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\documentclass[finalversion]{FPSAC2018}
\articlenumber{51}
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\usepackage{tikz,graphicx}
\graphicspath{{.}{./figures/}}

\theoremstyle{plain}
\newtheorem{mainthm}{Theorem}\renewcommand{\themainthm}{\Alph{mainthm}}
\newtheorem{thm}{Theorem}[section]
\newtheorem*{thm*}{Theorem}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{corollary}[thm]{Corollary}
\newtheorem{prop}[thm]{Proposition}
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\newtheorem{observation}[thm]{Observation}
\newtheorem{reminder}[thm]{Reminder}
\newtheorem{obs}[thm]{Observation}

\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
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\newtheorem{example}[thm]{Example}
\newtheorem{question}[thm]{Question}
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\newtheorem{algo}[thm]{Algorithm}

\theoremstyle{remark}
\newtheorem{rem}[thm]{Remark}
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% definitions
\newcommand{\lr}{\ell_R}
\newcommand{\lrp}{\ell_{R'}}
\newcommand{\lrz}{\ell_{R_0}}
\newcommand{\lro}{\ell_{R_1}}
\newcommand{\lrt}{\ell_{R_2}}
\newcommand{\ls}{\ell_S}
\newcommand{\rell}{\ell_R}  % reflection lenth in \aW
\newcommand{\rello}{\ell_{R_0}}  % reflection lenth in \sW
\newcommand{\sell}{\ell_S}  % standard lenth
\newcommand{\one}{\mathbf{1}}
\newcommand{\zero}{\mathbf{0}}

\newcommand{\rk}{\mathrm{rank}} % rank of a Coxeter group = number of Coxeter generators
\newcommand{\aut}{\mathrm{Aut}} % Automorphismengruppe
\newcommand{\parabolic}{\mathrm{A}} %parabolic closure of a subgroup

\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\F}{\mathbb{F}}

\newcommand{\define}{\mathrel{\mathop:}=}
\newcommand{\size}[1]{\ensuremath{\vert #1 \vert}}
\newcommand{\onto}{\twoheadrightarrow}
\newcommand{\into}{\hookrightarrow}
\newcommand{\vvert}{\;\vert\;}  % vertical bar inside a math bracket 

\newcommand{\RS}{\Phi}%Root system
\newcommand{\id}{\mathrm{1}} % identity element in a group
\newcommand{\sW}{W_0} %spherical Weyl group	
\newcommand{\aW}{W} %affine Weyl group

\newcommand{\sph}{\mathbb{S}}
\newcommand{\euc}{\mathbb{E}}
\newcommand{\sdim}{\dim_\sph}
\newcommand{\edim}{\dim_\euc}
\newcommand{\nll}{\nu}
\newcommand{\ncplx}{\textsc{NullCplx}}
\newcommand{\nprt}{\Pi^0}
\newcommand{\spn}{\textsc{Span}}
\newcommand{\zspn}{\Z\textrm{-}\textsc{Span}}
\newcommand{\arr}{\textsc{Arr}}
\newcommand{\prt}{\Pi}
\newcommand{\cyc}{\textsc{cyc}}
\newcommand{\blo}{\textsc{blocks}}
\newcommand{\Pos}{\textsc{Pos}}
\newcommand{\Neg}{\textsc{Neg}}
\newcommand{\Basic}{\textsc{Basic}}

\newcommand{\aff}[1]{\widetilde{#1}}
\newcommand{\mov}{\textsc{Mov}}
\newcommand{\fix}{\textsc{Fix}}
\newcommand{\im}{\textsc{Im}}
\newcommand{\Ker}{\textsc{Ker}}
\newcommand{\isom}{\textsc{Isom}}
\newcommand{\Hy}{\mathcal{H}}
\newcommand{\Sym}{\mathfrak{S}}
\newcommand{\affS}{\widetilde{\mathfrak{S}}}
\newcommand{\typeA}{\mathrm{A}}

\DeclareMathOperator{\wt}{wt}
\DeclareMathOperator{\cy}{cyc}

%% define your title in the usual way
\title[Computing reflection length in an affine Coxeter group]{Computing
reflection length in \\ an affine Coxeter group}


%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Joel Brewster
Lewis\thanks{\href{mailto:jblewis@gwu.edu}{jblewis@gwu.edu}. Work of
Lewis supported by NSF grant DMS-1401792.}\addressmark{1}, Jon
McCammond\thanks{\href{mailto:jon.mccammond@math.ucsb.edu}{jon.mccammond@math.ucsb.edu}}\addressmark{2},
T. Kyle
Petersen\thanks{\href{mailto:tpeter21@depaul.edu}{tpeter21@depaul.edu}. Work of Petersen
supported by a Simons Foundation collaboration travel
grant.}\addressmark{3},\\ \and Petra
Schwer\thanks{\href{mailto:petra.schwer@kit.edu}{petra.schwer@kit.edu}.
Work of Schwer supported by DFG grant SCHW 1550 4-1 within SPP
2026.}\addressmark{4}}

%\address{Petra Schwer, Department of Mathematics, Karlsruhe Institute
%of Technology, Englerstrasse 2, 76133 Karlsruhe, Germany}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}George Washington University \\
\addressmark{2}University of California at Santa Barbara \\
\addressmark{3}DePaul University \\
\addressmark{4}Karlsruhe Institute of Technology}

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\received{\today}

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{ In any Coxeter group, the conjugates of elements in its Coxeter
  generating set are called reflections and the reflection length of
  an element is its length with respect to this expanded generating
  set.  In this article we give a simple formula that computes the
  reflection length of any element in any affine Coxeter group. In the affine symmetric group, we have a combinatorial formula that generalizes D\'enes' formula for reflection length in the symmetric group. }

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Coxeter groups, affine reflection groups, reflection length}

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\usepackage[backend=bibtex]{biblatex}
\addbibresource{51-Lewis-McCammond-Petersen-Schwer.bib}
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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

\section*{Introduction}

In any Coxeter group $W$, the conjugates of elements in its standard
Coxeter generating set are called \emph{reflections} and we write $R$
for the set of all reflections in $W$.  The reflections generate $W$
and the associated \emph{reflection length} function $\lr\colon W \to \Z_{\geq 0}$
records the length of $w$ with respect to this expanded generating set.
When $W$ is spherical, i.e., finite, reflection length can be given an
intrinsic, geometric definition, as follows. Define a \emph{root subspace} to be
any space spanned by a subset of the corresponding root system and define the
\emph{dimension} $\dim(w)$ of an element $w$ in $W$ to be the minimum dimension
of a root subspace that contains the move-set of $w$. (See
Section~\ref{sec:dim} for the details.) The following result is due to Carter \cite{Carter72}. 

\begin{thm*}[{\cite[Lemma~2]{Carter72}}]
Let $W$ be a spherical Coxeter group and let $w \in W$.  Then $\lr(w) = \dim(w)$.
\end{thm*}

When $W$ is infinite, much less is known \cite{McPe-11, Duszenko12,
  MST-affine-deligne-lusztig}, even in the affine case.  In this
work, we give a simple, analogous formula that computes the
reflection length of any element in any affine Coxeter group.\footnote{This note is a condensed version of \cite{LMPS-arxiv}, in which we provide a simple uniform proof of Theorem \ref{main:dim}. Indeed, proofs of all results are omitted here but can be found in the full paper.} 

\newcommand{\maindim}{0}
\begin{mainthm}[Formula]\label{main:dim}
  Let $W$ be an affine Coxeter group and let $p\colon W \onto W_0$ be
  the projection onto its associated spherical Coxeter group.  For any
  element $w\in W$, its reflection length is 
  \[\lr(w) = 2 \cdot \dim(w) - \dim(p(w)) = 2d + e,\] 
  where $e = \dim(p(w))$ and $d = \dim(w)-\dim(p(w))$.
\end{mainthm}

We call $e = e(w) = \dim(p(w))$ the \emph{elliptic dimension} of $w$
and we call $d = d(w) = \dim(w) - \dim(p(w))$ the \emph{differential
  dimension} of $w$.  Both statistics are geometrically meaningful.
For example, $e(w) = 0$ if and only if $w$ is a translation (that is,
if it sends every point $x$ to $x+\lambda$ for some fixed vector
$\lambda$), and $d(w)=0$ if and only if $w$ is elliptic (that is, if
it fixes a point) -- see Proposition~\ref{prop:stat-geo}. In \cite{LMPS-arxiv}, we prove Theorem~\ref{main:dim} by showing the claimed value is both a lower bound and an upper bound for the reflection length of $w$.

The translation and elliptic versions of our formula were already
known (see \cite{McPe-11} and \cite{Carter72}, respectively) and these
special cases suggest that the formula in Theorem~\ref{main:dim} might
correspond to a carefully chosen factorization of $w$ as a product of
a translation and an elliptic element.  This is indeed the case.

\newcommand{\mainfact}{1}
\begin{mainthm}[Factorization]\label{main:fact}
  Let $W$ be an affine Coxeter group.  For every element $w\in W$, there
  is a translation-elliptic factorization $w = t_\lambda  u$ such that
  $\lr(t_\lambda) = 2d(w)$ and $\lr(u) = e(w)$.  
  In particular, $\lr(w) = \lr(t_\lambda) + \lr(u)$ for
  this factorization of $w$.
\end{mainthm}

The proof of Theorem~\ref{main:fact} relies on a nontrivial technical
result recently established by Vic Reiner and the first author
\cite[Corollary~1.4]{LewisReiner-16}. Elements in an affine Coxeter
group typically have many different potential translation-elliptic
factorizations and the most common way to find one is to view the
group as a semidirect product.  For an affine Coxeter group $W$ that
naturally acts cocompactly on an $n$-dimensional euclidean space, the
set of translations in $W$ forms a normal abelian subgroup $T$
isomorphic to $\Z^n$, and the quotient $W_0 = W/T$ is the spherical
Coxeter group associated with $W$.  An identification of $W$ as a
semidirect product $T \rtimes W_0$ corresponds to a choice of an
inclusion map $i\colon W_0 \into W$ that is a section of the
projection map $p\colon W \onto W_0$.  There is a unique point $x$
fixed by the subgroup $i(W_0)$ and every element has a unique
factorization $w = t_\lambda u$ where $t_\lambda$ is a translation in
$T$ and $u$ is an elliptic element in the copy of $W_0$ that fixes
$x$.  However, for some elements $w$ none of the
translation-elliptic factorizations that come from an identification
of $W$ as a semidirect product satisfy Theorem~\ref{main:fact} -- see
Example 2.4 of \cite{LMPS-arxiv}.

In the particular case of the symmetric group $\Sym_n$ (the spherical
Coxeter group of type A), reflection length also has a natural
combinatorial characterization: $\lr(w) = n - c(w)$, where $c(w)$ is
the number of \emph{cycles} of the permutation $w$ \cite{Denes59}.
When $W$ is the affine symmetric group $\affS_n$, we show that the
differential and elliptic dimensions of an element can also be given a
combinatorial interpretation, leading to the very similar formula in
Theorem \ref{thm:lr-affS}.

