplanes (i.e. $\op{codim}(V^t)=1$). We further say that $W$ is {\it irreducible} if it has no stable linear subspaces apart from $V$ and $\{0\}$. 

Shephard and Todd classified irreducible complex reflection groups into an infinite $3$-parameter family $G(r,p,n)$ and 34 exceptional cases indexed $G_4$ to $G_{37}$. In what follows we will silently assume that our reflection groups $W$ are in fact irreducible. This simplifies the argument but imposes no serious restrictions on our results, see \cite[\S\,5.1]{full-paper}.

We denote by $\mathcal{R}$ the set of reflections of $W$ and we write $\mathcal{A}$ for the associated arrangement of fixed hyperplanes. For such a hyperplane $H$, let $W_H$ be its pointwise stabilizer. It consists of the identity and the reflections that fix $H$. Furthermore, because unitary reflections are semisimple, $W_H$ is cyclic.

    
Now, if $e_H:=|W_H|$ is the size of this cyclic group and $t_H$ is one of its generators, the set of reflections $\mathcal{R}$ can be partitioned as:\begin{equation} \mathcal{R}=\bigcup_{H\in\mathcal{A}}\{ t_H,\cdots, t_H^{e_H-1} \}.\label{EQ: Reflection set partition into cyclis}\end{equation}
The reflection group $W$ acts on $\mathcal{A}$ determining orbits of hyperplanes which we will denote by $\mathcal{C}\in\mathcal{A}/W$. The size $\omega_{\mathcal{C}}$ of an orbit $\mathcal{C}$ is given by $\omega_{\mathcal{C}}:=[W:N_W(H)]$ (for any $H\in \mathcal{C}$). All elements $H\in\mathcal{C}$ have conjugate stabilizers $W_H$ and we write $e_{\mathcal{C}}$ for their common order. With this notation, the cardinalities of the set of reflections $\mathcal{R}$ and of the set of reflecting hyperplanes $\mathcal{A}$ are given by $$|\mathcal{R}|=\sum_{\mathcal{C}\in\mathcal{A}/W}\omega_{\mathcal{C}}(e_{\mathcal{C}}-1)\quad\quad\text{ and }\quad\quad |\mathcal{A}|=\sum_{\mathcal{C}\in\mathcal{A}/W}\omega_{\mathcal{C}}.$$
Notice that if some $e_{\mathcal{C}}\neq 2$, then $|\mathcal{R}|$ and $|\mathcal{A}|$ are not equal.

\subsubsection*{Braid groups and the full twist}

We say that a vector $v\in V$ is {\it regular} if it is not contained in any reflection hyperplane and we write $V^{\op{reg}}:=V\setminus \mathcal{A}$ for the set of regular vectors. We define the {\it pure braid group} $P(W):=\pi_1(V^{\op{reg}})$ to be the fundamental group of the regular space $V^{\op{reg}}$. It is a theorem of Steinberg that the action of $W$ on $V$ is free precisely on $V^{\op{reg}}$. 

Steinberg's theorem implies that the restriction of the quotient map $\rho: V\rightarrow V/W$ on $V^{\op{reg}}$ is a Galois covering. We define the {\it \color{blue} braid group} $B(W):=\pi_1(V^{\op{reg}}/W)$ to be the fundamental group of the base of this covering and use the following short exact sequence  to obtain a surjection $\pi:B(W)\twoheadrightarrow W$:
\begin{equation}\label{1PBW1}
1\rightarrow{\pi_1(V^{\op{reg}})}\lhook\joinrel\xrightarrow{\ \displaystyle \rho_*\ }
{\pi_1(V^{\op{reg}}/W)}
\xtwoheadrightarrow{\ \displaystyle \pi} W\rightarrow 1.
\end{equation}
Given a choice of a basepoint $x_0\in V^{\op{reg}}$, a loop $\bm b\in B(W)$ lifts to a path that connects $x_0$ to $\bm b_*(x_0)$ (we call this the {\it Galois action} of $b$). Then, we define $w:=\pi(\bm b)$ to be the {\it unique} element $w\in W$ such that $w\cdot x_0=b_*(x_0)$. 


Brou{\'e}-Malle-Rouquier considered \cite[Notation~2.3]{broue-malle-rouquier-braid-hecke} a particular element $\bm\pi$ of the pure braid group $P(W)$. It is given as the geometric circle $[0,1]\ni t\rightarrow e^{2\pi i t}\cdot x_0$.


\begin{definition}\label{Defn: The full twist} We call $\bm\pi$ the {\it\color{blue} full twist}. It is central in $B(W)$ and lies in $P(W)$ \cite[\S\,2.2]{full-paper}.
\end{definition}

Although our initial purpose for this project was to give a uniform proof of the Chapuy-Stump formula \eqref{EQ: Chapuy-Stump formula} which regards Coxeter elements, it soon became clear that the techniques developed (see \cref{Lemma: only multiples of |g| contribute}) apply to the larger class of Springer-regular elements. The crucial property these elements share is that they lift to roots of $\bm \pi$:

\begin{definition}\cite{lehrer-taylor-book} \label{Defn: regular element}
We say that an element $g\in W$ is {\color{blue} $\zeta$-regular} if it has a regular $\zeta$-eigenvector. Its order $d:=|g|$ is equal to the order of $\zeta$ and is called a {\it \color{blue} regular number}.
\end{definition}



\begin{proposition}\cite[Prop.~5.24]{broue-book-braid-groups}\label{Prop: regular elements lift to roots of the full twist}
Let $\zeta=\op{exp}(2\pi i l/d)$ be a primitive $d^{\text{th}}$ root of unity, and let $g$ be a $\zeta$-regular element of $W$. Then, $g$ has a lift $\bm g\in B(W)$ such that $\bm g^d=\bm \pi^l$.
\end{proposition}

