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%%%%% Auteur

%%%1
\author{\firstname{Christian} \lastname{Gaetz}}

\address{Department of Mathematics\\ 
Massachusetts Institute of Technology\\ 
Cambridge, MA\\
USA}

\email{gaetz@mit.edu} 

\thanks{C.~G. was partially supported by an NSF Graduate Research Fellowship under grant no. 1122374.}

%%%2
\author{\firstname{Yibo} \lastname{Gao}}

\address{Department of Mathematics\\ 
Massachusetts Institute of Technology\\ 
Cambridge, MA\\
USA}

\email{gaoyibo@mit.edu}


%%%%% Sujet

\keywords{Absolute order, Sperner property, antichain, normalized flow, reflection group.}

\subjclass{20F55, 06A11, 06A07}


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\DOI{10.5802/alco.114}
\datereceived{2019-10-01}
\daterevised{2020-02-06}
\dateaccepted{2020-02-12}

%%%%% Titre et résumé

\title[On the Sperner property for the absolute order]
{On the Sperner property for the absolute order on complex reflection groups}

\begin{abstract}
Two partial orders on a reflection group $W$, the \emph{codimension order} and the \emph{prefix order}, are together called the \emph{absolute order} $\abs(W)$ when they agree. We show that in this case the absolute order on a complex reflection group has the strong Sperner property, except possibly for the Coxeter group of type $D_n$, for which this property is conjectural. The Sperner property had previously been established for the \emph{noncrossing partition lattice $NC_W$}~\cite{Muhle, Reiner}, a certain maximal interval in $\abs(W)$, but not for the entire poset, except in the case of the symmetric group~\cite{Harper2019}. We also show that neither the codimension order nor the prefix order has the Sperner property for general complex reflection groups.
\end{abstract}

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\begin{document}

\maketitle

\section{Introduction} \label{sec:intro}

A ranked poset $P$ with rank decomposition $P_0 \sqcup P_1 \sqcup \cdots \sqcup P_r$ is \emph{$k$-Sperner} if no union of $k$ antichains of $P$ is larger than the union of the largest $k$ ranks of $P$ (see~\cite{Engel} for an introduction to Sperner theory). It is \emph{strongly Sperner} if it is $k$-Sperner for $k=1,2,\ldots ,r+1$. 

The \emph{absolute order} on a Coxeter group $W$ has two equivalent descriptions: one in terms of the reflection lengths of its elements, and the other in terms of the codimensions of their fixed spaces. The maximal intervals $[\id, c]$ in $\abs(W)$, where $c$ is a \emph{Coxeter element}, are known as the \emph{noncrossing partition lattices} $NC_W$. They appeared in work of Brady and Watt~\cite{Brady} on $K(\pi,1)$'s for Artin braid groups and have been studied combinatorially by Reiner~\cite{Reiner} and Armstrong~\cite{Armstrong}, among others. The posets $NC_W$ are known to be strongly Sperner~\cite{Muhle, Reiner}; this paper follows recent work of Harper and Kim~\cite{Harper2019} in studying the problem of whether the whole absolute order $\abs(W)$ is strongly Sperner. 

We choose to work in the setting of general complex reflection groups, rather than just Coxeter groups. In this generality, the two orders (the \emph{prefix} and \emph{codimension} orders) do not always agree. We reserve the term \emph{absolute order} for the situation when they do agree (these cases have been classified by Foster-Greenwood~\cite{Foster-Greenwood}, see Proposition~\ref{prop:codim-equals-prefix}). It is a common feature of work in this area that the two orders are better-behaved when they agree; and Theorem~\ref{thm:main-theorem} and Conjecture~\ref{conj:type-D} below make the case that this is true in regard to the Sperner property.

Our main result follows; see Section~\ref{sec:background} for background and definitions, Section~\ref{sec:proof} for the proof, and Section~\ref{sec:counterexamples} for examples showing that the strong Sperner property does not hold in general for either the prefix order or codimension order when they disagree. Conjecture~\ref{conj:type-D} is discussed in Section~\ref{sec:type-D}.

\begin{theorem} \label{thm:main-theorem}
Let $W=W_1 \times \cdots \times W_k$ be a finite complex reflection group, where each $W_i$ is in the family $G(m,1,n)$ or is an irreducible Coxeter group of type other than type $D$; then $\abs(W)$ is strongly Sperner.
\end{theorem}

\begin{remark}
Theorem~\ref{thm:main-theorem} in the case where all $W_i$ are symmetric groups follows from a result of Harper and Kim~\cite{Harper2019}. The type $B$ case was proven independently by Harper, Kim, and Livesay~\cite{Harper2019b} while this paper was in preparation.
\end{remark}

\begin{conj} \label{conj:type-D} \text{}
\begin{enumerate}[(\alph*)] 
 \item %[\normalfont{(a)}] 
 Let $W$ be the Coxeter group of type $D_n$, then $\abs(W)$ has a normalized flow with $\nu \equiv 1$ (see Section~\ref{sec:normalized-flow}). And, therefore,
 \item %[\normalfont{(b)}] 
 $\abs(W)$ is strongly Sperner for any finite complex reflection group $W$ such that the prefix order $\pre(W)$ is equal to the codimension order $\co(W)$.
\end{enumerate}
\end{conj}