In this brief note, we highlight the key definition at play in Theorem \ref{main:dim}, namely the notion of dimension of an element. We assume the reader is familiar with Coxeter groups at the level of Humphreys \cite{Humphreys-90}. Full details can be found in \cite{LMPS-arxiv}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Dimensions in affine Coxeter groups}\label{sec:dimensions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%In this section we recall the relevant background on reflection length
%and basic notions for spherical and affine Coxeter groups. We then
%develop a notion of the dimension of an element in an affine Coxeter
%group. We assume the reader is familiar with Coxeter groups at the
%level of Humphreys \cite{Humphreys-90}.
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Reflection length}\label{sec:lr}
%
%This section reviews basic facts about reflection length in arbitrary
%Coxeter groups.  
%
%\begin{defn}[Reflection length]\label{def:lr}
%  Given a Coxeter group $W$ with standard Coxeter
%  generating set $S$, a \emph{reflection} is any element conjugate
%  to an element in $S$.  Let $R$ denote the set of all reflections in
%  $W$; note that $R$ is infinite whenever the
%  Coxeter group $W$ is infinite.  The \emph{reflection length} $\lr(w)$
%  of an element $w$ is the minimum integer $k$ such that
%  there exist reflections $r_i \in R$ with $w = r_1 r_2 \cdots r_k$.
%  The identity has reflection length $0$.
%\end{defn}
%
%\begin{rem}[Prior results]\label{rem:prior}
%  When $W$ is a spherical Coxeter group, Carter's lemma mentioned in
%  the introduction establishes that the reflection length of an
%  element $w \in W$ may be computed in terms of the dimension of its
%  move-set or fixed space (see Section~\ref{sec:finite Coxeter groups}
%  below for the definitions): $\lr(w) = \dim (\mov(w)) = \dim(V) -
%  \dim (\fix(w))$. When $W$ is an affine Coxeter group acting
%  cocompactly on an $n$-dimensional euclidean space, the second and
%  third authors have shown that $\lr(w)$ is bounded above by $2n$ for
%  every element $w$ in $W$, and established an exact formula for the
%  reflection length of a translation \cite[Theorem~A and
%    Proposition~4.3]{McPe-11}.  Mili\'{c}evi\'{c}, Thomas and the
%  fourth author found better bounds and explicit formulas in some
%  special cases as part of their work on affine Deligne--Lusztig
%  varieties \cite[Theorem~11.3 and
%    Corollary~11.5]{MST-affine-deligne-lusztig}.  Finally, when $W$ is
%  neither spherical nor affine, Duszenko has shown that the reflection
%  length function is always unbounded \cite{Duszenko12}.
%\end{rem}
%
%\begin{rem}[Basic properties]\label{rem:lr-basic}
%  If $W$ is a reducible Coxeter group $W = W_1 \times W_2$, the
%  reflection length of an element $w \in W$ is just the sum of the
%  reflection lengths of its factors \cite[Proposition~1.2]{McPe-11}.
%  Thus, it is sufficient to study \emph{irreducible} Coxeter groups,
%  for which there is a well-known classification \cite[Section
%    6.1]{Humphreys-90}.  Since the set $R$ of reflections is a union
%  of conjugacy classes, the length function $\lr \colon W \to \Z_{\geq
%    0}$ is constant on conjugacy classes.  This implies, in
%  particular, that $\lr(uv) = \lr(vu)$.  The function $\lr$ can also
%  be viewed as the natural combinatorial distance function on the
%  Cayley graph of $W$ with respect to the generating set $R$, which
%  means that $\lr$ satisfies the triangle inequality $\lr(u)-\lr(v)
%  \leq \lr(uv) \leq \lr(u)+\lr(v)$.  Finally, for every Coxeter group
%  there is a homomorphism onto $\Z/2\Z$ that sends every reflection to
%  the non-identity element.  As a consequence, reflection length has a
%  parity restriction: $\lr(uv) = \lr(u) + \lr(v) \mod 2$.  These facts
%  combine to show that $\lr(rw)=\lr(wr)=\lr(w)\pm 1$ for every element
%  $w \in W$ and reflection $r\in R$.
%\end{rem}

%One standard result about reflection factorizations is that they can
%be rewritten in many different ways.
%
%\begin{lemma}[Rewriting reflection factorizations]\label{lem:hurwitz}
%  Let $w = r_1 r_2 \cdots r_k$ be a reflection factorization of an
%  element $w$ of a Coxeter group.  For any selection $1 \leq i_1 < i_2
%  < \cdots < i_m \leq k$ of positions, there is a length-$k$
%  reflection factorization of $w$ whose first $m$ reflections are
%  $r_{i_1} r_{i_2} \cdots r_{i_m}$ and another length-$k$ reflection
%  factorization of $w$ where these are the last $m$ reflections.
%\end{lemma}

%\begin{proof}
%  Because the set of reflections is closed under conjugation, we have
%  for any reflections $r$ and $r'$ that the elements $r'' = r r' r$
%  and $r''' = r' r r'$ are also reflections, satisfying $r r' = r'' r
%  = r' r'''$.  Thus in any reflection factorization one may replace a
%  consecutive pair $rr'$ of factors with the pair $r''r$ (moving $r$
%  to the right one position) or $r'r'''$ (moving $r'$ to the left one
%  position) without changing the length of the factorization.
%  Iterating these rewriting operations allows one to move any subset
%  of reflections into the desired positions.
%\end{proof}
%
%\begin{rem}[Hurwitz action]
%  The individual moves in the proof of Lemma~\ref{lem:hurwitz} are
%  called \emph{Hurwitz moves}.  Globally, they correspond to an
%  action, called the \emph{Hurwitz action}, of the $k$-strand braid
%  group on the set of all length-$k$ reflection factorizations of a
%  given word $w$.
%\end{rem}
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Spherical Coxeter groups}\label{sec:finite Coxeter groups}
%
%In this section, we discuss the spherical Coxeter groups and their
%connection to root systems.
%
%\begin{defn}[Spherical Coxeter groups]\label{def:spherical coxeter group}
%  A \emph{euclidean vector space} $V$ is an $n$-dimensional real
%  vector space equipped with a positive definite inner product
%  $\langle \cdot, \cdot \rangle$.  A \emph{crystallographic root
%    system $\Phi$} is a finite collection of vectors that span a real
%  euclidean vector space $V$ satisfying a few elementary properties --
%  see \cite{Humphreys-90} for a precise definition. (While there are
%  non-crystallographic root systems as well, it is only the
%  crystallographic root systems that arise in the study of affine
%  Coxeter groups.) The elements of $\Phi$ are called \emph{roots}.
%  Each crystallographic root system corresponds to a finite (or
%  \emph{spherical}) Coxeter group $W_0$, as follows: for each $\alpha$
%  in $\Phi$, $H_\alpha$ is the hyperplane through the origin in $V$
%  orthogonal to $\alpha$, and the unique nontrivial isometry of $V$
%  that fixes $H_{\alpha}$ pointwise is a \emph{reflection} that we
%  call $r_{\alpha}$.  The collection $R = \{r_{\alpha} \mid \alpha \in
%  \Phi\}$ generates the spherical Coxeter group $W_0$, and $R$ is its
%  set of reflections in the sense of Definition~\ref{def:lr}. 
%  In
%  Figure \ref{fig:hyperplanes-finite} we see the hyperplanes for each
%  irreducible crystallographic root system of rank two.
%\end{defn}

%\begin{figure}
%  \begin{tabular}{c c c}
%    \begin{tikzpicture}[cm={1,0,.5,.8660254,(0,0)}, >=stealth,baseline=2.6cm]
%      \draw[very thick] (0,3)--(4,3);
%      \draw[very thick] (2,1)--(2,5);
%      \draw[very thick] (0,5)--(4,1);
%    \end{tikzpicture}
%    &
%    \begin{tikzpicture}[>=stealth,baseline=0,scale=.85]
%      \draw[very thick] (-2,2)--(2,-2);
%      \draw[very thick] (-2,-2)--(2,2);
%      \draw[very thick] (0,-2)--(0,2);
%      \draw[very thick] (-2,0)--(2,0);
%    \end{tikzpicture}
%    &
%    \begin{tikzpicture}[cm={1,0,.5,.8660254,(0,0)}, >=stealth,baseline=2.6cm]
%      \draw[very thick] (0,3)--(4,3);
%      \draw[very thick] (2,1)--(2,5);
%      \draw[very thick] (0,5)--(4,1);
%      \draw[very thick] (1,2)--(3,4);
%      \draw[very thick] (0,4)--(4,2);
%      \draw[very thick] (1,5)--(3,1);
%    \end{tikzpicture}
%    \\
%    (a) & (b) & (c)
%  \end{tabular}
%  \caption{ Reflecting hyperplanes associated to the root systems of
%    types (a) $A_2$, (b) $B_2$ and $C_2$, and (c) $G_2$.}
%  \label{fig:hyperplanes-finite}
%\end{figure}
%
%In a spherical Coxeter group constructed as above, each element is
%associated to two fundamental subspaces, its fixed space and its
%move-set.
%
%\begin{defn}[Fixed space and move-set]\label{def:linear-fixed-space}
%  Given an orthogonal transformation $w$ of $V$, its \emph{fixed
%    space} $\fix(w)$ is the set of vectors $\lambda \in V$ such that
%  $w(\lambda) = \lambda$.  In other words, it is the kernel $\Ker(w -
%  1)$.  Its \emph{move-set} $\mov(w)$ is the set of vectors $\mu$ in
%  $V$ for which there exists some $\lambda$ in $V$ such that
%  $w(\lambda) = \lambda + \mu$.  In other words, it is the image
%  $\im(w - 1)$.
%\end{defn}
%
%\begin{remark}[Orthogonal transformations]\label{rem:orthogonal complement}
%  Every element in a spherical Coxeter group, constructed from a root
%  system as above, is an orthogonal transformation, and so the
%  move-set and fixed space are orthogonal complements with respect to
%  $V$.
%\end{remark}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Points, vectors, and affine Coxeter groups}\label{sec:pt-vec}

Before we define affine Coxeter groups, it is helpful to establish notions for spherical Coxeter groups.

\begin{defn}[Spherical Coxeter groups]\label{def:spherical coxeter group}
  A \emph{euclidean vector space} $V$ is an $n$-dimensional real
  vector space equipped with a positive definite inner product
  $\langle \cdot, \cdot \rangle$.  A \emph{crystallographic root
    system $\Phi$} is a finite collection of vectors that span a real
  euclidean vector space $V$ satisfying a few elementary properties --
  see \cite{Humphreys-90} for a precise definition. (While there are
  non-crystallographic root systems as well, it is only the
  crystallographic root systems that arise in the study of affine
  Coxeter groups.) The elements of $\Phi$ are called \emph{roots}.
  Each crystallographic root system corresponds to a finite (or
  \emph{spherical}) Coxeter group $W_0$, as follows: for each $\alpha$
  in $\Phi$, $H_\alpha$ is the hyperplane through the origin in $V$
  orthogonal to $\alpha$, and the unique nontrivial isometry of $V$
  that fixes $H_{\alpha}$ pointwise is a \emph{reflection} that we
  call $r_{\alpha}$.  The collection $R = \{r_{\alpha} \mid \alpha \in
  \Phi\}$ generates the spherical Coxeter group $W_0$, and $R$ is its
  set of reflections.
  % in the sense of Definition~\ref{def:lr}. 
%  In
%  Figure \ref{fig:hyperplanes-finite} we see the hyperplanes for each
%  irreducible crystallographic root system of rank two.
\end{defn}
  
In the same way that doing linear algebra using fixed coordinate
systems can obscure an underlying geometric elegance, working in
affine Coxeter groups with a predetermined origin can have an
obfuscating effect.  One way to avoid making such a choice is to
distinguish between \emph{points} and \emph{vectors}, as in
\cite{SnapperTroyer-89} and \cite{BradyMcCammond-15}.

\begin{defn}[Points and vectors]\label{def:pt-vec}
  Let $V$ be a euclidean vector space.  A \emph{euclidean space} is a
  set $E$ with a uniquely transitive $V$-action, i.e., for every
  ordered pair of points $x,y \in E$ there exists a unique vector
  $\lambda \in V$ such that the $\lambda$ sends $x$ to $y$.  When this
  happens we write $x + \lambda = y$.  The preceding sentences
  illustrate two conventions that we adhere to throughout the paper:
  the elements of $E$ are \emph{points} and are denoted by Roman
  letters, such as $x$ and $y$, while the elements of $V$ are
  \emph{vectors} and are denoted by Greek letters, such as $\lambda$
  and $\mu$.
\end{defn}

The main difference between $E$ and $V$ is that $V$ has a well-defined
origin, but $E$ does not.  If we select a point $x \in E$ to serve as
the origin, then $V$ and $E$ can be identified by sending each vector
$\lambda$ to the point $x+\lambda$.  We use this identification in the
construction of the affine Coxeter groups.