\section{Frobenius lemma via Coxeter numbers}
\label{Section: Frobenius lemma via Coxeter numbers}

The lemma of Frobenius gives a representation theoretic formula for enumerating factorizations of group elements, when the factors belong to given conjugacy classes:

\begin{theorem}\cite[App.~A.1.3]{LZ-graphs-on-surfaces} \label{Thm: Frobenius lemma}
Let $G$ be a finite group and $A_i\subset G$, $i=1\dots l$, subsets that are closed under conjugation. Then the number of factorizations $t_1\cdots t_l=g$ of an element $g\in W$, where each factor $t_i$ belongs to $A_i$, is given by$$
\dfrac{1}{|G|}\sum_{\chi\in \widehat{G}}\chi(1)\cdot\chi(g^{-1})\cdot\dfrac{\chi(A_1)}{\chi(1)}\cdots\dfrac{\chi(A_l)}{\chi(1)},$$ where $\widehat{G}$ denotes the (complete) set of irreducible characters of $G$ and $\chi(A):=\sum_{g\in A}\chi(g)$.  
\end{theorem}

For a reflection group $W$, the set of reflections $\mathcal{R}$ is indeed closed under conjugation. If we write $\op{Fact}_{W,g}(l)$ for the cardinality of the following set of factorizations$$
\op{Fact}_{W,g}(l):=\#\{(t_1,\cdots,t_l)\in\mathcal{R}^l\ |\ t_1\cdots t_l=g\},$$ then the lemma of Frobenius implies that $$
\op{Fact}_{W,g}(l)=\dfrac{1}{|W|}\sum_{\chi\in\widehat{W}}\chi(1)\cdot\chi(g^{-1})\cdot \Big[\frac{\chi(\mathcal{R})}{\chi(1)}\Big]^l.$$ After this, the exponential generating function for reflection factorizations of $g$ equals:\begin{equation}
\op{FAC}_{W,g}(t):=\sum_{l\geq 0}\op{Fact}_{W,g}(l)\cdot\frac{t^l}{l!}=\dfrac{1}{|W|}\sum_{\chi\in\widehat{W}}\chi(1)\cdot\chi(g^{-1})\cdot \op{exp}\Big[t\cdot\frac{\chi(\mathcal{R})}{\chi(1)}\ \Big].\label{Eq: Frob formula no Cox} \end{equation}

Now, a priori the evaluations $\chi(\mathcal{R})$ are complex numbers, but the decomposition \eqref{EQ: Reflection set partition into cyclis} of the set $\mathcal{R}$ forces them to in fact be integers:

\begin{proposition}\label{Prop: bounds for chi(R)}
The numbers $\chi(\mathcal{R})$ are integers, and they further respect the tight bounds: $$-|\mathcal{A}|\cdot\chi(1)\ \leq\ \chi(\mathcal{R})\  \leq\ |\mathcal{R}|\cdot\chi(1).$$
\end{proposition}
\begin{proof}
We keep the notation from \eqref{EQ: Reflection set partition into cyclis} and choose a generator $t_H$ for each of the cyclic groups $W_H$ and write $e_H:=|W_H|$ for its order.

For each eigenvalue $\lambda$ of $t_H$ in the representation $U_{\chi}$ associated to $\chi$, the contribution of the set of reflections $\{t_H,\cdots,t_H^{e_H-1}\}$ in the evaluation of $\chi(\mathcal{R})$ equals $\sum_{k=1}^{e_H-1}\lambda^k$. Since $\lambda^{e_H}=1$, this quantity is either $e_H-1$ or $-1$ depending on whether $\lambda$ itself is $1$ or not. 

This implies the integer property as well as the inequalities, after noticing that the multiset of eigenvalues of $t_H$ acting on $U_{\chi}$ has $\chi(1)$-many elements. In particular, in order to recover the second inequality we use, after \eqref{EQ: Reflection set partition into cyclis}, that $\sum_{H\in\mathcal{A}}(e_H-1)=|\mathcal{R}|$.

For the tightness statement, the higher bound is achieved when each eigenvalue of each $t_H$ equals $1$; of course this happens only in the trivial representation. For the lower bound, we need all $\lambda\neq 1$, which happens for instance in the det representation. 
\end{proof}


The character values $\chi(\mathcal{R})$ on the sum of reflections are related to a statistic of the associated representation called the {\it Coxeter number} and denoted by $c_{\chi}$. We postpone to \cref{Section: Local Coxeter numbers and Malle's character permutation} the discussion about its origin and for now we only give the definition:

\begin{definition}\cite[\S1.3]{gordon-griffeth-catalan-numbers}\label{Defn: Coxeter numbers}
We define the Coxeter number $c_{\chi}$ associated to the character $\chi$, as the normalized trace of the central element $\sum_{t\in\mathcal{R}}(\mathbf{1}-t)$. That is, $$c_{\chi}:=\dfrac{1}{\chi(1)}\cdot\big(|\mathcal{R}|\chi(1)-\chi(\mathcal{R})\big)=|\mathcal{R}|-\dfrac{\chi(\mathcal{R})}{\chi(1)}.$$ 
\end{definition}


%For the reflection representation $\chi_V$ of $W$, the number $c_{\chi_V}$ equals $(|\mathcal{R}|+|\mathcal{A}|)/n$ and, as we saw in \S\ref{Section:Regular elements and roots of the full twist}, is called the {\it Coxeter number of $W$}. It is denoted by $h$, because when $W$ is well-generated it agrees with the order of the Coxeter element.