\section{Background and definitions} \label{sec:background}
\subsection{Complex reflection groups} \label{sec:reflection-groups}

For $V$ an $n$-dimensional complex vector space, a finite group $W \subset GL(V)$ is a \emph{complex reflection group} of \emph{rank $n$} if it is generated by its set of \emph{reflections} $T=\{w \in W \: | \: \dim(V^w)=n-1\}$, where $V^w$ denotes the fixed subspace of $w$. We say $W$ is \emph{irreducible} if $V$ is irreducible as a representation of $W$. Any finite reflection group is a product of irreducible reflection groups. The finite irreducible complex reflection groups were famously classified by Shephard and Todd~\cite{Shephard}. They consist of the following groups:
\begin{itemize}
 \item A 3-parameter infinite family of groups $G(m,p,n)$ where $p | m$ and $n,m \geq 1$, with the exception that $G(2,2,2)$ is reducible. $G(m,p,n)$ has rank $n$ except when $m=1$, in which case $G(1,1,n)\cong S_n$ has rank $n-1$.
 \item 34 exceptional groups usually denoted $G_4,G_5,...,G_{37}$.
\end{itemize}

Among these, the groups which can be realized over a \emph{real} vector space $V$ are exactly the finite Coxeter groups. These too have a familiar classification into Cartan--Killing types:
\begin{itemize}
 \item Type $A_{n-1}$: the symmetric groups $S_n = G(1,1,n)$,
 \item Type $B_n$: the hyperoctahedral groups $(\Z/2 \Z) \wr S_n =G(2,1,n)$,
 \item Type $D_n$: the groups $G(2,2,n)$, index-2 subgroups of the hyperoctahedral groups,
 \item Type $I_2(m)$: the dihedral groups $G(m,m,2)$, and
 \item The exceptional finite Coxeter groups of types $H_3, H_4, F_4, E_6, E_7, E_8$ (which coincide with $G_{23},G_{30},G_{28},G_{35},G_{36}$ and $G_{37}$ respectively).
\end{itemize}

\subsection{The absolute order on a reflection group} \label{sec:absolute-order}

Given $W$ a reflection group, the \emph{reflection length} $\ell_R(w)$ of an element $w \in W$ is defined to be the smallest $k$ such that $w=t_1 \cdots t_k$ with each $t_i \in T$, where $T$ denotes the set of all reflections in $W$. In this case, we say $t_1\cdots t_k$ is a \textit{reduced word} or \textit{reduced decomposition} for $w$. The \emph{prefix order} $\pre(W)$ is the partial order on $W$ such that $u \leq v$ if and only if
\begin{equation} \label{eq:prefix-order}
\ell_R(u)+\ell_R(u^{-1}v) = \ell_R (v).
\end{equation}
$\pre(W)$ is a ranked poset with rank function $\ell_R$.

\begin{remark}
The prefix order should not be confused with the \emph{weak order} on $W$ when $W$ is a Coxeter group. The weak order is defined by an equation similar to~\eqref{eq:prefix-order}, but with the reflection length $\ell_R$ replaced by the more classical length $\ell$, which records the shortest decomposition of an element of $W$ as a product of \emph{simple} reflections. The strong Sperner property for the weak order in type $A_n$ was previously established by the authors~\cite{weak-order-sperner}. Another related order, the (strong) Bruhat order, is known to be strongly Sperner for all Coxeter groups by work of Stanley~\cite{Stanley1980}.
\end{remark}

The \emph{codimension order} $\co(W)$ is defined so that $u \leq v$ if and only if 
\begin{equation} \label{eq:codimension-order}
\codim(V^u)+\codim(V^{u^{-1}v})=\codim(V^v),
\end{equation}
where $V^w$ denotes the subspace of $V$ fixed by the action of $w$ given by the inclusion $W \subset GL(V)$.

\looseness-1
It was first proven by Carter~\cite{Carter} that when $W$ is a Coxeter group, we have $\ell_R(w)=\codim(V^w)$ for all $w \in W$, so that in particular $\co(W)=\pre(W)$. Foster-Greenwood has classified the complex reflection groups for which this coincidence continues to hold:

\begin{prop}[Foster-Greenwood~\cite{Foster-Greenwood}] \label{prop:codim-equals-prefix}
For $W$ an irreducible complex reflection group, $\co(W)=\pre(W)$ if and only if $W$ is a Coxeter group or is in the family $G(m,1,n)$.
\end{prop}

We follow Huang, Lewis, and Reiner's convention~\cite{Huang} by using the term \emph{absolute order} to refer to the codimension and prefix orders when they agree (in other parts of the literature, ``absolute order'' is used to refer to what we call the prefix order).