\begin{defn}[Affine Coxeter groups]\label{def:affine}
  Let $E$ be a euclidean space, whose associated vector space $V$
  contains the crystallographic root system $\Phi$.  An affine Coxeter
  group $W$ can be constructed from $\Phi$, as follows.  Fix a point
  $x$ in $E$, to temporarily identify $V$ and $E$.  For each $\alpha
  \in \Phi$ and $j\in \Z$, let $H_{\alpha,j}$ denote the (affine)
  \emph{hyperplane} in $E$ of solutions to the equation $\langle
  v,\alpha \rangle = j$, where the brackets denote the standard inner
  product (treating $x$ as the origin).  The unique nontrivial
  isometry of $E$ that fixes $H_{\alpha,j}$ pointwise is a
  \emph{reflection} that we call $r_{\alpha,j}$.  The collection $R =
  \{r_{\alpha,j} \mid \alpha \in \Phi, j \in \Z\}$ generates the
  affine Coxeter group $W$ and $R$ is its set of reflections.
  % in the sense of Definition~\ref{def:lr}.  
  A standard minimal generating set
  $S$ can be obtained by restricting to those reflections that reflect
  across the facets of a certain polytope in $E$. 
%  The irreducible
%  affine hyperplane arrangements of rank $2$ are shown in Figure
%  \ref{fig:hyperplanes-infinite}.
\end{defn}

%\begin{figure}[!h]
%  \begin{tabular}{c c}
%    (a) &
%    \begin{tikzpicture}[cm={1,0,.5,.8660254,(0,0)}, >=stealth,baseline=2.6cm, scale=.8]
%      \foreach \x in {1,...,6}{
%        \draw (\x-1,.8)--(\x-1,6.2);
%      }
%      \draw (-1,2.8)--(-1,6.2);
%      \draw (-2,4.8)--(-1-1,6.2);
%      \draw (6,.8)--(6,4.2);
%      \draw (7,.8)--(7,2.2);
%      \foreach \x in {2,...,6}{
%        \draw (1.2+\x,.8)--(-4.2+\x,6.2);
%      }
%      \draw (2.2,.8)--(-2.2,5.2);
%      \draw (1.2,.8)--(-1.2,3.2);
%      \draw (.2,.8)--(-.2,1.2);
%      \draw (7.2,1.8)--(2.8,6.2);
%      \draw (6.2,3.8)--(3.8,6.2);
%      \foreach \x in {1,...,6}{
%        \draw (.2-.5*\x,\x)--(8.2-.5*\x,\x);
%      }
%    \end{tikzpicture}
%    \\
%    \\
%    (b) & 
%    \begin{tikzpicture}[>=stealth,baseline=2.5cm,scale=.68]
%      \foreach \x in {1,...,10}{
%        \draw (\x,.8)--(\x,6.2);
%      }
%      \foreach \x in {1,...,6}{
%        \draw (.8,\x)--(10.2,\x);
%      }
%      \foreach \x in {1,...,3}{
%        \draw (.8,2*\x+.2)--(2*\x+.2,.8);
%        \draw (.8,2*\x-.2)--(7.2-2*\x,6.2);
%        \draw (2*\x+3.8,.8)--(10.2,7.2-2*\x);
%      }
%      \foreach \x in {1,...,2}{
%        \draw (2*\x+.8,6.2)--(2*\x+6.2,.8);
%        \draw (2*\x+4.8,6.2)--(10.2,2*\x+.8);
%        \draw (2*\x+5.2,6.2)--(2*\x-.2,.8);
%      }
%    \end{tikzpicture}
%    \\
%    \\
%    (c)
%    &
%    \begin{tikzpicture}[cm={1,0,.5,.8660254,(0,0)}, >=stealth,baseline=2.6cm,scale=.8]
%      \foreach \x in {1,...,6}{
%        \draw (\x-1,.8)--(\x-1,6.2);
%      }
%      \draw (-1,2.8)--(-1,6.2);
%      \draw (-2,4.8)--(-1-1,6.2);
%      \draw (6,.8)--(6,4.2);
%      \draw (7,.8)--(7,2.2);
%      \foreach \x in {2,...,6}{
%        \draw (1.2+\x,.8)--(-4.2+\x,6.2);
%      }
%      \draw (2.2,.8)--(-2.2,5.2);
%      \draw (1.2,.8)--(-1.2,3.2);
%      \draw (.2,.8)--(-.2,1.2);
%      \draw (7.2,1.8)--(2.8,6.2);
%      \draw (6.2,3.8)--(3.8,6.2);
%      \foreach \x in {1,...,6}{
%        \draw (.2-.5*\x,\x)--(8.2-.5*\x,\x);
%      }
%      \draw (.8,.8)--(5.5,5.5);
%      \draw (3.2,6.2)--(-.8,2.2);
%      \draw (0.2,6.2)--(-1.8,4.2);
%      \draw (3.8,.8)--(6.5,3.5);
%      \draw (6.8,.8)--(7.5,1.5);
%      \draw (-2.6,5.8)--(7.4,.8);
%      \draw (-1.6,3.8)--(4.4,.8);
%      \draw (-.6,1.8)--(1.4,.8);
%      \draw (-.4,6.2)--(7,2.5);
%      \draw (2.6,6.2)--(6,4.5);
%      \draw (1.1,.8)--(-1.6,6.2);
%      \draw (2.6,.8)--(-.1,6.2);
%      \draw (4.1,.8)--(1.4,6.2);
%      \draw (5.6,.8)--(2.9,6.2);
%      \draw (7.1,.8)--(4.4,6.2);
%    \end{tikzpicture}
%  \end{tabular}
%  \caption{ Affine hyperplanes for the root system of type (a) $A_2$,
%    (b) $B_2$, and (c) $G_2$.}
%  \label{fig:hyperplanes-infinite}
%\end{figure}

The affine Coxeter group $W$ associated to a finite crystallographic
root system $\Phi$ is closely related to the spherical Coxeter group
$W_0$.

\begin{defn}[Subgroups and quotients]\label{def:sub-quo}
  Let $W$ be an affine Coxeter group constructed as in
  Definition~\ref{def:affine}.  The map sending $r_{\alpha,j}$ in $W$
  to $r_\alpha$ in $W_0$ extends to a group homomorphism $p\colon W
  \onto W_0$.  The kernel of $p$ is a normal abelian subgroup $T$,
  isomorphic to $\Z^n$, whose elements are called \emph{translations},
  and $W_0 \cong W/T$.  When $x$ is the point used in the construction
  of $W$, the map $i\colon W_0 \into W$ sending $r_\alpha$ to
  $r_{\alpha, 0}$ is a section of the projection $p$, identifying
  $W_0$ with the subgroup of all elements of $W$ that fix $x$. (Note
  also the composition $i\circ p$ sends reflection $r_{\alpha,j}$ to
  $r_{\alpha}$ to $r_{\alpha,0}$.) Thus $W$ may be identified as a
  semidirect product $W \cong T \rtimes W_0$.
\end{defn}
  
Of course, the identification of $W_0$ with a subgroup of $W$ is not
unique: conjugation by elements of $T$ gives an infinite family of
such subgroups.

%\begin{figure}[h]
%  \begin{tikzpicture}[>=stealth]
%    \draw (0,0) node{
%      \begin{tikzpicture}[baseline=2.5cm,scale=.8]
%        \draw[draw=none,fill=white!80!blue] (3,4)--(4,4)--(4,3)--(3,4);
%        \foreach \x in {1,...,5}{
%          \draw (\x,1.8)--(\x,6.2);
%          \draw (.8,\x+1)--(5.2,\x+1);
%        }
%        \foreach \x in {1,...,3}{
%          \draw (.8,2*\x+.2)--(2*\x-.8,1.8);
%          \draw (.8,2*\x-.2)--(7.2-2*\x,6.2);
%        }
%        \foreach \x in {1,...,2}{
%          \draw (2*\x+.8,6.2)--(5.2,2*\x+1.8);
%          \draw (5.2,6.2-2*\x)--(2*\x+.8,1.8);
%        }
%        \draw[very thick] (.5,6.5)--(5.5,1.5);
%        \draw[very thick] (0,4)--(6,4);
%        \draw (5,4) node[circle,inner sep=2, fill=black] {};
%        \draw (5,6) node[circle,inner sep=2, fill=black] {};
%        \draw (5,2) node[circle,inner sep=2, fill=black] {};
%        \draw (3,6) node[circle,inner sep=2, fill=black] {};
%        \draw (3,4) node[circle,inner sep=2, fill=black] {};
%        \draw (3,2) node[circle,inner sep=2, fill=black] {};
%        \draw (1,2) node[circle,inner sep=2, fill=black] {};
%        \draw (1,4) node[circle,inner sep=2, fill=black] {};
%        \draw (1,6) node[circle,inner sep=2, fill=black] {};
%      \end{tikzpicture}
%    };
%    \draw (0,0) circle (4cm);
%    \foreach \x in {1,...,8}{
%      \draw (45*\x:4cm) node[circle,inner sep=2, fill=black] {}; 
%      \draw[dashed] (45*\x:3.25cm)--(45*\x:4.5cm);
%    }
%    \draw[dashed,->] (.8,-.25) node[inner sep=1,left] {$A$} --
%    (-20:4.5cm) node[below right,inner sep=1] {$p(A)=C$}; 
%  \end{tikzpicture}
%  \caption{Elements of $W$ can be put in bijection with alcoves in
%    tessellated plane and the orbit of $x$ under translations in $W$
%    is illustrated with the fat vertices of these alcoves.  Elements
%    of $W_0$ are in bijection with chambers on the boundary sphere.
%    The projection map $p$ maps an alcove $A$ to the chamber $C$ at
%    infinity that points ``in the same direction''. For details see
%    Remark~\ref{rem:alcoves}. }\label{fig:projection}
%\end{figure}

%\begin{remark}[Geometric interpretation of $W$, $W_0$ and $p$]\label{rem:alcoves}
%  Given $W$, an \emph{alcove} is (the closure of) a maximal
%  connected component in the complement of the reflection hyperplanes
%  $H_{\alpha,j}$ in the euclidean space $E$. These are the small
%  triangles in Figures \ref{fig:hyperplanes-infinite} and
%  \ref{fig:projection}. Each of these triangles can play the role of a
%  fundamental domain of the action of $W$ on $E$.
%%
%  The choice of a point $x$ as origin and a fundamental alcove $c_x$
%  having $x$ as a vertex determines a bijection between elements of
%  $W$ and alcoves. Here the fundamental alcove corresponds to the
%  identity and every element $w\in W$ corresponds to the image of
%  $c_x$ under $w$.
%
%  The boundary of $E$ is a sphere $S$ whose points are the parallelism
%  classes of geodesic rays. Its simplicial structure is induced by the
%  tessellation of the plane. A parallelism class of hyperplanes in $E$
%  corresponds to a reflection hyperplane, i.e., an equator, in the
%  sphere $S$.  We call the maximal connected components in the
%  complement of the hyperplanes in $S$ \emph{chambers}.  Each chamber
%  is also the parallelism class of simplicial cones in $E$. Namely, for
%  any one of the black vertices in Figure \ref{fig:projection}, 
%  say $y$, the complement of all the
%  hyperplanes going through $y$ decomposes into simplicial cones. We
%  call them \emph{Weyl cones} based at $y$. The parallelism classes of
%  Weyl cones are in bijection with the chambers in $S$.  Moreover, the
%  chambers in $S$ are in a natural bijection with elements of $W_0$, as
%  follows: Each element $w\in W$ is an affine motion of $E$ that
%  obviously maps parallel rays to parallel rays. Hence it induces an
%  isometry of (the tessellation of) $S$ where translations on $E$
%  induce the identity on $S$. If we want the bijection between $W_0$
%  and the chambers in $S$ to be compatible with these induced
%  isometries of $W$ on $S$ we need to map the identity in $W_0$ to the
%  parallelism class of the unique Weyl cone based at $x$ that contains
%  the alcove $c_x$.
%
%  From this perspective, the projection map $p: W \onto W_0$ can be
%  understood geometrically as the map that sends an element $w\in W$
%  to the induced isometry on $S$. In geometric terms this means that
%  an alcove $A$ with vertex $y$ is mapped to the chamber $C$ at
%  infinity that is the parallelism class of the cone based at $y$ that
%  contains $A$. This is indicated by the dotted arrow in
%  Figure~\ref{fig:projection}. One can think of this as ``walking to
%  infinity in the direction of $A$ at $y$''.  More about the interplay
%  between faces of alcoves and faces of chambers is studied in work of
%  Marcelo Aguiar and the third author \cite{AguiarPetersen-15}.
%\end{remark}

We now define two fundamental affine subspaces associated to any element $w$.

\begin{defn}[Move-sets]\label{def:move-sets}
  The \emph{motion} of a point $x \in E$ under a euclidean isometry
  $w$ is the vector $\lambda \in V$ such that $w(x) = x + \lambda$.
  The \emph{move-set} of $w$ is the collection of all motions of the
  points in $E$; by \cite[Proposition~3.2]{BradyMcCammond-15}, it is
  an affine subspace of $V$.
  In symbols, $\mov(w) = \{ \lambda \mid w(x) = x+\lambda
  \textrm{ for some } x \in E \} \subset V$.
\end{defn}

\begin{defn}[Fixed space]\label{def:fixed-space}
  The \emph{fixed space} $\fix(w)$ of an isometry $w$ is the subset of
  points $x \in E$ such that $w(x) = x$.  Equivalently, $\fix(w)$
  consists of all points whose motion under $w$ is the vector $0$.
  When $\fix(w)$ is nonempty, it is an (affine) subspace of $E$.
\end{defn}

When $w$ is an element of a spherical Coxeter group, it is just an orthogonal transformation of $V$, so $\fix(w) = \Ker(w-1)$ and $\mov(w)=\im(w-1)$. In particular, $\fix(w)$ and $\mov(w)$ are orthogonal complements in $V$. However, this complementarity between $\mov(w)$ and $\fix(w)$ does not hold for isometries of
euclidean space.  It can be recovered if the fixed space is
replaced with the min-set of points that are moved a minimal distance
under $w$.  Then the space of directions for the move-set and the
space of directions for the min-set give an orthogonal decomposition
of $V$ -- see \cite[Lemma 3.6]{BradyMcCammond-15}.