We record the following as an immediate corollary of \cref{Thm: Frobenius lemma}:

\begin{corollary}
The exponential generating function $\op{FAC}_{W,g}(t)$ for arbitrary length reflection factorizations of an element $g\in W$ is given by:\begin{equation}
\op{FAC}_{W,g}(t)=\dfrac{e^{t|\mathcal{R}|}}{|W|}\sum_{\chi\in\widehat{W}}\chi(1)\cdot \chi(g^{-1})\cdot e^{-t\cdot c_{\chi}}.\label{EQ: Frobenius via Coxeter numbers}\end{equation}
\end{corollary}

The following lemma is the main technical ingredient for the proof of \cref{Thm: Main not weighted}; we postpone an argument until \cref{Section: The weighted enumeration}, where we prove the more general \cref{Lemma: weighted enumeration only multiples of |g| contribute}. It relies on a cyclic action on the set $\op{Irr}(W)$ of irreducible representations of $W$ which is induced by a Galois action (see \cref{Defn: Malle's character permutations}) on the modules of the Hecke algebra $\mathcal{H}(W)$. 

\begin{lemma}\label{Lemma: only multiples of |g| contribute}
For a complex reflection group $W$, and a \textbf{regular} element $g\in W$, the total contribution in \eqref{EQ: Frobenius via Coxeter numbers} of those characters $\chi\in\widehat{W}$ for which $c_{\chi}$ is not a multiple of $|g|$ is $0$.
\end{lemma}


\begin{theorem}\label{Thm: Main not weighted}
For a complex reflection group $W$, and a regular element $g\in W$, the exponential generating function $\op{FAC}_{W,g}(t)$ of reflection factorizations of $g$ takes the following form:$$
\op{FAC}_{W,g}(t)=\dfrac{e^{t|\mathcal{R}|}}{|W|}\cdot \Big[ (1-X)^{l_R(g)}\cdot\Phi(X)\Big]\Big|_{X=e^{-t|g|}}\ .$$ Here $l_R(g)$ is the reflection length of $g$ and $\Phi(X)$ is a polynomial of degree $\frac{|\mathcal{R}|+|\mathcal{A}|}{|g|}-l_R(g)$ in $X$, which is not further divisible by $(1-X)$ and has constant term equal to $1$.
\end{theorem}
\begin{proof}
After \cref{Lemma: only multiples of |g| contribute} we only consider terms of the form $\chi(1)\cdot\chi(g^{-1})\cdot e^{-t\cdot k|g|},\,k\in\ZZ$ in the evaluation of \eqref{EQ: Frobenius via Coxeter numbers}. Furthermore, rephrasing \cref{Prop: bounds for chi(R)} in terms of the Coxeter numbers (via \cref{Defn: Coxeter numbers}) forces $k\in \{0,\dots,\frac{|\mathcal{R}|+|\mathcal{A}|}{|g|}\}$. This means that if we set $X=e^{-t|g|}$, we can rewrite \eqref{EQ: Frobenius via Coxeter numbers} as $$\op{FAC}_{W,g}(t)=\dfrac{e^{t|\mathcal{R}|}}{|W|}\cdot \tilde{\Phi}(X),$$ where $\tilde{\Phi}(X)$ is a priori a polynomial in $\CC[X]$ of degree $(|\mathcal{R}|+|\mathcal{A}|)/|g|$. From the proof of \cref{Lemma: only multiples of |g| contribute} we have that the constant term of $\tilde{\Phi}(X)$ is equal to $\chi_{\op{triv}}(1)\cdot\chi_{\op{triv}}(g^{-1})=1$.

%To see how $\tilde{\Phi}(X)$ actually has integer coefficients notice that, if we write $\tilde{\Phi}(X)=\sum_k a_k\cdot X^k$, we have after \eqref{EQ: Frobenius via Coxeter numbers} that $$a_k=\sum_{\substack{\chi\in\widehat{W}\\c_{\chi}=k|g|}}\chi(1)\chi(g^{-1}).$$ The numbers $a_k$ are therefore algebraic integers and we only need to show that $a_k\in\QQ$. Now, the coefficient of $t^r/r!$ in $\op{FAC}_{W,g}(t)$ is by definition a natural number and can be written as:$$\Bigg[\dfrac{t^r}{r!}\Bigg]\op{FAC}_{W,g}(t)=\dfrac{1}{|W|}\sum_k a_k\big( |\mathcal{R}|-k|g|\big)^r.$$
%The right hand side of this expression has to be an integer for all $r\geq 0$. 


Now, since $\tilde{\Phi}(X)$ essentially encodes the generating function $\op{FAC}_{W,g}(t)$, the combinatorial properties of the latter impose restrictions on its structure. In particular, consider the root factorization of the polynomial:$$
\tilde{\Phi}(X)=a(\alpha_1-X)(\alpha_2-X)\cdots (\alpha_r-X).$$

If we revert to $X=e^{-t|g|}$, each of the linear terms above has a Taylor expansion that starts with $(\alpha_i-1)+t|g|+\cdots$. This means that it contributes to the leading term of $\op{FAC}_{W,g}(t)$ either by a factor of $(\alpha_i-1)$ or by a factor of $t|g|$, depending on whether $\alpha_i$ equals $1$ or not.