The functions $\ell_R(w)$ and $\codim(V^{w})$ are both \emph{subadditive}; this means that for any $u,v \in W$
\begin{align} \label{eq:subadditive}
\ell_R(uv) &\leq \ell_R(u) + \ell_R(v), \\
\codim(V^{uv}) & \leq \codim(V^u) + \codim(V^v).
\end{align}

Now, for a reducible reflection group $W \times W' \subset GL(V \oplus V')$ it is clear that $\ell_R((w,w'))=\ell_R(w)+\ell_R(w')$ and $\codim((V \oplus V')^{(w,w')})=\codim(V^w)+\codim(V^{w'})$. This implies that for either the prefix or codimension order, if $u \leq v$ and $u' \leq v'$ then $(u,u') \leq (v,v')$. The reverse implication then follows by subadditivity. Together these facts imply:

\begin{prop} \label{prop:product-of-groups-has-product-order}
Let $W \times W' \subset GL(V \oplus V')$ be a reducible reflection group. Then $\pre(W \times W') \cong \pre(W) \times \pre(W')$ and $\co(W \times W') \cong \co(W) \times \co(W')$.
\end{prop}

\subsection{Rank generating functions} \label{subsec:rank-functions}
For $P$ a finite ranked poset, we let 
\[
F(P,q)=\sum_{i=0}^{\rank(P)} |P_i| \cdot q^i
\]
denote the rank generating function of $P$. It is known that for any complex reflection group $W$ the codimension generating function
\begin{equation} \label{eq:codimension-generating-function}
C(W,q)=\sum_{i=0}^{\rank(W)} |\{w \in W | \codim(V^w)=i\}| \cdot q^i
\end{equation}
is equal to $(1+e_1q)\cdots (1+e_nq)$ where the $e_i$'s are positive integer invariants of $W$ called the \emph{exponents} (see Solomon~\cite{Solomon} for a uniform proof). For general complex reflection groups, the rank of $w$ in $\co(W)$ is not necessarily equal to $\codim(V^w)$ (for example, Foster-Greenwood~\cite{Foster-Greenwood} gives examples of elements in rank one of $\co(W)$ with fixed-space codimension two), so that $C(W,q) \neq F(\co(W),q)$ in general. However, $\pre(W)$ is always ranked by $\ell_R$, and thus when $\pre(W)=\co(W)$ we have 
\begin{equation} \label{eq:rank-generating-function}
F(\abs(W),q)=\prod_{i=1}^{n} \left(1+e_iq \right).
\end{equation}
This fact demonstrates the common theme that both $\pre(W)$ and $\co(W)$ are more tractable when they agree.

\subsection{The normalized flow property} \label{sec:normalized-flow}
The main tool that we will be using to establish the Sperner property of a poset is the theory of normalized flows, developed by Harper~\cite{Harper1974}, and we will be mainly following his notation in this section. 

Let $G=(V=A \sqcup B,E)$ be a bipartite graph, equipped with a weight function $\nu:V\rightarrow\R_{\geq0}$. We consider $\nu$ as a measure. Namely, for any subset $ X \subset V$, let $\nu(X):=\sum_{x\in X}\nu(x)$. A \textit{normalized flow} on $G$, with respect to $\nu$, is a map $f:E\rightarrow \R_{\geq0}$ defined on the set of edges of $G$, such that for any $a\in A$ we have 
\[
\sum_{b\in D(a)}f(a,b)=\nu(a)/\nu(A),
\]
and for any $b\in B$ we have 
\[
\sum_{a\in D(b)}f(a,b)=\nu(b)/\nu(B),
\] 
where $D(x)$ denote the set of neighbors of $x$. 

Now let $P$ be a ranked poset with rank decomposition $P_0 \sqcup P_1 \sqcup \cdots \sqcup P_r$, with a weight function $\nu:P\rightarrow\R_{\geq0}$. We say that $f:E\rightarrow\R_{\geq0}$ is a \textit{normalized flow} on $P$ with respect to $\nu$, if the restriction of $f$ to the bipartite graph consisting of $P_i$ and $P_{i+1}$ and the covering relations between them is a normalized flow for each $i$. Normalized flows will be useful to us thanks to the following theorem:

\begin{theorem}[Corollary to Theorem III of~\cite{Harper1974}]\label{thm:nfsperner}
If a ranked poset $P$ has a normalized flow with respect to the weights $\nu \equiv 1$, then $P$ is strongly Sperner.
\end{theorem}
An important advantage of using normalized flows is that they behave well under product. Let $P$ and $Q$ be two ranked posets with weight functions $\nu_P$ and $\nu_Q$ respectively. Their \textit{(Cartesian) product} $P\times Q$ is $\{(p,q):p\in P,\ q\in Q\}$ where the partial order is defined as $(p,q)\leq (p',q')$ if $p\leq p'$ in $P$ and $q\leq q'$ in $Q$, with a weight function $\nu_{P\times Q}((p,q))=\nu_P(p)\cdot \nu_Q(q)$. We say that a ranked poset $P$ with rank decomposition $P_0 \sqcup P_1 \sqcup \cdots \sqcup P_r$ is \textit{log-concave} with respect to a weight function $\nu$, if $\nu(P_i)^2\geq \nu(P_{i-1})\nu(P_{i+1})$ for all $i$.
\begin{theorem}[Theorem I.C of~\cite{Harper1974}]\label{thm:nfproduct}
Let $P$ and $Q$ be two ranked posets that are log-concave with respect to weight functions $\nu_P$ and $\nu_Q$. If both of them have normalized flows, then their product also has a normalized flow and is log-concave.
\end{theorem}