%\begin{figure}
%  \begin{tikzpicture}[>=stealth,baseline=2.5cm,scale=1]
%    \foreach \x in {1,...,10}{
%      \draw (\x,.8)--(\x,6.2);
%    }
%    \foreach \x in {1,...,6}{
%      \draw (.8,\x)--(10.2,\x);
%    }
%    \foreach \x in {1,...,3}{
%      \draw (.8,2*\x+.2)--(2*\x+.2,.8);
%      \draw (.8,2*\x-.2)--(7.2-2*\x,6.2);
%      \draw (2*\x+3.8,.8)--(10.2,7.2-2*\x);
%    }
%    \foreach \x in {1,...,2}{
%      \draw (2*\x+.8,6.2)--(2*\x+6.2,.8);
%      \draw (2*\x+4.8,6.2)--(10.2,2*\x+.8);
%      \draw (2*\x+5.2,6.2)--(2*\x-.2,.8);
%    }
%    \draw[very thick] (5,0)--(5,7) node[right] {$s$};
%    \draw[very thick] (0,4)--(11,4) node[right] {$r$};
%    \draw[very thick] (4,7) node[left] {$t$}--(11,0);
%    \draw (5,4) node[circle,inner sep=2, fill=black] {} node[xshift=-8pt,yshift=-8pt,inner sep=1,fill=white] {$z$};
%    \draw (5,5) node[circle,inner sep=2, fill=black] {} node[xshift=-8pt,inner sep=1,fill=white] {$y$};
%    \draw (6,4) node[circle,inner sep=2, fill=black] {} node[yshift=-8pt,inner sep=1,fill=white] {$x$};
%    \draw (3,2) node[circle,inner sep=2, fill=black] {} node[xshift=-8pt,yshift=-12pt,inner sep=1,fill=white] {$w(z)$};
%    \draw (4,2) node[circle,inner sep=2, fill=black] {} node[yshift=-12pt,inner sep=1,fill=white] {$w(y)$};
%    \draw (3,3) node[circle,inner sep=2, fill=black] {} node[xshift=-16pt,inner sep=1,fill=white] {$w(x)$};
%  \end{tikzpicture}
%  \caption{The action of $w = rst$ in the group of affine type $B_2$
%    on the points $x, y, z$ in the euclidean plane, as in
%    Example~\ref{ex:B_2}.}\label{fig:B2example}
%\end{figure}

%\begin{example}[The move-set of an element in affine $B_2$]\label{ex:B_2}
%  To make the notion of a move-set concrete, we describe the move-set
%  of an element in affine $B_2$. Suppose $w = rst$, where $r$, $s$,
%  and $t$ are reflections across the lines pictured in Figure
%  \ref{fig:B2example}, and suppose $x, y,$ and $z$ are the points
%  labeled. We have also labeled the locations of $w(x)$, $w(y)$ and
%  $w(z)$.
%  
%  The standard basis vectors $\varepsilon_1$ and $\varepsilon_2$ in
%  $\R^2$ are the vectors that send $z$ to $x$ and $z$ to $y$,
%  respectively. By abuse of notation, we will write $\varepsilon_1 =
%  x-z$ and $\varepsilon_2 = y-z$. Having made this identification, we
%  can express any point $u$ in $E\cong \R^2$ as 
%  \[ 
%  u = z + a(x-z) + b(y-z) 
%  \] 
%  for some $(a,b)\in \R^2$. Using this coordinate system, we have
%  \begin{align*}
%    w(x) &= z-2(x-z)-(y-z) = x-3(x-z)-(y-z),\\
%    w(y) &=z-(x-z)-2(y-z) =y -(x-z)-3(y-z), \quad \mbox{and}\\
%    w(z) &=z-2(x-z)-2(y-z).
%  \end{align*}
%  We see that the motion of $x$ is the vector $(-3,-1)$, the motion of
%  $y$ is $(-1,-3)$, and the motion of $z$ is $(-2,-2)$.
%
%  In general, we use linearity to compute
%  \begin{align*}
%    w(u) &= w(z) +a w(x-z) + bw(y-z),\\
%    &=w(z) +a(w(x)-w(z)) + b(w(y)-w(z)),\\
%    &=z-2(x-z)-2(y-z) + a(y-z) + b(x-z),\\
%    &=u+(b-a-2)(x-z) + (a-b-2)(y-z).
%  \end{align*}
%  Thus, a generic motion is $\lambda = (-2,-2) + (b-a)\cdot (1,-1)$,
%  and the move-set of this element $w$ is an affine line:
%  \[
%  \mov(w) = (-2,-2)+\R(1,-1).
%  \]
%\end{example}

Move-sets for general euclidean plane isometries are given in
Example~\ref{ex:euc-plane}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Elliptics and translations}\label{sec:elliptic}

In this section, we record some basic facts about special kinds of
elements in an affine Coxeter group.

%\begin{defn}[Roots and reflections]\label{def:roots}
%  For a reflection $r$ whose fixed space is the hyperplane $H$,
%  the motion of any point under $r$ is in a direction orthogonal to $H$,
%  and $\mov(r)$ is the line through the origin in $V$ in this direction.
%  A vector $\alpha \in V$ is called a
%  \emph{root} of $r$ if $\mov(r) = \spn(\alpha) = \R \alpha$.
%  For affine Coxeter groups, the fixed spaces of the reflections come
%  in finitely many parallel families.  Within each family, the fixed
%  hyperplanes are equally spaced and the length of their common root
%  $\alpha$ can be chosen to encode additional information such as the
%  distance between adjacent parallel fixed hyperplanes.  Once the
%  roots are normalized in this way, the result is 
%  the usual crystallographic root system $\Phi$ inside $V$. The
%  $\Z$-span of $\Phi$ in $V$ is called the \emph{root lattice}.
%  It is an abelian group (under vector addition) 
%  isomorphic to $\Z^n$ where $n = \dim(V)$.
%\end{defn}

\begin{defn}[Elliptic elements and elliptic part]\label{def:ell-part}
  An element $w$ in an affine Coxeter group $W$ is called
  \emph{elliptic} if its fixed space is non-empty.  Equivalently,
  these are exactly the elements of $W$ of finite order.
  Given an arbitrary element $w \in W$, its \emph{elliptic part}
  $w_e = p(w)$ is its image under the projection $p: W \onto W_0$.
  (In particular, the elliptic part is an element of $W_0$, acting naturally
  on $V$ rather than on $E$.)
\end{defn}

%One can characterize elliptic elements in a variety of ways.
%
%\begin{lem}[Elliptic elements]\label{lem:ell}
%  For a euclidean isometry $w$, the following are equivalent: (1) $w$
%  is elliptic, (2) $\mov(w) \subset V$ is a linear subspace, and (3)
%  $\mov(w)$ contains the origin.
%\end{lem}
%
%\begin{proof}
%  \cite[Proposition~3.2 and Definition~3.3]{BradyMcCammond-15}.
%\end{proof}

\begin{defn}[Coroots]\label{def:coroots}
  Let $\Phi \subset V$ be a crystallographic root system. For each root
  $\alpha \in \Phi$, its \emph{coroot} is the vector $\alpha^\vee =
  \frac{2}{\langle \alpha, \alpha \rangle} \alpha$.  The collection of
  these coroots is another crystallographic root system $\Phi^\vee =
  \{ \alpha ^\vee \mid \alpha \in \Phi\}$.  The $\Z$-span $L(\Phi^\vee)$ of
  $\Phi^\vee$ in $V$ is the \emph{coroot lattice}; as an abelian 
  group, it is also isomorphic to $\Z^n$ where $n = \dim(V)$.
\end{defn}

\begin{defn}[Translations]\label{def:trans}
  For every vector $\lambda \in V$ there is a euclidean isometry
  $t_\lambda$ of $E$ called a \emph{translation} that sends each point $x
  \in E$ to $x + \lambda$.  Let $W$ be an affine Coxeter group acting
  on $E$ with root system $\Phi \subset V$.  An element of $W$ is a
  translation in this sense if and only if it is in the kernel $T$ of the
  projection $p\colon W \onto W_0$.  Moreover, the set of vectors in
  $V$ that define the translations in $T$ is identical to the set of
  vectors in the coroot lattice $L(\Phi^\vee)$.
\end{defn}

%The next result follows immediately from the definitions.
%
%\begin{prop}[Translated move-sets]
%\label{prop:translated move-set}
%  For any euclidean isometry $w$ and any translation $t_\lambda$, 
%  $\mov(t_\lambda w) = \lambda + \mov(w)$.
%\end{prop}

\begin{defn}[Translation-elliptic factorizations]\label{def:trans-ell}
  Let $W$ be an affine Coxeter group.  There are many ways to write an
  element $w \in W$ as a product of a translation $t_\lambda \in W$
  and an elliptic $u \in W$.  We call any such factorization $w =
  t_\lambda u$ a \emph{translation-elliptic factorization} of $w$.
  In any such factorization we call
  $t_\lambda$ the \emph{translation part} and $u$ the \emph{elliptic
    part}.
\end{defn}

\begin{defn}[Normal forms]\label{def:semidirect}
  Let $W$ be an affine Coxeter group acting cocompactly on a euclidean
  space $E$.  An identification of $W$ as a semidirect product $T
  \rtimes W_0$ corresponds to a choice of an inclusion map $i\colon
  W_0 \into W$ that is a section of the projection map $p\colon W \onto
  W_0$.  The unique point $x \in E$ fixed by the subgroup $i(W_0)$
  serves as our origin and every element has a unique factorization $w
  = t_\lambda u$ where $t_\lambda$ is a translation in $T$ and $u$ is
  an elliptic element in $i(W_0)$. In
  particular, $u = i(w_e)$ is the image under $i$ of the elliptic part
  of $w$ (Definition~\ref{def:ell-part}).  For a fixed choice of a
  section $i$, we call this unique factorization $w = t_\lambda u$ the
  \emph{normal form} of $w$ under this identification.
\end{defn}

If $w = t_\lambda u$ is a translation-elliptic factorization then $u_e
= w_e \in W_0$ and $t_\lambda$ is in the kernel of $p$.  Some
translation-elliptic factorizations come from an identification of $W$
and $T \rtimes W_0$ as in Definition~\ref{def:semidirect}, but not all
of them do: see Example 2.4 of \cite{LMPS-arxiv}.