On the other hand, the combinatorial definition of $\op{FAC}_{W,g}(t)$ in \eqref{Eq: Frob formula no Cox} implies that its leading term is a multiple of $t^{l_R(g)}$. Therefore, exactly $l_{R}(g)$-many of the roots of $\tilde{\Phi}$ must be equal to $1$ and this completes the proof. The statements about the degree and the constant term follow from the analogous results for $\tilde{\Phi}$ described previously. 
\end{proof}

\begin{remark}
In the previous argument, the existence of a reflection length and therefore the knowledge that the first few terms of the generating function $\op{FAC}_{W,g}(t)$ are zero, came for free but was very useful nonetheless. It is hoped that similar ideas might apply to other groups with natural length functions, such as $\op{GL}_n(\mathbb{F}_q)$. Moreover, one might construct special length functions to support different enumerative questions (as we pursue in \cref{Prop: transitive factorizations enum.} and in \cref{Defn: special reflection lengths n_C}).  
\end{remark}

\begin{corollary}\label{Cor: Chapuy-Stump for dn}
For a complex reflection group $W$ and a regular element $g\in W$ of order $|g|=d_n$, the exponential generating function for reflection factorizations of $g$ is given by:$$
\op{FAC}_{W,g}(t)=\dfrac{e^{t|\mathcal{R}|}}{|W|}\cdot\big(1-e^{-t|g|}\big)^{l_R(g)}.$$ 
\end{corollary}
\begin{proof}[Sketch:] This is essentially due to a theorem of Bessis \cite[Prop.~4.2]{bes-zariski} which states that when $d_n$ is regular, the minimal number of reflections needed to generate $W$ is $(|\mathcal{R}|+|\mathcal{A}|)/d_n$. The remaining details appear in \cite[Corol.~3.9]{full-paper}.
\end{proof}

\begin{remark}
When $W$ is a well-generated group and $c$ a Coxeter element of $W$, we always have $|c|=d_n$. The previous corollary therefore completes a proof of the Chapuy-Stump formula \eqref{EQ: Chapuy-Stump formula} and extends it to the groups for which $d_n$ is regular.
\end{remark}

We finish this section with an example where we can push the previous ideas slightly further by considering a different length function, the transitive factorization length:

\begin{proposition}\cite[Prop.~3.11]{full-paper} \label{Prop: transitive factorizations enum.}
The exponential generating function for transitive reflection factorizations of the regular element $g=(12\cdots n-1)(n)\in S_n$ is given by $$\op{TR-FAC}_{S_n,g}(t)=\dfrac{e^{t\binom{n}{2}}}{n!}\cdot \big( 1-e^{-t(n-1)}\big)^{n}. $$
\end{proposition}


\section{Hecke algebras and the technical lemma}
\label{Section: Hecke algebras and the technical lemma}


The following definition of Hecke algebras, which recovers the usual Iwahori-Hecke algebras when $W$ is a Coxeter group, is due to Brou\'e, Malle, and Rouquier, and was introduced in their seminal paper \cite{broue-malle-rouquier-braid-hecke}.

Let $\mathcal{C}\in\mathcal{A}/W$ denote an orbit of hyperplanes, and $e_{\mathcal{C}}$ the common order of the pointwise stabilizers $W_H$ (for $H\in\mathcal{C}$). Consider now a set of $\sum_{\mathcal{C}\in\mathcal{A}/W}e_{\mathcal{C}}$ many variables $\bm u:=(u_{\mathcal{C},j})_{(\mathcal{C}\in\mathcal{A}/W,0\leq j\leq e_{\mathcal{C}}-1)}$ and let $\ZZ[\bm u,\bm u^{-1}]$ be the Laurent polynomial ring on them. 

\begin{definition}\cite[Defn.~4.21]{broue-malle-rouquier-braid-hecke}\label{Defn: Hecke algebra}
The {\it generic Hecke algebra} $\mathcal{H}(W)$ associated to $W$ is the quotient of the group ring $\ZZ[\bm u,\bm u^{-1}]B(W)$ of the braid group, over the ideal generated by the elements of the form \begin{equation}
(\bm s-u_{\mathcal{C},0})(\bm s-u_{\mathcal{C},1})\cdots (\bm s-u_{\mathcal{C},e_{\mathcal{C}}-1}),\label{EQ: deformed order relations}\end{equation}
which we call {\it deformed order relations}. Here $\bm s$ runs over all possible braid reflections (these are a set of topological generators of $B(W)$ \cite[\S\,2.1]{full-paper}) around the stratum $\mathcal{C}$ of $\mathcal{H}$. 
\end{definition}

Brou{\'e}, Malle, Rouquier also made various conjectures about these Hecke algebras, the most important of which was until recently known as \textbf{\textit{``The BMR freeness conjecture"}}:


\begin{theorem*}\cite[after Thm.~3.5]{chlouveraki-et-al-arxiv-BMR}
The algebra $\mathcal{H}(W)$ is a free $\ZZ[\bm u,\bm u^{-1}]$-module of rank $|W|$.
\end{theorem*} 


The Hecke algebra is by construction a deformation of the group algebra of $W$.  Indeed, for $\zeta_n:=\op{exp}(2\pi i /n)$, the specialization  $\sigma:\ u_{\mathcal{C},j}\rightarrow \zeta_{e_{\mathcal{C}}}^j$ turns the defining relations \eqref{EQ: deformed order relations} to order relations of the form $\bm s^{e_{\mathcal{C}}}=1$, where we have $W\cong B(W)/\langle \bm s^{e_{\mathcal{C}}}\rangle$, \cite[\S\,2.1]{full-paper}. 