Another useful property of normalized flow is described as ``the Fundamental Lemma'' by Harper~\cite{Harper1974}; we reformulate it here: 

\begin{theorem}[Lemma I.B of~\cite{Harper1974}]\label{thm:fundlemma}
Let $\ph:P\rightarrow Q$, with weight functions $\nu_P$ on $P$ and $\nu_Q$ on $Q$, be a surjective, measure-preserving and rank-preserving map of ranked posets. If $Q$ admits a normalized flow with weights $\nu_Q$ and for each covering relation $q\lessdot q'$ on $Q$, the induced bipartite graph on $\ph^{-1}(\{q\})$ and $\ph^{-1}(\{q'\})$ admits a normalized flow with weights $\nu_P$, then $P$ admits a normalized flow with weights $\nu_P$.
\end{theorem}

\section{Proof of Theorem~\ref{thm:main-theorem}} \label{sec:proof}

\subsection{The generalized symmetric groups \texorpdfstring{$G(m,1,n)$}{G(m,1,n)}}\label{sec:m1n}
We first review some background on the complex reflection groups $G(m,p,n)$. Any $w\in G(m,p,n)$ can be expressed in the form $w=[a_1,\ldots,a_n|\sigma]$ where each $a_i \in\Z/m\Z$ and $p$ divides $\sum_{i=1}^na_i$. Note that we always require $p|m$. Naturally, we can view such $w=[a_1,\ldots,a_n|\sigma]$ as an element in $GL(\C^n)$ that sends the $k^{th}$ coordinate vector $v_k$ to $\exp(\frac{2\pi\sqrt{-1}a_k}{m})v_{\sigma(k)}.$ Correspondingly, we can also think of such $w$ as a permutation on $(\Z/m\Z)\times[n]$ such that $w(b,k)=(b+a_k,\sigma(k))$. There are two types of reflections in $G(m,p,n)$, which we call \textit{type (1)} and \textit{type (2)}:
\begin{enumerate}
\item $\sigma=(i,j)$ is a transposition for some $i<j$, $a_i=-a_j$ and $a_k=0$ for $k\neq i,j$;
\item \label{item2} $\sigma=\id $, $p|a_i\neq0$ for some $i\in[n]$ and $a_k=0$ for $k\neq i$.
\end{enumerate}

Shi~\cite{shi} gives explicit formulae for the reflection length $\ell_R$ in $G(m,p,n)$ and we will be using his results for the special case $p=1$. For $w=[a_1,\ldots,a_n|\sigma]\in G(m,1,n)$, we can write $\sigma=\sigma^{(1)}\cdots\sigma^{(r)}$ in disjoint cycle notation. For $i=1,\ldots,r$, define the \textit{sign} of the cycle $\sigma^{(i)}$ to be $\sgn(\sigma^{(i)})=\sum_{j\in\sigma^{(i)}}a_j\in\Z/m\Z$. Let $t_0(w)=\#\{i\in[r]:\sgn(\sigma^{(i)})=0\}$ be the number of cycles of $\sigma$ with sign 0.
\begin{prop}[Theorem 2.1 of~\cite{shi}]\label{prop:lenm1n}
For $w\in G(m,1,n)$, 
\[
\ell_R(w)=n-t_0(w).
\]
\end{prop}

We note the following subword property of the prefix order of any complex reflection group, which follows from~\cite{Huang} (or from~\cite{Armstrong} in the case of Coxeter groups).
\begin{prop}\label{prop:subword}
If $w=t_1\cdots t_{\ell}$ is a reduced word for $w$, then any subword $u=t_{j_1}\cdots t_{j_k}$ where $1\leq j_1<\cdots<j_k\leq\ell$ is reduced and $u\leq w$ in $\pre(W)$.
\end{prop}

We define the \textit{claw poset} of order $n$, denoted $C_n$, to be the ordinal sum of a single element with an antichain of size $n-1$. In other words, elements of $C_n$ are given by $x_0,x_1,\ldots,x_{n-1}$ and their orders relations are defined to be $x_0<x_i$ for all $i=1,\ldots,n-1$. For $n\geq2$, $C_n$ is a ranked poset of rank 2, which clearly has a normalized flow (with respect to $\nu\equiv1$). We are now ready to state the main result of the section.

\begin{theorem}\label{thm:m1n}
For $W=G(m,1,n)$, $\abs(W)$ has a coarsening (on the same underlying set $W$) isomorphic to $C_m\times C_{2m}\times\cdots\times C_{nm}$.
\end{theorem} 
\begin{proof}
We partition all reflections of $G(m,1,n)$ into sets $T_1\sqcup\cdots\sqcup T_n$, where $T_j$ consists of the following reflections $t=[a_1,\ldots,a_k|\sigma]$:
\begin{enumerate}
\item\label{un} $\sigma=(i,j)$ for some $i<j$, $a_i+a_j=0$ and $a_k=0$ for $k\neq i,j$, and 
\item\label{deux} $\sigma=\id $, $a_j\neq0$, $a_k=0$ for $j\neq k$.
\end{enumerate}
In $T_j$, there are $(j-1)m$ reflections of type~\eqref{un} and $m-1$ reflections of type~\eqref{deux}. So its cardinality is $|T_j|=jm-1$. 