%\begin{prop}[Recognizing elliptics]\label{prop:ell-recog}
%  Let $W$ be an affine Coxeter group and let $w = t_\lambda u$ be a
%  translation-elliptic factorization of an element $w \in W$.  The
%  following are equivalent: (1) $w$ is elliptic, (2) $\lambda \in
%  \mov(u)$, (3) $\mov(w) = \mov(u)$, and (4) $\mov(w) = \mov(w_e)$.
%\end{prop}
%
%\begin{proof}
%  By Proposition~\ref{prop:translated move-set}, $\mov(w) = \lambda +
%  \mov(u)$.  Since $u$ is elliptic, $\mov(u)$ is a linear subspace of
%  $V$ (Lemma~\ref{lem:ell}).  Thus, $\mov(w)$ is a linear subspace of
%  $V$ if and only if $\lambda \in \mov(u)$, so (1) and (2) are
%  equivalent.  Moreover, $\lambda \in \mov(u)$ if and only if $\mov(u)
%  = \lambda + \mov(u) = \mov(w)$, so (2) and (3) are equivalent.
%  Finally, for any elliptic element $w$, choose a point $x$ fixed by
%  $w$; then $w(x + \lambda) = x + w_e(\lambda)$ for all vectors
%  $\lambda$.  Thus $w(x + \lambda) = (x + \lambda) + (w_e(\lambda) -
%  \lambda)$, so $\mu$ belongs to $\mov(w)$ if and only if $\mu$ is of
%  the form $w_e(\lambda) - \lambda$, i.e., if $\mu$ belongs to
%  $\mov(w_e)$.  Thus (1) and (4) are equivalent.
%\end{proof}

%When isometries are composed, the new move-set is contained in the
%vector sum of the affine subspaces that are the individual move-sets.
%
%\begin{prop}[Move-set addition]\label{prop:move-add}
%  If $w_1$ and $w_2$ are elements of an affine Coxeter group $W$, then
%  $\mov(w_1 \cdot w_2) \subset \mov(w_1) + \mov(w_2)$.  Moreover, if
%  $r$ is a reflection and its root $\alpha$ is not in $\mov(w_e)$, 
%  then $\mov(wr) = \mov(rw) = \mov(r) + \mov(w)$.
%\end{prop}

%\begin{proof}
%  The motion vector of a point $x$ under the product $w_1w_2$ is the
%  motion vector of $x$ under $w_2$ plus the motion vector of $w_2(x)$
%  under $w_1$.  The second assertion is part of
%  \cite[Proposition~6.2]{BradyMcCammond-15}.
%\end{proof}
%
%This quickly leads to Carter's elegant geometric characterization of
%the reflection length in spherical Coxeter groups \cite{Carter72} and
%its extension to reflection length for elliptic elements in affine
%Coxeter groups.
%
%\begin{lemma}[Factoring elliptics]\label{lem:red-ell}
%  Let $w =r_1 r_2 \cdots r_k$ be a product of reflections in an affine Coxeter 
%  group, where reflection $r_i$ is through an affine hyperplane $H_i$ 
%  orthogonal to the root $\alpha_i$.  If $w$ is elliptic and $\lr(w)=k$ then the
%  roots $\alpha_i$ are linearly independent.  Conversely, if the roots
%  $\alpha_i$ are linearly independent then $w$ is elliptic, $\lr(w)=k$, 
%  $\fix(w) = H_1 \cap \cdots \cap H_k$, and $\mov(w) =
%  \spn(\{\alpha_1,\alpha_2,\ldots,\alpha_k\})$.
%\end{lemma}
%
%\begin{proof}
%  In \cite[Lemmas~3.6 and~6.4]{BradyMcCammond-15}, these facts are
%  proved in the full isometry group of the euclidean space $E$
%  generated by all possible reflections, but the statements and their
%  proofs easily restrict to the case where every reflection is in the
%  set $R \subset W$ and reflection length is computed with respect to
%  this collection of reflections.
%\end{proof}

\begin{rem}[Maximal elliptics]\label{rem:max-ell}
  Let $W$ be an affine Coxeter group acting cocompactly on an
  $n$-dimensional euclidean space $E$.  When $u$ is an elliptic
  element of reflection length~$n = \dim(E)$ (such as a Coxeter
  element of a maximal parabolic subgroup of $W$), its move-set
  $\mov(u)$ is all of $V$, 
%  (Lemma~\ref{lem:red-ell}). By the
%  equivalence of parts (1) and (2) in
%  Proposition~\ref{prop:ell-recog}, 
  the elements $t_\lambda u$ are
  elliptic for all choices of translation $t_\lambda\in T$ and, in
  particular, $\lr(t_\lambda u) = \lr(u) = n$ for every $t_\lambda \in
  T$.
\end{rem}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Dimension of an element}\label{sec:dim}

This section shows how to assign a dimension to each element in a
spherical or affine Coxeter group.  It is based on the relationship
between move-sets and root spaces.

\begin{defn}[Root spaces]\label{def:move-set-arr}
  Let $V$ be a euclidean vector space with root system $\Phi$.  A
  subset $U \subset V$ is called a \emph{root space} if it is the span
  of the roots it contains.  In symbols, $U$ is a root space when $U =
  \spn(U \cap \Phi)$.  Equivalently, $U$ is a root space when $U$ is a
  linear subspace of $V$ that is spanned by a collection of roots, or
  when $U$ has a basis consisting of roots.  Since $\Phi$ is a finite
  set, there are only finitely many root spaces; the collection of all
  root spaces in $V$ is called the \emph{root space arrangement}
  $\arr(\Phi) = \{ U \subset V \mid U = \spn(U \cap \Phi)\}$.
\end{defn}

\begin{defn}[Root dimension]\label{def:root-dim}
  For any subset $A \subset V$, we define its \emph{root dimension}
  $\dim_\Phi(A)$ to be the minimal dimension of a root space in
  $\arr(\Phi)$ that contains $A$.  Since $V$ itself is a root space,
  $\dim_\Phi(A)$ is defined for every subset $A$ in $V$.
\end{defn}

\begin{defn}[Dimension of an element]\label{def:dim-elt}
  When $w$ is an element of a spherical or affine Coxeter group, its
  move-set is contained in a euclidean vector space $V$ that also
  contains the corresponding root system $\Phi$.  The \emph{dimension}
  $\dim(w)$ of such an element is defined to be the root dimension of
  its move-set.  In symbols, $\dim(w) = \dim_\Phi(\mov(w))$.  Let $W$
  be an affine Coxeter group acting on a euclidean space $E$ and let
  $p\colon W \onto W_0$ be its projection map.  For each element $w
  \in W$ we can compute the dimension of $w$ and the dimension of its
  elliptic part $w_e = p(w) \in W_0$. We call $e = e(w) = \dim(w_e)$
  the \emph{elliptic dimension} of $w$.  Instead of focusing on the
  dimension of $w$ itself, we focus on the number $d = d(w) = \dim(w)
  - \dim(w_e)$, which we call the \emph{differential dimension} of
  $w$.  Note that $\dim(w) = d+e$.
\end{defn}

The statistics $d(w)$ and $e(w)$ carry geometric meaning.

\begin{prop}[Statistics and geometry]\label{prop:stat-geo}
  Let $W$ be an affine Coxeter group.  An element $w\in W$ is a
  translation if and only if $e(w) = 0$, and $w$ is elliptic if and
  only if $d(w) = 0$.
\end{prop}

%\begin{proof}
%  If $w$ is a translation, then it is in the kernel of the projection $p$,
%  $w_e$ is the identity, and $e(w) = 0$.  Conversely, if $e(w) = 0$
%  then $\mov(w_e) = \{0\}$, $w_e$ is the identity, $w$ is in the
%  kernel of $p$ and thus is a translation.  
%  
%  If $w$ is elliptic, then by Proposition~\ref{prop:ell-recog}
%  $\mov(w) = \mov(w_e)$, so $\dim(w) = \dim(w_e)$ and $d(w) = 0$.
%  Conversely, if $d(w) =0$ then $\dim(w) = \dim(w_e)$.  By
%  Lemma~\ref{lem:red-ell}, $\mov(w_e)$ is itself a root subspace, so
%  we must have $\mov(w) = \mov(w_e)$.  Then $w$ is elliptic by
%  Proposition~\ref{prop:ell-recog}.
%\end{proof}

Thus one way to interpret these numbers is that, roughly speaking,
$d(w)$ measures how far $w$ is from being an elliptic element and
$e(w)$ measures how far $w$ is from being a translation.  

%We record one
%more elementary fact for later use.
%
%\begin{lem}[Separation]\label{lem:sep}
%  Let $M$ and $U$ be linear subspaces of a vector space $V$ and let
%  $\lambda$ be a vector.  The space $M$ contains $\lambda + U$ if and
%  only if $M$ contains both $U$ and $\lambda$.
%\end{lem}
%
%\begin{proof}
%  If $M$ contains $\lambda + U$ then it contains $\lambda$ (since $U$
%  contains the origin) and it contains $-\lambda$ (since $M$ is closed
%  under negation).  Thus it contains $(\lambda + U) + (-\lambda) = U$.
%  The other direction is immediate.
%\end{proof}
%
%The next two propositions record the basic relationships between
%these dimensions and reflection length.
%
%\begin{prop}[Inequalities]\label{prop:inequal}
%  Let $W$ be an affine Coxeter group.  For every element $w \in W$,
%  $\lr(w) \geq \dim(w) \geq \dim(w_e) = \dim(U)$ where $U = \mov(w_e)$
%  is the move-set of the elliptic part of $w$.
%\end{prop}
%
%\begin{proof}
%  If $w$ can be written as a product of $k$ reflections, then
%  $\mov(w)$ is contained in the (linear) span of their roots
%  (Proposition~\ref{prop:move-add}).  Thus $k \geq \dim(w)$, and
%  choosing a minimum-length reflection factorization for $w$ shows
%  that $\lr(w) \geq \dim(w)$.  Next, if $w = t_\lambda u$ is a
%  translation-elliptic factorization of $w$, then $\mov(w) = \lambda +
%  U$, where $U = \mov(u) = \mov(w_e)$
%  (Proposition~\ref{prop:translated move-set}).  By
%  Lemma~\ref{lem:sep}, any root space $M$ that contains $\mov(w) =
%  \lambda + U$ also contains $\mov(w_e) = U$, which proves $\dim(w)
%  \geq \dim(w_e)$.  Finally, since $U$ is itself a root subspace by
%  Lemma~\ref{lem:red-ell}, we have $\dim(w_e) = \dim(U)$.
%\end{proof}
%
%\begin{prop}[Elliptic equalities]\label{prop:equal}
%  Let $W$ be an affine Coxeter group.  When $w$ is elliptic, $\lr(w) =
%  \dim(w) = \dim(w_e) = \dim(U)$ where $U = \mov(w) = \mov(w_e)$.
%\end{prop}
%
%\begin{proof}
%  By Proposition~\ref{prop:ell-recog}, $\mov(w) = \mov(w_e) = U$.  By 
%  Lemma~\ref{lem:red-ell}, $U$ is
%  a root space and $\lr(w) = \dim(U)$.  Proposition~\ref{prop:inequal}
%  completes the proof.
%\end{proof}

\begin{table}
  \begin{center}
  \begin{tabular}{|c|c|cc|c|}
    \hline
    $w$ & $\mov(w)$ & $d$ & $e$ & $\lr$\\
    \hline
    identity & the origin & 0 & 0 & 0 \\
    reflection & a root line & 0 & 1 & 1 \\
    rotation & the plane & 0 & 2 & 2 \\
    \hline
    translation & an affine point & 1 or 2 & 0 & 2 or 4\\
    \hline
    glide reflection & an affine line & 1 & 1 & 3 \\
    \hline
  \end{tabular}
  \end{center}
  \vspace*{1em}
  \caption{Basic invariants for the $5$ types of elements in an affine
    Coxeter group acting on the euclidean plane.}\label{tab:euc-plane}
\end{table}

\begin{example}[Euclidean plane]\label{ex:euc-plane}
  The affine Coxeter groups that act on the euclidean plane have five
  different types of move-sets (see Table~\ref{tab:euc-plane}).  
  Among the elliptic elements, the move-set of the identity is the
  point at the origin, the move-set of a reflection $r$ with root
  $\alpha \in \Phi$ is the root line $\R\alpha$, and the move-set of
  any non-trivial rotation is all of $V=\R^2$.  For these elements, 
  $d(w)= 0$ and $\lr(w) = \dim(w) = \dim(w_e) = e(w)$
  where this common value is $0$, $1$, or $2$, respectively.  The
  move-set of a non-trivial translation $t_\lambda$ is the
  single nonzero vector $\{\lambda\}$.  Its elliptic
  dimension is $0$ and its differential dimension is either $1$ 
  (when $\lambda$ is contained in a root line $\R\alpha$) or $2$.  By
  \cite[Proposition~4.3]{McPe-11}, $\lr(t_\lambda)$ is twice its
  dimension. Finally, when $w$ is a glide reflection, $\mov(w)$ is a
  line not through the origin, so $e = \dim(w_e) = 1$, $\dim(w) = 2$, 
  $d=2-1=1$, and $\lr(w) = 3$.
\end{example}