Any ring map $\theta:\ZZ[\bm u,\bm u^{-1}]\rightarrow R$ defines an $R$-module structure on the Hecke algebra and will be called a {\it specialization} of it. We will further say that $\theta$ is an {\it\color{blue} admissible} specialization if it factors through $\sigma$ (i.e. if there is a map $f$ such that $f\circ\theta(u_{\mathcal{C},j})=\zeta_{e_{\mathcal{C}}}^j$).


Two particular specializations are fundamental in what follows. We first pick a set of parameters $\bm x:=(x_{\mathcal{C}})_{\mathcal{C}\in\mathcal{A}/W}$ and the single parameter $x$ and define the ring maps:\begin{align}
\theta_{\bm x}:\ZZ[\bm u,\bm u^{-1}]&\rightarrow \ZZ[\bm x,\bm x^{-1}] \quad\quad \quad\quad\text{ and }& \theta_x:\ZZ[\bm u,\bm u^{-1}]&\rightarrow \ZZ[x,x^{-1}]\nonumber \\
\theta_{\bm x}(u_{\mathcal{C},j})&=\begin{cases} x^{\phantom{N_W}}_{\mathcal{C}}&\text{if }j=0\\ \zeta_{e_{\mathcal{C}}}^j&\text{if }j\neq 0\end{cases}  &\theta_x(u_{\mathcal{C},j})&=\begin{cases} x&\text{if }j=0\\ \zeta_{e_{\mathcal{C}}}^j&\text{if }j\neq 0\end{cases} \label{EQ: admissible specializations}
\end{align}

Both $\theta_{\bm x}$ and $\theta_x$ are admissible specializations (as seen by further sending $x_{\mathcal{C}}$ or $x$ to $1$). We write $\mathcal{H}_{\bm x}(W)$ and $\mathcal{H}_x(W)$ for the corresponding Hecke algebras; by Tits' deformation theorem \cite[\S\,4.1]{full-paper} (and because of the BMR-freeness theorem) there is a spectra-preserving bijection between their representations and those of $W$. Therefore, given a character $\chi$ of $W$, we will write $\chi_{\bm x}$ and $\chi_x$ for the corresponding characters of $\mathcal{H}_{\bm x}(W)$ and $\mathcal{H}_x(W)$ respectively.

\subsection{Local Coxeter numbers and Malle's character permutation}
\label{Section: Local Coxeter numbers and Malle's character permutation}

We are now going to define a local version of Coxeter numbers (see \cref{Defn: Coxeter numbers}) and study how they naturally appear in our key technical lemma (\cref{Prop: The key technical lemma}). 


\begin{definition}\label{Defn: Local Coxeter numbers} 
We define the {\it local} Coxeter number $c_{\chi_{,\mathcal{C}}}$ associated to the character $\chi$ and the hyperplane orbit $\mathcal{C}\in\mathcal{A}/W$, as the normalized trace $$c_{\chi_{,\mathcal{C}}}:=\dfrac{1}{\chi(1)}\cdot\chi\Big( \sum_{V^{t}\in\mathcal{C}}(\bm 1-t)\Big).$$ Here, the sum is taken over all reflections $t$ whose fixed hyperplane belongs to the orbit $\mathcal{C}$. Notice that these numbers refine the Coxeter numbers in the sense that $c_{\chi}=\sum c_{\chi_{,\mathcal{C}}}$.
\end{definition}


The following statement is proved similarly to \cref{Prop: bounds for chi(R)} (see \cite[Corol.~4.16]{full-paper}). The integer property is \cite[Corol.~4.17]{broue-michel-french-paper}.

\begin{proposition}\label{Corol: local cox num are integers andbounds}
The numbers $c_{\chi_{,\mathcal{C}}}$ are integers and they satisfy $\quad\displaystyle 0\leq c_{\chi_{,\mathcal{C}}}\leq e_{\mathcal{C}}\cdot\omega_{\mathcal{C}}$.
\end{proposition}

The local Coxeter numbers $c_{\chi_{,\mathcal{C}}}$ appear in the following character calculation from \cite{broue-michel-french-paper} (and we have reinterpreted their formulas, but one can find more details in \cite[\S\,4.2-4.3]{full-paper}). Let $\bm\pi$ be the full twist as in \cref{Defn: The full twist} and $w$ a regular element. By \cref{Prop: regular elements lift to roots of the full twist} $w$ has a lift $\bm w$ in $B(W)$, that satisfies $\bm w^d=\bm\pi ^l$ for suitable $l$ and $d$. Then, for the characters $\chi_{\bm x}$ of $\mathcal{H}_{\bm x}(W)$ (see after \eqref{EQ: admissible specializations}), and the elements $T_{\bm \pi}$ and $T_{\bm w}$ of the Hecke algebra we have: \begin{equation}
\chi_{\bm x}(T_{\bm \pi})=\chi(1)\prod_{\mathcal{C}\in\mathcal{A}/W}x_{\mathcal{C}}^{e_{\mathcal{C}}\omega_{\mathcal{C}}-c_{\chi_{,\mathcal{C}}}}\quad\text{ and }\quad\chi_{\bm x}(T_{\bm w})=\chi(w)\prod_{\mathcal{C}\in\mathcal{A}/W}x_{\mathcal{C}}^{(e_{\mathcal{C}}\omega_{\mathcal{C}}-c_{\chi_{,\mathcal{C}}})l/d}.\label{EQ: character values via local Coxeter}\end{equation}