We claim that for each $w\in G(m,1,n)$, there is a unique way to write $w=t_{i_1}t_{i_2}\cdots t_{i_k}$ such that $1\leq i_1<\cdots<i_k\leq n$ and $t_{i_j}\in T_{i_j}$. Moreover, such decomposition of $w$ is reduced, meaning $\ell_R(w)=k$. Proceed by induction on $n$. The claim is clear when $n=1$. Now assume $n\geq2$. All reflections in $T_1,\ldots,T_{n-1}$, viewed as permutations on $(\Z/m\Z)\times[n]$, keep $(b,n)$ fixed for any (or a specific) $b\in\Z/m\Z$, while all reflections in $T_n$ do not. Therefore, if $w$ fixes $(b,n)$, we cannot choose any $t_n\in T_n$, and induction hypothesis takes care of the rest of the argument. If $w$ doesn't fix $(b,n)$, we have to choose $t_n\in T_n$ such that $wt_n^{-1}$ fixes $(b,n)$. Let $w^{-1}(b,n)=(a+b,n')\neq(b,n)$ for some $a\in\Z/m\Z$, and all $b\in\Z/m\Z$. Then $t_n$ must map $(b,n)$ to $(a+b,n')$. It's not hard to see such $t_n=[a_1,\ldots,a_n|\sigma]$ is unique: if $n'\neq n$, then $t_n$ must be of type~\eqref{un} with $\sigma=(n',n)$, $a_n=a$, $a_{n'}=-a$; if $n'=n$, then $a\neq0$ and $t_n$ must be of type~\eqref{deux} with $a_n=a$. Then $w'=wt_n^{-1}$ fixes $(b,n)$ and lies in $G(m,1,n-1)$. By induction, the uniqueness and existence of such decomposition are established. To see that the expression obtained in this way is reduced, let's consider the signs of cycles in $w$, as in Proposition~\ref{prop:lenm1n}. If $t_n$ is of type~\eqref{un}, multiplication by $t_n^{-1}$ on $w$ splits the cycle containing $n$ into two cycles, keeping the overall sign. But as one of the cycles is a singleton containing $n$ with sign 0, we know that $t_0(w')=t_0(w)+1$ so $\ell_R(w')=\ell_R(w)-1$ by Proposition~\ref{prop:lenm1n}. If $t_n$ is of type~\eqref{deux}, then in $w$, $n$ is a singleton with nonzero sign. After multiplication by $t_n^{-1}$, $n$ becomes a singleton with sign 0 so similarly, $\ell_R(w')=\ell_R(w)-1$. The rest of the argument follows by induction.

For each $j=1,\ldots,n$, label the unique minimum element of the claw poset $C_{jm}$ by $\id $ and its elements in rank 1 by the elements of $T_j$, recalling that $|T_j|=jm-1$. Every element in $P=C_{m}\times C_{2m}\times\cdots\times C_{nm}$ is then labeled by a tuple $x=(x_1,x_2,\ldots,x_n)$ where $x_j\in T_j\sqcup\{\id \}$. We identify this element as $w(x)=x_1x_2\cdots x_n\in G(m,1,n)$. By the uniqueness and existence argument in the last paragraph, we obtain a bijection between elements in $P$ and $W=G(m,1,n)$. Moreover, the rank of $x$ in $P$ equals the number of $x_j$'s in $T_j$, which is precisely $\ell_R(w(x))$ as shown above. Finally, $x'\leq x$ in $P$ precisely means that $x_1'\cdots x_n'$ is a subword of $x_1\cdots x_n$, after ignoring identity terms, which implies that $w(x')\leq w(x)$ by the subword property (Proposition~\ref{prop:subword}). As a result, $P$ is contained in $\abs(W)$ as desired.
\end{proof}
The following corollary is immediate.
\begin{coro} \label{cor:G(m,1,n)-normalized-flow}
The absolute order of $G(m,1,n)$ has a normalized flow with $\nu\equiv1$, and is thus strongly Sperner.
\end{coro}
\begin{proof}
Each $C_k$ is trivially log-concave and has a normalized flow, with $\nu\equiv1$. By Theorem~\ref{thm:nfproduct}, $C_m\times\cdots\times C_{nm}$ is log-concave and has a normalized flow. But $C_m\times\cdots\times C_{nm}\subset \abs(W)$, for $W=G(m,1,n)$ by Theorem~\ref{thm:m1n} so $\abs(W)$ has a normalized flow (taking the weights of all edges not contained in the product to be zero) and is thus strongly Sperner by Theorem~\ref{thm:nfsperner}.
\end{proof}