We finish this section with a pair of remarks about computing $e$ and
$d$ in general.  Let $W$ be an affine Coxeter group with a fixed
identification of $W$ with $T \rtimes W_0$ and let $w \in W$ be an
element that is given in its semidirect product normal form $w =
t_\lambda u$ for some vector $\lambda$ in the coroot lattice
$L(\Phi^\vee)$ and some elliptic element $u$. Computing the elliptic
dimension $e(w)$ is straightforward.

\begin{rem}[Computing elliptic dimension]\label{rem:computing-e}
  To compute the elliptic dimension $e(w)$ it is sufficient to simply
  compute the dimension of the move-set of the elliptic part $u$ of
  its normal form.  Indeed, by Definition~\ref{def:dim-elt},
  $e(w) = \dim(w_e)$, but since $\mov(w_e)$ is
  itself a root subspace,
  % by Lemma~\ref{lem:red-ell},
  $\dim_\Phi(\mov(w_e)) = \dim(\mov(w_e))$.  Finally, since $p(u) =
  p(w) = w_e$, $\mov(w_e) = \mov(u)$ 
  %by Proposition~\ref{prop:equal}
  and so $\dim(\mov(w_e)) = \dim(\mov(u))$.
\end{rem}

Computing the differential dimension $d(w)$ is more complicated but it
can be reduced to computing the dimension of a point in a simpler
arrangement of subspaces in a lower dimensional space.  How much lower
depends on the elliptic dimension $e(w)$.

\begin{rem}[Computing differential dimension]\label{rem:computing-d}
  By Definition~\ref{def:dim-elt}, to compute the differential 
  dimension $d(w)$, we need to find the minimal dimension of a root 
  subspace containing $\mov(w)$ and then subtract $e(w)$ from this
  value.  Since $w = t_\lambda u$, $\mov(w) = \lambda + U$ where $U =
  \mov(u) = \mov(w_e)$.  
  %Moreover, by Lemma~\ref{lem:sep} 
  As shown in \cite{LMPS-arxiv}, we only need
  to consider root spaces that contain $\lambda$ and $U$ or,
  equivalently, root spaces that contain $\lambda + U$ and $U$.  Let
  $q\colon V \onto V/U$ be the natural quotient linear transformation
  whose kernel is $U$.  Under the map $q$, the coset $\lambda + U$ is
  sent to a point in $V/U$ that we call $\lambda/U$ and the subspaces in
  $\arr(\Phi)$ containing $U$ are sent to a collection of subspaces in
  $V/U$ that we call $\arr(\Phi/U)$.  Let $\dim_{\Phi/U}(\lambda/U)$
  be the minimal dimension of a subspace in $\arr(\Phi/U)$ that
  contains the point $\lambda/U$.  Since the dimensions involved have
  all been diminished by $e(w) = \dim(U)$, we have that
  $\dim_{\Phi/U}(\lambda/U) = d(w)$ is the differential dimension of
  $w$.
\end{rem}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Proofs of main theorems}\label{sec:mains}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%In the following subsections we prove our main results.
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Proof of Theorem~\ref{main:dim}}\label{subsec:A}
%
%In this subsection, we prove Theorem~\ref{main:dim} by showing the
%claimed value is both a lower bound and an upper bound for the
%reflection length of $w$.  Let $W$ be an affine Coxeter group with
%root system $\Phi$ and projection map $p\colon W \onto W_0$.  For each
%element $w \in W$ we write $w_e = p(w)$ for the elliptic part of $w$,
%$e = e(w) = \dim(w_e)$ for its elliptic dimension, and $d = d(w) =
%\dim(w) - \dim(w_e)$ for its differential dimension.
%
%\begin{prop}[Lower bound]\label{prop:thmA-lb}
%  For every $w \in W$, $\lr(w) \geq 2d + e$.
%\end{prop}
%
%\begin{proof}
%  Let $w = r_1 r_2 r_3 \cdots r_k$ be a reflection factorization of
%  $w$. For each reflection $r_i$, let $\alpha_i$ be one of its
%  roots.  Let $M$ be the span of this set of vectors $\alpha_i$ and
%  let $m = \dim(M)$.  By Proposition~\ref{prop:move-add}, 
%  $\mov(w) \subset M$ and thus $d + e = \dim(w) \leq m$.
%
%  Next, pick a basis for $M$ from among the $\alpha_i$.  By
%  Lemma~\ref{lem:hurwitz}, we may assume that the reflections
%  corresponding to our chosen basis are the reflections $r_1, \ldots,
%  r_m$.  Define $u = r_1 r_2\cdots r_m$ 
%  and $v = r_{m+1} \cdots r_{k-1}r_k$, so that $w = uv$ and $u =
%  wv^{-1}$.
%
%  By Lemma~\ref{lem:red-ell}, $u$ is elliptic and $\mov(u) =\mov(u_e) =
%  M$, and so $\dim(u) = \dim(u_e) = m$ (Proposition~\ref{prop:equal}).
%  By Proposition~\ref{prop:inequal}, $ k-m \geq \dim(v) \geq \dim(v_e)
%  = \dim(v_e^{-1})$ because $v$ is a product of $k-m$ reflections.
%  Projecting $u = wv^{-1}$ gives $u_e = w_e v_e^{-1}$ and thus
%  $\mov(u_e) \subset \mov(w_e) + \mov(v_e^{-1})$ by
%  Proposition~\ref{prop:move-add}.  Taking dimensions shows that $m
%  \leq e + (k - m)$.  Thus, $k \geq 2m - e \geq 2(d+e)-e = 2d + e$, as
%  claimed.
%\end{proof}
%
%
%\begin{prop}[Upper bound]\label{prop:thmA-ub}
%  For every $w\in W$, $\lr(w) \leq 2d+e$.
%\end{prop}
%
%\begin{proof}
%  Let $w = t_\lambda u$ be the normal form of $w$ in $W$ for some
%  identification of $W$ and $T \rtimes W_0$ and let $U = \mov(u)$. By
%  Propositions~\ref{prop:translated move-set} and~\ref{prop:ell-recog}, 
%  $\mov(w) = \lambda + U$, $\mov(w_e) = U$, and
%  $e = \dim(U)$.  Let $m = \dim(w) = d+e$.  By definition, there
%  exists an $m$-dimensional root subspace $M$ containing the affine
%  subspace $\mov(w)$.  By Lemma~\ref{lem:sep}, $M$ must contain both
%  $\lambda$ and $U$.  Since $M$ has a basis of roots, there exists a
%  relative root basis $\{\alpha_1, \alpha_2, \ldots, \alpha_d\}$ for
%  $M$ over $U$, i.e., $M = \spn(\{\alpha_1, \ldots, \alpha_d\}) \oplus
%  U$ (and in particular, $\spn(\{\alpha_1, \ldots, \alpha_d\}) \cap U
%  = \{0\}$).
%
%  For each $i$, let $r_i$ be any reflection with root $\alpha_i$.
%  Consider the element $v=w r_1 r_2 \cdots r_d$.  Iteratively applying
%  Proposition~\ref{prop:move-add} one reflection at a time shows that
%  $\mov(v) = \spn(\{\alpha_1, \ldots, \alpha_d\}) + \mov(w)$.  Since
%  $\mov(w) = \lambda + U$, it follows that $\mov(v) = \spn(\{\alpha_1,
%  \ldots, \alpha_d\}) + \lambda + U = \lambda + M = M$.  Since $M$ is
%  a linear subspace of $V$, $v$ is elliptic and $\lr(v) = \dim(M) =
%  d+e$.  Since $w = v r_d \cdots r_2 r_1$, $\lr(w) \leq \lr(v) + d =
%  (d+e) + d = 2d+e$, as claimed.
%\end{proof}
%
%Together Propositions~\ref{prop:thmA-lb} and~\ref{prop:thmA-ub} prove
%Theorem~\ref{main:dim}.
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Proof of Theorem~\ref{main:fact}}\label{subsec:B}
%
%
%In this subsection, we prove Theorem~\ref{main:fact} using a recent
%technical result about inefficient factorizations in spherical Coxeter
%groups due to the first author and Vic Reiner
%\cite[Corollary~1.4]{LewisReiner-16}.
%
%\begin{prop}[Inefficient factorizations]\label{prop:inefficient}
%  Let $W_0$ be a spherical Coxeter group and let $w$ be an element of
%  $W_0$.  If $\lr(w) = \ell$, then every factorization of $w$ into $k$
%  reflections lies in the Hurwitz orbit of some factorization $w = r_1
%  r_2 \cdots r_k$ where $r_1 = r_2$, $r_3 = r_4$, \ldots $r_{k−\ell−1}
%  = r_{k−\ell}$, and $r_{k−\ell+1} \cdots r_k$ is a minimum-length
%  reflection factorization of $w$.
%\end{prop}
%
%We use this proposition about inefficient factorizations of an element
%in a spherical Coxeter group to find a way to rewrite an efficient
%factorization of an element in an affine Coxeter group into a
%particular form.  This is our second main result.
%
%\setcounter{mainthm}{\mainfact}
%\begin{mainthm}[Factorization]
%  Let $W$ be an affine Coxeter group.  For any element $w\in W$ there
%  is a translation-elliptic factorization $w = t_\lambda u$ such that
%  $\lr(t_\lambda) = 2d(w)$ and $\lr(u) = e(w)$.  
%  In particular, $\lr(w) = \lr(t_\lambda) + \lr(u)$ for
%  this factorization of $w$.
%\end{mainthm}
%
%\begin{proof}
%  Let $k = 2d +e = \lr(w)$ and let $w = r_1' r_2' \cdots r_k'$ be a
%  minimum-length reflection factorization of $w$.  The projection
%  $p\colon W \onto W_0$ sends this efficient factorization of $w \in
%  W$ to a (not necessarily efficient) factorization $w_e = p(w) =
%  p(r_1') p(r_2') \cdots p(r_k')$ of $w_e \in W_0$.  By Carter's lemma
%  (see Remark~\ref{rem:prior}) and Proposition~\ref{prop:ell-recog},
%  minimum-length reflection factorizations of $w_e$ all have length $e
%  = e(w)$.
%
%  By Proposition~\ref{prop:inefficient}, there is a sequence of
%  Hurwitz moves on the given $W_0$-factorization that produces a
%  factorization of a special form.  The exact sequence of Hurwitz
%  moves applied to the factorization in $W_0$ can be mimicked on the
%  original factorization in $W$; since the Hurwitz action is easily
%  seen to be compatible with $p$, the result is an $R$-factorization
%  $w = r_1 r_2 \cdots r_k$ of $w$ such that $p(r_1) = p(r_2)$, $p(r_3)
%  = p(r_4)$, \ldots, $p(r_{2d-1}) = p(r_{2d})$, and $p(r_{2d+1})
%  \cdots p(r_k)$ is a minimum-length reflection factorization of
%  $p(w)$.  This means that $r_1$ and $r_2$ are reflections through
%  parallel hyperplanes, and so $t_{\lambda_1} = r_1 r_2$ is a
%  (possibly trivial) translation. Similarly $t_{\lambda_i} = r_{2i -
%    1} r_{2i}$ is a translation for $i = 1, \ldots, d$.  Thus the
%  product of these first $2d$ reflections is a translation $t_\lambda
%  = t_{\lambda_1} t_{\lambda_2} \cdots t_{\lambda_{d}}$ and
%  $\lr(t_\lambda) \leq 2d$.
%
%  Let $u = r_{2d+1} \cdots r_k$ be the product of the remaining
%  $e$ reflections, so that $w = t_\lambda u$.  
%  Because the given factorization of $u$ projects to a
%  minimum-length reflection factorization of $u_e = w_e$ in $W_0$, the
%  roots of $p(r_{2d+1})$, \ldots, $p(r_k)$ must be linearly
%  independent (Lemma~\ref{lem:red-ell}), which means that the same is
%  true for $r_{2d+1}$, \ldots, $r_k$ in $W$.  By Lemma~\ref{lem:red-ell},
%  $u$ is elliptic and $\lr(u) = e$.  Finally, by Theorem~\ref{main:dim}
%  and the triangle inequality (Remark~\ref{rem:lr-basic}), we must have 
%  $\lr(t_\lambda) = 2d$.
%\end{proof}
%
%It is important to note that this theorem does \emph{not} assert that
%there exists an identification of $W$ with $T \rtimes W_0$ such that
%the corresponding normal form $w = t_\lambda u$ satisfies the
%conclusion of the theorem.  In fact this stronger assertion is
%demonstrably false.
%
%\begin{example}[Normal forms are insufficient]\label{ex:insuff}
%  In all irreducible affine Coxeter groups other than the affine
%  symmetric groups, there is a maximal parabolic subgroup $W'$ that is not
%  isomorphic to $W_0$.  Let $w$ be a Coxeter element for one of these
%  alternative maximal parabolic subgroups. 
%  
%  Being a Coxeter element of a spherical Coxeter group, $w$ has a
%  unique fixed point; denote it by $x'$. Reflections across the
%  hyperplanes passing through $x'$ generate $W'$. On the other hand,
%  reflections across the hyperplanes passing through any choice of
%  origin used in the construction of $W$ (i.e., the $r_{\alpha,0}$
%  reflecting across the $H_{\alpha,0}$ as in Definitions
%  \ref{def:affine} and \ref{def:sub-quo}) must generate a group
%  isomorphic to $W_0$. As $W'$ is not isomorphic to $W_0$, this means
%  $x'$ cannot be the origin for any such identification of $W$ with $T
%  \rtimes W_0$.
%  
%  Now fix an identification of $W$ and $T \rtimes W_0$ having origin
%  $x\neq x'$, and consider the corresponding normal form $w =
%  t_\lambda u$. By Remark~\ref{rem:max-ell}, $\lr(w) = \lr(u) =
%  n$. However, since $w(x)\neq x = u(x)$, we have $t_{\lambda}\neq 1$,
%  i.e., the normal form for $w$ includes a nontrivial translation.
%  Thus, for this element $w$, $\lr(w) < \lr(t_\lambda) + \lr(u)$.
%  
%  Figure \ref{fig:B2notnormal} provides an illustration.  Here we see
%  a portion of the affine line arrangement for $B_2$. The nodes in
%  black correspond to all possible identifications of an origin to
%  form the semidirect product.
%  
%  Let $r$ and $s$ be the reflections across the lines indicated in
%  bold, and let $w=rs$. Let $x'$ denote the point of intersection for
%  these lines. (The reflections $r$ and $s$ generate the maximal
%  parabolic subgroup of type $A_1\times A_1$.) We have $\lr(w) =
%  \lr(w_e)=2$. However, if we choose an origin $x\neq x'$ and write
%  $w=t_{\lambda}u$ with respect to this origin, then the motion $u$
%  will be a $\pi$-rotation about $x$ which requires at least $2$
%  reflections on its own, i.e. $\lr(u) = 2$.  But since $x$ and $x'$
%  are not equal and $w$ does not fix $x$, $t_\lambda$ cannot be
%  trivial.  Thus $\lr(t_\lambda) >0$ and $\lr(w) < \lr(t_\lambda) +
%  \lr(u)$.
%\end{example}
%  
%\begin{figure}
%  \begin{tikzpicture}[>=stealth,baseline=2.5cm,scale=1]
%    \foreach \x in {1,...,10}{
%      \draw (\x,.8)--(\x,6.2);
%    }
%    \foreach \x in {1,...,6}{
%      \draw (.8,\x)--(10.2,\x);
%    }
%    \foreach \x in {1,...,3}{
%      \draw (.8,2*\x+.2)--(2*\x+.2,.8);
%      \draw (.8,2*\x-.2)--(7.2-2*\x,6.2);
%      \draw (2*\x+3.8,.8)--(10.2,7.2-2*\x);
%    }
%    \foreach \x in {1,...,2}{
%      \draw (2*\x+.8,6.2)--(2*\x+6.2,.8);
%      \draw (2*\x+4.8,6.2)--(10.2,2*\x+.8);
%      \draw (2*\x+5.2,6.2)--(2*\x-.2,.8);
%    }
%    \draw[very thick] (6,0)--(6,7) node[right] {$r$};
%    \draw[very thick] (0,4)--(11,4) node[right] {$s$};
%    \draw (6,4) node[inner sep=2,circle,draw=black,fill=white] {}
%    node[inner sep = 1,xshift=8pt,yshift=8pt] {$x'$}; 
%    \foreach \x in {1,...,5}{
%      \foreach \y in {1,2,3}{
%        \draw (2*\x-1,2*\y) node[circle,inner sep=2, fill=black] {};
%        \draw (2*\x,2*\y-1) node[circle,inner sep=2, fill=black] {};
%      }
%    }
%  \end{tikzpicture}
%  \caption{The element $w=rs$ has $\ell_R(w) = \ell_R(w_e)=2$. The
%    origin can only be identified with the black nodes, whereas $x'$
%    is the unique fixed point of $w$. Thus every normal form for $w$
%    has a nontrivial translation.}\label{fig:B2notnormal}
%\end{figure}
%
%In an affine symmetric group, all maximal parabolic subgroups are
%isomorphic and every vertex (maximal intersection of hyperplanes)
%in the hyperplane arrangement can play the role of the origin in an
%identification with a semidirect product.  This immediately
%establishes the following result.
%
%\begin{corollary}[Affine symmetric normal form]
%  Let $W = \affS_n$ be the affine symmetric group.  For each element
%  $w \in W$ there is an identification of $W$ and $T \rtimes W_0$ so
%  that $w$ has normal form $w = t_\lambda u$ and $\lr(w) =
%  \lr(t_\lambda) + \lr(u)$.
%\end{corollary}
%
%\begin{proof}
%  Let $w = t_\lambda u$ be the factorization from
%  Theorem~\ref{main:fact}.  Choose an identification of $W$ and
%  $T \rtimes W_0$ so that the role of the origin is played by one
%  of the vertices that is fixed by $u$.
%\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Local statistics}\label{sec:gen-fns}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In a spherical Coxeter group $W = W_0$, Shephard and Todd
\cite[Theorem~5.3]{ShephardTodd} showed that the generating function for reflection length has a particularly nice form:
\begin{equation}\label{eq:ST}
f_0(t) =\sum_{u \in W_0} t^{\ell_R(t)} = \prod_{i = 1}^n (1 + e_i t),
\end{equation}
where the numbers $e_i$ are positive integers called the
\emph{exponents} of $W_0$.  In earlier work \cite{McPe-11}, the second
and third authors asked whether there are similarly nice generating
functions associated to an affine Coxeter group.  In this section, we
explore this question.