Moreover, after the further specialization $x_{\mathcal{C}}\rightarrow x$ of $\theta_x$ from \eqref{EQ: admissible specializations} and for the characters $\chi_x$ of $\mathcal{H}_x(W)$, we have (recalling that $\sum e_{\mathcal{C}}\omega_{\mathcal{C}}=|\mathcal{R}|+|\mathcal{A}|$):
\begin{equation}
\chi_x(T_{\bm \pi})=\chi(1)\cdot x^{|\mathcal{R}|+|\mathcal{A}|-c_{\chi}}\quad\text{ and }\quad\chi_x(T_{\bm w})=\chi(w)\cdot x^{(|\mathcal{R}|+|\mathcal{A}|-c_{\chi})l/d}.\label{EQ: character values via Coxeter-not local}\end{equation}


%\subsection{Malle's character permutations and the technical lemma}
%\label{Section: Malle's character permutations and the technical lemma}

\subsubsection*{Malle's permutation is a Galois action on the characters}


The fake degree $P_{\chi}(q):=\sum q^{e_i(\chi)}$ of a character $\chi\in\widehat{W}$ is a polynomial that records its exponents $e_i(\chi)$ (see \cite[\S4.4]{lehrer-taylor-book}). Beynon and Lusztig \cite[Prop.~A]{beynon-lustzig-coxeter-numbers-genesis} had observed a remarkable reciprocity property for these polynomials. They satisfy $$P_{\chi}(q)=q^{c_{\chi}}P_{\iota(\chi)}(q^{-1}),$$where $c_{\chi}$ is the Coxeter number as given in \cref{Defn: Coxeter numbers} and $\iota$ is a permutation on $\op{Irr}(W)$ that for Weyl groups is the identity apart from two characters of $E_7$ and four of $E_8$. 

Malle later on \cite[Thm.~6.5]{malle-fake-degrees} extended this reciprocity result for all complex reflection groups, defining a permutation $\Psi$ on $\op{Irr}(W)$ that is induced by a Galois action on the irreducible characters of the Hecke algebra (the two permutations satisfy $\iota(\chi)=\Psi(\chi^*)$). This permutation of Malle  is the missing ingredient for \cref{Lemma: only multiples of |g| contribute}; the characters $\chi$ for which $|g|$ does not divide $c_{\chi}$ are grouped together by $\Psi$ and their contributions cancel.

Recall from \eqref{EQ: admissible specializations} the specializations of the Hecke algebra $\mathcal{H}_{\bm x}(W)$ and $\mathcal{H}_x(W)$. It is a theorem of Malle \cite[Thm.~5.2]{malle-fake-degrees} that one can find split extensions $\CC(\bm y)/ \CC(\bm x)$ and $\CC(y)/\CC(x)$ of the coefficient fields, by introducing parameters $\bm y:=(y_{\mathcal{C}})_{\mathcal{C}\in\mathcal{A}/W}$ and $y$, which satisfy $y_{\mathcal{C}}^{N_W}=x_{\mathcal{C}}$ and $y^{N_W}=x$ for some number $N_W$. This means that all representations of $\mathcal{H}_{\bm x}(W)$ and $\mathcal{H}_x(W)$ are realizable over $\CC(\bm y)$ and $\CC(y)$ respectively.

\begin{definition}\label{Defn: Malle's character permutations}
We consider the permutations $\Psi_{\mathcal{C}}$ and $\Psi$ on the sets $\op{Irr}(\mathcal{H}_{\bm x}(W))$ and $\op{Irr}(\mathcal{H}_x(W))$ that are induced by the Galois automorphisms $\Phi_{\mathcal{C}}$ (for $\mathcal{C}\in\mathcal{A}/W$) and $\Phi$:\begin{align*}
\Phi_{\mathcal{C}}&\in\op{Gal}\big(K(\bm y)/K(\bm x)\big) &\Phi &\in\op{Gal}\big(K(y)/K(x)\big)    \\
y_{\mathcal{C}}&\rightarrow e^{2\pi i/N_W}\cdot y_{\mathcal{C}} & y &\rightarrow e^{2\pi i/N_W}\cdot y
\end{align*} 
In particular, they are defined via $\Psi_{\mathcal{C}}(\chi_{\bm y})(T_{\bm g}):=\Phi_{\mathcal{C}}\big( \chi_{\bm y}(T_{\bm g})\big)$ and similarly for $\Psi$. By Tits' deformation theorem, they induce permutations on the set $\widehat{W}$ of irreducible characters of $W$, which we also denote by $\Psi_{\mathcal{C}}$ and $\Psi$.
\end{definition}