\subsection{The dihedral groups \texorpdfstring{$G(m,m,2)$}{G(m,m,2)}}

The following proposition was also stated without proof by Harper, Kim, and Livesay in their independent work~\cite{Harper2019b}. We provide a short proof here for the sake of completeness.

\begin{prop} \label{prop:dihedral-normalized-flow}
Let $W=G(m,m,2)$ for some $m \geq 2$, then $\abs(W)$ admits a normalized flow with $\nu \equiv 1$.
\end{prop}
\begin{proof}
The non-identity elements of $W$ are either reflections, with fixed-space codimension 1, or rotations, with fixed-space codimension 2. The product of a reflection and a rotation is a reflection. This implies that every reflection is covered by every rotation in the absolute order, so the Hasse diagram of $\abs(W)$ is a complete bipartite graph between both pairs of consecutive ranks, which clearly admits the desired normalized flow.
\end{proof}

\subsection{Exceptional Coxeter groups}

In order to verify the normalized flow property for the exceptional Coxeter groups, we take advantage of the large automorphism group of $\abs(W)$. Since the set of reflections in $W$ is invariant under the conjugation action of $W$, it follows that the conjugation action of $W$ on $\abs(W)$ is by poset automorphisms. 

For $P$ a poset and $G \subset \Aut(P)$ a group of poset automorphisms, the \emph{quotient poset} $P/G$ has as elements the orbits of the action of $G$ on the set $P$. For two orbits $\mc{O},\mc{O}'$ we have $\mc{O} \leq \mc{O}'$ in $P/G$ if and only if there exists some $x \in \mc{O}$ and some $x' \in \mc{O}'$ such that $x \leq_P x'$ (equivalently, for \emph{all} $x \in \mc{O}$ there exists such an $x' \in \mc{O}'$).

\begin{prop} \label{prop:flow-on-quotient}\looseness-1
Let $P$ be a finite ranked poset and let $G \subset \Aut(P)$, then $P$ has a normalized flow with $\nu_P \equiv 1$ if and only if $P/G$ has a normalized flow with $\nu_{P/G}(\mc{O})=|\mc{O}|$.
\end{prop}
\begin{proof}
First suppose that $P$ has a normalized flow with $\nu_P \equiv 1$. Then it is immediate from the definitions that $P/G$ has a normalized flow with $\nu_{P/G}(\mc{O})=|\mc{O}|$ and edge weights 
\[
f_{P/G}(\mc{O},\mc{O}')=\sum_{\substack{x \in \mc{O} \\ y \in \mc{O}'}} f_P(x,y).
\]

Next, suppose that $P/G$ has a normalized flow with $\nu_{P/G}(\mc{O})=|\mc{O}|$. Then the natural map $\ph: P \sur P/G$ given by $x \mapsto G \cdot x$ is clearly surjective, rank preserving, order preserving, and measure preserving. Thus, by Theorem~\ref{thm:fundlemma} it only remains to check that the induced subposet of $P$ on the elements $\ph^{-1}(\mc{O}) \cup \ph^{-1}(\mc{O}')$ admits a normalized flow with $\nu \equiv 1$ for any covering relation $\mc{O} \lessdot \mc{O}'$ in $P/G$. We claim that the corresponding subgraph of the Hasse diagram of $P$ is in fact a biregular graph (regular on each side of the bipartition). Indeed, let $x \in \mc{O}$ and suppose $y_1,...,y_k$ are the upper covers of $x$ which lie in $\mc{O}'$. Given any other element $gx \in \mc{O}$ for $g \in G$, the upper covers of $gx$ which lie in $\mc{O'}$ are exactly $gy_1,...,gy_k$, and similarly for lower covers, so we have proven biregularity. As observed by Harper~\cite{Harper1974}, any biregular graph admits a normalized flow simply by taking all edge weights to be equal and scaled appropriately. 
\end{proof}

Proposition~\ref{prop:flow-on-quotient} makes it feasible to construct normalized flows on $\abs(W)$ for the exceptional Coxeter groups of types $H_3, H_4, F_4, E_6, E_7,$ and especially $E_8$. For example, the Coxeter group $W$ of type $E_8$ has approximately $7 \times 10^8$ elements, and the absolute order $\abs(W)$ has approximately $4 \times 10^{10}$ cover relations. Explicitly checking for a normalized flow on a poset of this size is unfeasible; however $\abs(W)/W$ has only 112 elements and 449 cover relations, allowing for a simple computer check for a normalized flow using SageMath~\cite{Sage}. In this way explicit normalized flows (with $\nu(\mc{O})=|\mc{O}|$) have been constructed for $\abs(W)/W$ for all exceptional Coxeter groups $W$. Together with Proposition~\ref{prop:flow-on-quotient}, this proves that these posets $\abs(W)$ admit normalized flows (with $\nu \equiv 1$). The above discussion proves the following proposition:

\begin{prop} \label{prop:exceptional-types-normalized-flow}
Let $W$ be an irreducible exceptional Coxeter group (type $H_3$, $H_4$, $F_4$, $E_6$, $E_7$, or $E_8$), then $\abs(W)$ admits a normalized flow with $\nu \equiv 1$. In particular, $\abs(W)$ is strongly Sperner.
\end{prop}