For an affine Coxeter group $W$, reflection length is bounded and
$|W|$ is infinite, so the naive generating function is not defined.  A
natural fix is to consider only a finite piece of $W$.

\begin{defn}[Local generating function]
  Given an element $\lambda$ of the coroot lattice $L(\Phi^\vee)$,
  define the bivariate generating function
  \[
  f_{\lambda}(s, t)
  = \sum_{u \in W_0} s^{d(t_\lambda u)} \cdot t^{e(t_\lambda  u)}
  = \sum_{u \in W_0} s^{\dim(t_\lambda  u)} \cdot (t/s)^{\dim(u)}
  \] 
  that tracks the statistics of differential and elliptic dimension.
  By Theorem~\ref{main:dim}, we have
  \[
  f_{\lambda}(t^2, t) = \sum_{u \in W_0} t^{\lr(t_\lambda \cdot u)}.
  \]
  By mild abuse of notation, we let $f_{\lambda}(t) =
  f_{\lambda}(t^2,t)$ denote this local reflection length generating
  function.
\end{defn}

The term ``local'' here makes sense geometrically, once the elements
of $W$ have been identified with alcoves in the reflecting hyperplane
arrangement for $W$.
% as in Remark \ref{rem:alcoves}. 
The alcoves
neighboring the origin (each of which is identified with a unique
element of $W_0$) form a $W_0$-invariant polytope $P$. The set $\{
t_\lambda u \,|\, u \in W_0\}$ corresponds to the set the of alcoves
in $t_\lambda \cdot P$, i.e., those alcoves neighboring $\lambda$. In
Figure \ref{fig:graylocal}(a) we have shaded the alcoves in affine $B_2$ according to
reflection length, while in Figure \ref{fig:graylocal}(b) we have colored the translates $t_\lambda \cdot P$ according to
$f_\lambda(t)$. In Figure \ref{fig:graylocal}(a), the identity element is
identified with the black alcove, and lighter colored cells have greater
reflection length. The coroots are highlighted in white.

\begin{figure}
  \[
  \begin{array}{cc}
    \includegraphics[trim={2cm 3cm 2.5cm 3.5cm}, clip, scale=.8]{B2graylocal}
    &
    \includegraphics[scale=.5]{B2local}
    \\
    (a) & (b)
  \end{array}
  \]
  \caption{For the affine Coxeter group of type $B_2$, (a) alcoves shaded by reflection length  and (b) the translates $t_\lambda\cdot P$ shaded by local generating function.}\label{fig:graylocal}
\end{figure}

We collect here some easy facts about the local generating function.

\begin{prop}[Properties of local generating functions]\label{prop:simple gf facts}
  Let $\lambda$ be a vector in the coroot lattice $L(\Phi^\vee)$.
  \begin{enumerate}
  \item (The origin) If $\lambda=0$, then $f_{\lambda}(s, t) = f_0(t)$
    is the generating function for reflection length in $W_0$.
  \item (Generic) If $\lambda$ is generic, i.e., if it is not a member
    of any proper root subspace of $V$, then $f_\lambda(s, t) = s^{n}
    f_0(t/s) = \prod_{i = 1}^n (s + e_i t)$, where the $e_i$
    are the exponents of $W_0$.
  \item (Permutations) If $\lambda$ and $\lambda'$ belong to the same
    $W_0$-orbit (that is, there is some $w \in W_0$ such that
    $\lambda' = w(\lambda)$), then $f_{\lambda}(s, t) =
    f_{\lambda'}(s, t)$.
  \end{enumerate}
\end{prop}

%\begin{proof}
%  If $\lambda = 0$ then $t_\lambda$ is the identity, and so (i)
%  follows immediately from the definitions.
%
%  For (ii), since $u$ is elliptic we have $\lambda \in \mov(t_\lambda
%  u)$.  Thus, since $\lambda$ is generic, $\mov(t_\lambda u)$ is not
%  contained in any proper root subspace of $V$, and so $\dim(t_\lambda
%  u) = n$.  Substituting this into the definition of $f_\lambda$ gives
%  the first equality.  The second equality follows from Shephard and 
%  Todd's result.
%
%  For (iii), suppose that $\lambda' = w(\lambda)$.  Then $t_{\lambda'}
%  = w t_{\lambda} w^{-1}$ and so for every $u$ in $W_0$ we have
%  $t_{\lambda'}u = w( t_\lambda (w^{-1} u w)) w^{-1}$.  Conjugation of
%  group elements by $w$ acts on move-sets as multiplication by $w$.
%  Since $w \in W_0$, this preserves root dimensions, and so
%  $\dim(t_{\lambda'} u) = \dim(t_{\lambda} (w^{-1}uw))$.  Finally, as
%  $u$ runs over the spherical group $W_0$, $w^{-1}uw$ does as well,
%  and $\dim(w^{-1}uw) = \dim(u)$, so the joint distribution of
%  dimensions is the same over both sets.
%\end{proof}

%To get a feel for the local generating functions, Table
%\ref{tab:rank2gf} lists the different local generating functions for
%types $A_2$, $B_2$, and $G_2$. In Figure~\ref{fig:colorlocal} we
%revisit the alcove picture, but instead of individual alcoves, we have
%colored the translates $t_\lambda \cdot P$ according to
%$f_\lambda(t)$. In Figure \ref{fig:faces} we see affine $A_3$ with the
%translates $t_\lambda\cdot P$ colored according to $f_\lambda(t)$. In
%this case there are six different local generating functions, as
%listed in Table \ref{tab:A3gf}. (Only five are visible in Figure
%\ref{fig:faces} as the origin is hidden.) The important thing to know
%about the notation in Table \ref{tab:A3gf} is that $\alpha_1$ and
%$\alpha_3$ are orthogonal to each other, whereas neither is orthogonal
%to $\alpha_2$.