The following lemma is an almost immediate application of the character formulas \eqref{EQ: character values via local Coxeter} and \eqref{EQ: character values via Coxeter-not local}, in conjunction with Tits' deformation theorem, and \cref{Defn: Malle's character permutations}: 

\begin{proposition}[The key technical lemma]\cite[Prop.~4.19]{full-paper}\label{Prop: The key technical lemma}\ \newline
Let $g$ be a $\zeta$-regular element of $W$, with $\zeta=e^{2\pi il/d}$ of order $d$, $\chi\in\widehat{W}$ an irreducible character, and $\mathcal{C}\in\mathcal{A}/W$ an orbit of hyperplanes. Then, we have $$\Psi_{\mathcal{C}}(\chi)(g)=\op{exp}\big(-2\pi i \cdot \frac{lc_{\chi_{,\mathcal{C}}}}{d}\big) \cdot \chi(g) \quad\text{and}\quad\Psi(\chi)(g)=\op{exp}\big(-2\pi i \cdot \frac{lc_{\chi}}{d}\big) \cdot \chi(g).$$
\end{proposition}


\section{The weighted enumeration}
\label{Section: The weighted enumeration}

The following section studies the weighted enumeration of reflection factorizations as considered in \cite{reiner-delMas-Hameister}. It provides a uniform proof of their result and extends it in a similar direction as with the Chapuy-Stump formula \eqref{EQ: Chapuy-Stump formula}.

\begin{definition}\label{Defn: weight function}
Consider a set of variables $\bm w:=(w_{\mathcal{C}})_{(\mathcal{C}\in\mathcal{A}/W)}$ and a weight function $$\op{wt}:\mathcal{R}\rightarrow\{ w_{\mathcal{C}}\ |\ \mathcal{C}\in\mathcal{A}/W\},$$ \nopagebreak such that $\op{wt}(t)=w_{\mathcal{C}}$ if $\mathcal{C}$ is the orbit that contains the fixed hyperplane $V^t$. Then, for a given element $g\in W$, we count its weighted reflection factorizations via:
$$\op{FAC}_{W,g}(\bm w,z):=\sum_{\substack{(t_1,\cdots,t_N)\in\mathcal{R}^N \\t_1\cdots t_N=g}}\op{wt}(t_1)\cdots\op{wt}(t_N)\cdot \dfrac{z^N}{N!}. $$
\end{definition}

Because the sets $\mathcal{C}^{\op{ref}}:=\{t\in\mathcal{R}\ |\ V^t\in\mathcal{C}\}$ are closed under conjugation, the Lemma of Frobenius can again be used to express $\op{FAC}_{W,g}(\bm w,z)$ as a finite sum of exponentials. Assuming that there are $r=|\mathcal{A}/W|$ different orbits of hyperplanes, denoted $\mathcal{C}_1,\cdots, \mathcal{C}_r$, \cref{Thm: Frobenius lemma} implies that $$ 
\op{FAC}_{W,g}(\bm w,z)=\dfrac{1}{|W|}\sum_{\chi\in\widehat{W}}\chi(1)\cdot\chi(g^{-1})\cdot\op{exp}\Big[ zw_{\mathcal{C}_1}^{\phantom{l_1}}\cdot \dfrac{\chi(\mathcal{C}_1^{\op{ref}})}{\chi(1)}\Big]\cdots \op{exp}\Big[ zw_{\mathcal{C}_r}^{\phantom{l_r}}\cdot \dfrac{\chi(\mathcal{C}_1^{\op{ref}})}{\chi(1)}\Big].$$
By \cref{Defn: Local Coxeter numbers} we can rewrite the quantities in the exponentials in terms of local Coxeter numbers. Indeed, we have $c_{\chi_{,\mathcal{C}}}=|\mathcal{C}^{\op{ref}}|-\chi(\mathcal{C}^{\op{ref}})/\chi(1)$ and if we define $\op{wt}(\mathcal{R}):=\sum_{t\in\mathcal{R}}\op{wt}(t)$, the previous expression becomes a direct analog of \eqref{EQ: Frobenius via Coxeter numbers} :\begin{equation}
\op{FAC}_{W,g}(\bm w,z)=\dfrac{e^{z\cdot\op{wt}(\mathcal{R})}}{|W|}\sum_{\chi\in\widehat{W}}\chi(1)\cdot\chi(g^{-1})\cdot \big(e^{-zw_{\mathcal{C}_1}^{\phantom{l_1}}}\big)^{\displaystyle c_{\chi_{,\mathcal{C}_1}}}\cdots \big( e^{-zw_{\mathcal{C}_r}^{\phantom{l_r}}}\big)^{\displaystyle c_{\chi_{,\mathcal{C}_r}}}.\label{EQ: weighted enum via local Coxeter numbers}
\end{equation}