\subsection{Finishing the proof}

\begin{proof}[Proof Theorem~\ref{thm:main-theorem}]
Let $W=W_1 \times \cdots \times W_k$ be as described in the statement of Theorem~\ref{thm:main-theorem}. By Corollary~\ref{cor:G(m,1,n)-normalized-flow}, Proposition~\ref{prop:dihedral-normalized-flow}, and Proposition~\ref{prop:exceptional-types-normalized-flow}, the absolute order $\abs(W_i)$ of each factor admits a normalized flow with $\nu \equiv 1$. By Proposition~\ref{prop:product-of-groups-has-product-order}, 
\[
\abs(W) \cong \abs(W_1) \times \cdots \times \abs(W_k).
\]
By Equation~\eqref{eq:rank-generating-function} and the classical result that the coefficients of a real-rooted real polynomial are log-concave, we may apply Theorem~\ref{thm:nfproduct} to see that $\abs(W)$ admits a normalized flow (with $\nu \equiv 1$), implying that it is strongly Sperner.
\end{proof}

\begin{remark}
It is important to note that proving the stronger normalized flow property for the irreducible groups $W_i$ is necessary for our proof of Theorem~\ref{thm:main-theorem}. It is not in general true that the product of (even rank log-concave) strongly Sperner posets is Sperner~\cite{Proctor}.
\end{remark}

\section{Counterexamples} \label{sec:counterexamples}

In this section we show that neither $\co(W)$ nor $\pre(W)$ is Sperner for general complex reflection groups $W$.

\subsection{Codimension order}


Many small examples show that $\co(W)$ need not be Sperner when $\co(W) \neq \pre(W)$. For example, 

\begin{prop} \label{prop:codimension-sperner-counterexample}
$\co(G(4,2,2))$ is not Sperner. 
\end{prop}
\begin{proof}
Let $P=\co(G(4,2,2))$, then $P$ has rank sizes $|P_0|=1,|P_1|=8,$ and $|P_2|=7$. There are two maximal elements $x,y$ in rank one, thus $P_2 \cup \{x,y\}$ is an antichain of size 9, larger than any rank size. 
\end{proof}

In fact, $\co(W)$ need not even be \emph{ranked} in general: there is no consistent rank function for $\co(G(4,2,4))$. This means that, for some complex reflection groups $W$ anyway, it does not even make sense to ask whether $\co(W)$ is Sperner. We are not aware of an example $\co(W)$ which is ranked and Sperner, except in the cases $\co(W)=\pre(W)$. 

\begin{question}
Is there a complex reflection group $W$ such that $\co(W) \neq \pre(W)$, but $\co(W)$ is ranked and Sperner?
\end{question}

\subsection{Prefix order}

A finite poset is $\emph{graded}$ if all of its maximal chains have the same length. Although the posets $\pre(W)$ are always ranked, they are not in general graded when $\pre(W) \neq \co(W)$. Our strategy for finding a prefix order which is not Sperner is therefore to construct a reflection group $W$ such that $\pre(W)$ has a maximal element $m$ occurring in a rank below the largest rank. The antichain consisting of $m$ together with the largest rank would then violate the Sperner property.

\begin{prop} \label{prop:prefix-counterexample}
The prefix order on the reflection group $G(10,5,3)^{12}$ is not Sperner.
\end{prop}
\begin{proof}
Let $W=G(10,5,3)$ and $P=\pre(W)$; the following facts are all easily verified by computer:
\begin{itemize}
 \item $\rank(P)=5$,
 \item $P$ has a maximal element $m$ of rank 3, and 
 \item $F(P,q)=1+33q+287q^2+519q^3+314q^4+48q^5$.
\end{itemize}

Although $P$ itself is strongly Sperner, we observe that the largest coefficient in $F(P,q)^{12}$ is the coefficient of $q^{37}$. This means that the largest rank in $P^{12}$ is $(P^{12})_{37}$. However $(m,...,m)$ is a maximal element in $P^{12}$ of rank 36, so $P^{12} \cong \pre(W^{12})$ is not Sperner. 
\end{proof}

\section{A conjecture for type \texorpdfstring{$D_n$}{Dn}} \label{sec:type-D}
In this section, we discuss absolute order on the Weyl group $W$ of type $D_n$ for $n \geq 4$ ($W$ is also the group $G(2,2,n)$), and explain why the methods used to prove Theorem~\ref{thm:m1n} will not work in this case.