\begin{lemma}\label{Lemma: weighted enumeration only multiples of |g| contribute}
For a complex reflection group $W$, and a \textbf{regular} element $g\in W$, the total contribution in \eqref{EQ: weighted enum via local Coxeter numbers} of those characters $\chi\in\widehat{W}$ for which \textbf{any} $c_{\chi_{,\mathcal{C}}}$ is not a multiple of $|g|$ is $0$.
\end{lemma}
\begin{proof}
We have to start by ordering the orbits $\mathcal{C}\in\mathcal{A}/W$ (arbitrarily); we will then apply the following idea sequentially: We partition the set of irreducible characters $\chi\in\widehat{W}$ into orbits under the action of $\Psi_{\mathcal{C}_1}$. Pick a character $\chi$ whose orbit is {\it not} a singleton and let $k$ be the smallest number such that $\Psi_{\mathcal{C}_1}^k(\chi)=\chi$ (by \cref{Prop: The key technical lemma}, we will have $k=\frac{|g|}{\op{gcd}(c_{\chi_{,\mathcal{C}_1}},|g|)}).$ Now, the degrees of characters and the (local) Coxeter numbers are respected by $\Psi_{\mathcal{C}_1}$ (see \cite[Prop.~4.18]{full-paper}), and so it is enough to show that $$\sum_{j=1}^k\Psi^j_{\mathcal{C}_1}(\chi)(g^{-1})=0.$$ Indeed, this follows immediately from \cref{Prop: The key technical lemma} as $\Psi_{\mathcal{C}_1}^j(\chi)(g^{-1})=\xi^j\chi(g^{-1})$ for some $k^{\text{th}}$ root of unity $\xi$. Notice now that we can continue with the {\it remaining} characters and the orbit $\mathcal{C}_2$ without worrying that we might eventually cancel the same character twice.
\end{proof}

Before we proceed with our structural result for weighted enumeration formulas, we introduce the following combinatorial generalizations of the length function $l_R(g)$:

\begin{definition}\label{Defn: special reflection lengths n_C}
For an arbitrary element $g\in W$ and an orbit $\mathcal{C}\in\mathcal{A}/W$, we define $n_{\mathcal{C}}(g)$ to be the smallest number of reflections in $\mathcal{C}^{\op{ref}}$ that may appear in {\it any} reflection factorization of $g$ (i.e. not necessarily reduced).
\end{definition}

\begin{remark}
Notice that it is not always true that $\sum n_{\mathcal{C}}(g)=l_R(g)$. Indeed, the element $g:=(12\bar{1}\bar{2})=-\bm 1$ in $B_2$ (which is the square of the Coxeter element) can be written both as $g=(12)(1\bar{2})$ and as $g=(1\bar{1})(2\bar{2})$, so that $n_1(g)=n_2(g)=0$.
\end{remark}

\begin{theorem}\label{Thm: Main weighted}
For a complex reflection group $W$ and a regular element $g\in W$, the exponential generating function $\op{FAC}_{W,g}(\bm w,z)$ of weighted reflection factorizations of $g$ takes the form:$$
\op{FAC}_{W,g}(\bm w,z)=\dfrac{e^{z\cdot\op{wt}(\mathcal{R})}}{|W|}\cdot \Big[ \Phi({\bm X})\cdot\prod_{\mathcal{C}\in\mathcal{A}/W} (1-X_\mathcal{C})^{n^{\phantom{l}}_{\mathcal{C}}(g)}\Big]\Big|_{X_{\mathcal{C}}=e^{-zw_{\mathcal{C}}|g|}}.$$ Here, $\Phi({\bm X})$ is a polynomial of degree $(e_{\mathcal{C}}\cdot \omega_{\mathcal{C}})/|g|-n_{\mathcal{C}}(g)$ on each of its variables $X_{\mathcal{C}}$, it has constant term $\Phi(\bm 0)=1$, and it is not further divisible by any $(1-X_{\mathcal{C}})$. The exponents satisfy $$\dfrac{e_{\mathcal{C}}\omega_{\mathcal{C}}}{|g|}\ \geq\  n_{\mathcal{C}}(g)\ \geq\  l_R(g)-\dfrac{|\mathcal{R}|+|\mathcal{A}|-e_{\mathcal{C}}\omega_{\mathcal{C}}}{|g|}.$$
\end{theorem}
\begin{proof}[Remark:]
The proof \cite[Thm.~5.5]{full-paper} is very similar to that of \cref{Thm: Main not weighted}. One significant difference is that at some point it becomes necessary to write a generating function for factorizations (of arbitrary length) that have precisely $n_{\mathcal{C}}$ reflections from the orbit $\mathcal{C}$. This is done again through the Lemma of Frobenius.
\end{proof}

\begin{corollary}
For a complex reflection group $W$ and a regular element $g\in W$ of order $|g|=d_n$, the weighted reflection factorizations of $g$ are counted by the formula:
$$\op{FAC}_{W,g}(\bm w,z)=\dfrac{e^{z\cdot\op{wt}(\mathcal{R})}}{|W|}\cdot\prod_{\mathcal{C}\in\mathcal{A}/W}\big(1-e^{-zw_{\mathcal{C}}^{\phantom{l_r}}\!|g|}\big)^{n_\mathcal{C}(g)},$$
where the exponents are explicitly given by $n_{\mathcal{C}}(g)=(e_{\mathcal{C}}\omega_{\mathcal{C}})/|g|$.
\end{corollary}
\begin{proof}[Remark:]
This is essentially the same argument as in \cref{Cor: Chapuy-Stump for dn}.
\end{proof}

\begin{remark}
For well-generated groups $W$, we always have $|c|=d_n$ so that the previous statement recovers the main theorem of \cite{reiner-delMas-Hameister} and extends it to the groups with $d_n$ regular. Notice that while in well-generated groups we have at most two orbits of hyperplanes, the groups $G_{7},G_{11},G_{15},G_{19}$ have three orbits. For all of them but $G_{15}$, $d_n$ is regular.
\end{remark}


\acknowledgements{
We would like to thank Jean Michel, Maria Chlouveraki, Christian Stump, Guillaume Chapuy, and Ivan Marin, for fascinating discussions on this topic.}

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