It is well-known that the exponents $e_i$ for type $D_n$ are $1,3,5,\ldots,2n-3,n-1$. By the discussion in Section~\ref{sec:background}, we know that
\[
F(\abs(W),q)=(1+(n-1)q) \cdot \prod_{i=1}^{n-1}(1+(2i-1)q).
\]
Therefore it is natural to hope that $\abs(W)$ contains a a product of claw posets so that we might apply the techniques of Section~\ref{sec:m1n}. However, we will show that such strategy cannot work.

Following the setup from Section~\ref{sec:m1n}, every element in $G(2,2,n)$ can be written as $w=[a_1,\ldots,a_n|\sigma]$ with $a_k\in\Z/2\Z$ for $k\in[n]$, $\sum_{k=1}^n a_k=0$, and $\sigma\in S_n$. We can also view $w$ as a permutation on $1,2,\ldots,n,-1,-2,\ldots,-n$ (also called a \textit{signed permutation} on $[n]$) such that $w(i)=-w(-i)$ for all $i$, with the condition that there are an even number of $k\in[n]$ such that $w(k)<0$. For convenience, we write such $w$ in disjoint cycle notation for $\sigma$, and put a dot on $k\in[n]$ if $a_k=1\in\Z/2\Z$. For example, $(\dot1\dot2)$ refers to $w=[1,1,0,0,\ldots|(1,2)]\in G(2,2,n)$, which is also represented as $w(1)=-2$, $w(2)=-1$, $w(k)=k$ for $k=3,\ldots,n$, as a signed permutation.

We see that $G(2,2,n)$ doesn't have any reflections of type~\eqref{item2} (see Section~\ref{sec:m1n}). In fact, all of its reflections are of the form $(i,j)$ for some $i<j\in[n]$ or $(\dot i, \dot j)$ for some $i<j\in[n]$.
\begin{prop}\label{prop:DnoProd}
For $W=G(2,2,n)$ with $n\geq4$, $\abs(W)$ does not contain $C_2\times C_4\times\cdots\times C_{2n-2}\times C_{n}$.
\end{prop}
\begin{proof}
Assume for the sake of contradiction that $P=C_2\times C_4\times\cdots\times C_{2n-2}\times C_{n}\subset\abs(W)$. Each element of rank 1 in $P$ can be identified with a reflection in $W$, and the inclusion $P\subset\abs(W)$ partitions these elements into sets of size $1,3,5,\ldots,2n-3,n-1$. Denote these sets of reflections as $T_1,T_2,\ldots,T_n$ where $|T_k|=2k-1$ for $k=1,\ldots,n-1$ and $|T_n|=n-1$. Now, each element of $P$, and correspondingly each element of $W$, is labeled as $(t_1,t_2,\ldots,t_n)$ where $t_k\in T_k$ or $t_k=\id $. 

Focus on elements of rank 2. Each of them is labeled as $(t_1,t_2,\ldots,t_n)$, where all but two $t_k$'s are the identity. In particular, this element covers exactly two reflections in $P$, which belong to different $T_k$'s. Therefore, if $w\in\abs(W)$ has length 2, and covers exactly two reflections $t$ and $t'$ in $\abs(W)$, then $t\in T_k$, $t'\in T_{k'}$ for distinct $k\neq k'$. It is not hard to see that the following types of elements of $G(2,2,n)$ with absolute length 2 cover exactly two reflections:
\begin{itemize}
\item $tt'=t't$ with $t\in\{(i,j),(\dot i, \dot j)\}$ and $t'\in\{(i',j'),(\dot i',\dot j')\}$ such that $\{i,j\}\cap\{i',j'\}=\emptyset$;
\item $tt'=t't$ with $w=(\dot i)(\dot j)$, $t=(i,j)$ and $t'=(\dot i,\dot j)$.
\end{itemize}

Define a helper function $\phi:T\rightarrow \binom{[n]}{2}$, where $T=T_1\sqcup\cdots\sqcup T_n$ is the set of reflections, by $\phi((i,j))=\phi((\dot i,\dot j))=\{i,j\}$. By arguments above, $(i,j)$ and $(\dot i,\dot j)$ cannot be in the same $T_k$, meaning that $\phi|_{T_k}$ is injective for each $k\in[n]$. Moreover, for distinct $t\neq t'\in T_k$, $\phi(t)$ and $\phi(t')$ must intersect. Thus, the image of $\phi(T_k)$, having the same cardinality as $T_k$, is a set of pairwise intersecting 2-element subset of $[n]$. When $n\geq4$, there can be at most $n-1$ such sets, meaning $|T_k|\leq n-1$. But $|T_{n-1}|=2n-3>n-1$, a contradiction.
\end{proof}

Proposition~\ref{prop:DnoProd} shows that the approach used in Theorem~\ref{thm:m1n} doesn't work for the absolute order in type $D$. However, we still conjecture that the absolute order of type $D_n$ admits a normalized flow (Conjecture~\ref{conj:type-D}), and have verified this conjecture using computer search up to $n=8$.

\longthanks{The authors wish to thank Richard Stanley for suggesting this problem, and Gene B. Kim and Larry Harper for helpful correspondence.}

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