From the pictures it appears that the local generating functions line
up along faces of a hyperplane arrangement. The faces of the
arrangement are intersections of maximal root subspaces, and if two
coroots lie in the same face, they have the same local generating
function. However, note that while some of these intersections are
themselves root subspaces, not all of them are. For example, the white
balls in Figure \ref{fig:faces} lie along lines that are not of the
form $\R\alpha$ for any root $\alpha$.

%\begin{table}[t]
%  \[
%  \begin{array}{c}
%    A_2 \\
%    \hline
%    \begin{array}{cc|c}
%      & \lambda & f_\lambda(s,t)\\
%      \hline
%      \begin{tikzpicture} \draw(0,0) node[circle,inner sep=2, fill=blue,draw=black] {}; \end{tikzpicture} & 0
%      & (1+t)(1+2t) \\
%      \begin{tikzpicture} \draw(0,0) node[circle,inner sep=2, fill=white!60!blue,draw=black] {}; \end{tikzpicture}  & \alpha^{\vee} & (s+t)(1+2t) \\
%      \begin{tikzpicture} \draw(0,0) node[circle,inner sep=2, fill=white!90!blue,draw=black] {}; \end{tikzpicture}  & \text{generic} & (s+t)(s+2t)
%    \end{array}
%  \end{array}
%  \hspace{.5cm}
%  \begin{array}{c}
%    B_2 \\
%    \hline
%    \begin{array}{cc|c}
%      & \lambda & f_\lambda(s,t)\\
%      \hline
%      \begin{tikzpicture} \draw(0,0) node[circle,inner sep=2, fill=blue,draw=black] {}; \end{tikzpicture} & 0 & (1+t)(1+3t) \\
%      \begin{tikzpicture} \draw(0,0) node[circle,inner sep=2, fill=white!60!blue,draw=black] {}; \end{tikzpicture} & \alpha^{\vee} & (s+t)(1+3t) \\
%      \begin{tikzpicture} \draw(0,0) node[circle,inner sep=2, fill=white!90!blue,draw=black] {}; \end{tikzpicture} & \text{generic} & (s+t)(s+3t)
%    \end{array}
%  \end{array}
%  \]
%  \[
%  \begin{array}{c}
%    G_2 \\
%    \hline
%    \begin{array}{cc|c}
%      &\lambda & f_\lambda(s,t)\\
%      \hline
%      \begin{tikzpicture} \draw(0,0) node[circle,inner sep=2, fill=blue,draw=black] {}; \end{tikzpicture} & 0 & (1+t)(1+5t) \\
%      \begin{tikzpicture} \draw(0,0) node[circle,inner sep=2, fill=white!60!blue,draw=black] {}; \end{tikzpicture} & \alpha^{\vee} & (s+t)(1+5t) \\
%      \begin{tikzpicture} \draw(0,0) node[circle,inner sep=2, fill=white!90!blue,draw=black] {}; \end{tikzpicture} & \text{generic} & (s+t)(s+5t)
%    \end{array}
%  \end{array}
%  \]
%  \caption{Local generating functions for affine $A_2$, $B_2$, and $G_2$.}\label{tab:rank2gf}
%\end{table}
%
%
%\begin{figure}
%  \[
%  \begin{array}{cc}
%    \includegraphics[scale=.5]{A2local}
%    &
%    \includegraphics[scale=.5]{B2local}
%    \\
%    (a) & (b)
%  \end{array}
%  \]
%  \[
%  \begin{array}{c}
%    \includegraphics[scale=.5]{G2local}
%    \\
%    (c)
%  \end{array}
%  \]
%  \caption{The translates $t_\lambda\cdot P$ in (a) type $A_2$, (b)
%    type $B_2$, and (c) type $C_2$, colored according to the local
%    distribution of reflection length.}\label{fig:colorlocal}
%\end{figure}

\begin{figure}
  \begin{center}
    \includegraphics[scale=.7]{A3ball_cropped}
  \end{center}
  \caption{The translates $t_\lambda\cdot P$ in type $A_3$ colored
    according to the local distribution of reflection length.}
  \label{fig:faces}
\end{figure}

%\begin{table}[t]
%  \[
%  \begin{array}{c}
%    A_3 \\
%    \hline
%    \begin{array}{cc|c}
%      &\lambda & f_\lambda(s,t)\\
%      \hline
%      \begin{tikzpicture} \draw(0,0) node[circle,inner sep=2, fill=orange,draw=black] {}; \end{tikzpicture} & 0 & (1+t)(1+2t)(1+3t) \\
%      \begin{tikzpicture} \draw(0,0) node[circle,inner sep=2, fill=black,draw=black] {}; \end{tikzpicture} & \alpha^{\vee} & (s+t)(1+2t)(1+3t) \\
%      \begin{tikzpicture} \draw(0,0) node[circle,inner sep=2, fill=white!90!black, draw=black] {}; \end{tikzpicture} &  \alpha_1^\vee + \alpha_3^\vee & 2t^2 + 6t^3 + 4st+9st^2 + s^2+2s^2t \\
%      \begin{tikzpicture} \draw(0,0) node[circle,inner sep=2, fill=yellow,draw=black] {}; \end{tikzpicture} & \text{generic span of } \alpha_1^{\vee}, \alpha_2^{\vee} & (s+t)(s+2t)(1+3t) \\
%      \begin{tikzpicture} \draw(0,0) node[circle,inner sep=2, fill=red,draw=black] {}; \end{tikzpicture} & \text{generic span of } \alpha_1^{\vee}, \alpha_3^{\vee} & (s+t)(t+6t^2+s+4st) \\
%      \begin{tikzpicture} \draw(0,0) node[circle,inner sep=2, fill=green!60!black,draw=black] {}; \end{tikzpicture} & \text{generic} & (s+t)(s+2t)(s+3t) \\
%    \end{array}
%  \end{array}
%  \]
%  \caption{Local generating functions for affine $A_3$. Here
%    $\alpha_1$ and $\alpha_3$ are any two orthogonal roots, while
%    $\alpha_2$ and $\alpha_1$ are not orthogonal.}\label{tab:A3gf}
%\end{table}

We make the phenomenon precise here.

\begin{thm}[Equality of local generating functions]\label{thm:faces}
  Suppose that $\lambda$ and $\mu$ are two elements of the coroot
  lattice $L$.  Suppose furthermore that $\lambda$
  and $\mu$ belong to the same collection of root subspaces.
  Then $f_\lambda(s, t) = f_\mu(s, t)$.
\end{thm}

%\begin{proof}
%  In fact, something stronger is true: if $\lambda$ and $\mu$ are as
%  described, then $\dim(t_\lambda u) = \dim(t_\mu u)$ for every $u$ in
%  $W_0$.  Indeed, by Lemma~\ref{lem:sep}, for $u \in W_0$, a root
%  subspace $U$ contains $\mov(t_\lambda u)$ if and only if it contains
%  $\lambda$ and $\mov(u)$. But by hypothesis, every such $U$ contains
%  $\mu$ and $\mov(u)$, and so contains $\mov(t_\mu u)$.  Then the
%  result follows immediately from the definition of $f_\lambda$.
%\end{proof}

Unfortunately, while Theorem \ref{thm:faces} and Proposition
\ref{prop:simple gf facts} imply bounds on the number of local
generating functions in terms of the number of $W_0$-orbits of
intersections of root subspaces, it is probably intractable to compute
all $f_\lambda(s,t)$, or even all $f_\lambda(t)$, in general. We show
in the Appendix of \cite{LMPS-arxiv} that computing $d(t_\lambda)$ for an
element $\lambda$ of the type $A_n$ coroot lattice is essentially
equivalent to the NP-complete problem \texttt{SubsetSum}.

%A different approach to understanding the distribution of reflection
%length would be to introduce a new statistic that grows with $\lambda$
%and take a bivariate generating function, either over the whole group
%$W$ or over the elements with fixed elliptic part (that is, over a
%coset of the translations).  Thus, for a given element $u \in W_0$ one
%could consider the generating function
%\[
%g_{u}(q,t) = \sum_{\lambda \in L} t^{d(t_{\lambda} u)}q^{\operatorname{stat}(\lambda, u)}
%\]
%for some statistic ``$\operatorname{stat}$'', e.g., usual the Coxeter
%length of $t_\lambda$. By Theorem~\ref{main:dim}, the corresponding
%generating function for reflection length is $t^{\dim(u)} g_u(q,
%t^2)$.
%
%If $u$ has maximal reflection length in $W_0$, e.g., if $u$ is a
%Coxeter element, then by Remark \ref{rem:max-ell} we have
%$d(t_{\lambda} u)=0$, and this generating function simplifies to
%$\sum_{\lambda \in L} q^{\operatorname{stat}(\lambda, u)}$. However,
%at this point we have no good candidate for
%$\operatorname{stat}(\lambda, u)$ and have made little progress in
%this direction.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Affine symmetric groups}\label{sec:aff-sym}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section, we restrict our attention to the affine symmetric
groups and give simple combinatorial descriptions of the statistics
$d(w)$ and $e(w)$ used to define reflection length.  This provides an
affine analog of the formula for the symmetric group given by D\'enes
\cite{Denes59}.

%Throughout this part of the article, we can safely relax our
%distinction between ``points'' in euclidean space and ``vectors'' in a
%vector space.  To this end, fix a euclidean vector space $\R^n$ with a
%fixed ordered orthonormal basis $e_1, e_2,\ldots, e_n$.
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Permutations and affine permutations}\label{sec:perms}
%
%In this section, we describe the symmetric group and affine symmetric
%group from both the geometric and combinatorial perspectives.
%
%\begin{defn}[Permutations]\label{def:perms}
%  A \emph{permutation} of the set $[n]$ is a bijection $\pi\colon [n]
%  \to [n]$ and the set of all permutations under composition is the
%  \emph{symmetric group} $\Sym_n$.  A permutation $\pi$ may be
%  represented in \emph{one line notation} as the sequence $[\pi(1),
%    \ldots, \pi(n)]$ of its values.
%\end{defn}
%
%\begin{defn}[Root system]\label{def:affS-roots}
%  For every permutation $\pi \in \Sym_n$, there is an isometry
%  $u_\pi\colon \R^n \to \R^n$ that sends $(v_1, \ldots, v_n)$ to
%  $(v_{\pi(1)}, \ldots, v_{\pi(n)})$.  All of these isometries fix the
%  line $\R \one$ in direction $\one = (1, \ldots, 1)$.  Let $V$ be the
%  orthogonal complement of this line, consisting of all vectors whose
%  coordinate sum is $0$.  Then $\Sym_n$ is a spherical Coxeter group
%  acting on $V$, as in Definition~\ref{def:spherical coxeter group}.
%  The reflections are exactly the \emph{transpositions} that
%  interchange two values $i, j \in [n]$ while fixing all others.  For
%  the reflection that switches the $i$-th and $j$-th coordinates we
%  select the vectors $\pm(e_i - e_j)$ as its roots, and we denote by
%  $\Phi_n$ the full root system $\Phi_{n} = \{ e_i -e_j \mid i, j \in
%  [n], i\neq j\}$.
%\end{defn}
%
%We record a few elementary properties of $\Phi_n$ without proof.
%
%\begin{prop}[Coroots]\label{prop:coroots-affS}
%  Let $\Phi_n$ be the root system for $\Sym_n$. Roots are the same as
%  coroots, the root system spans $V$, and the lattice of (co)roots is
%  the set of vectors in $V$ with integer coordinates.  In symbols,
%  $\Phi_n = (\Phi_n)^\vee$, $\spn(\Phi_n) = V$ and $L(\Phi_n)
%  = V \cap \Z^n$.
%\end{prop}
%
%Following Definition~\ref{def:affine}, one can use the root system
%$\Phi_n$ to construct the \emph{affine symmetric group} $\affS_n$ as a
%group of isometries of $V$.  By Definition~\ref{def:sub-quo}, every
%\emph{affine permutation} $w\in \affS_n$ can be written as $w =
%t_\lambda u_\pi$ where $u_\pi$ is the elliptic isometry of $\R^n$
%indexed by a permutation $\pi \in \Sym_n$ and $t_\lambda$ is a
%translation by a vector $\lambda$ in the $\Z$-span of $\Phi_n$, i.e.,
%a vector with integer coordinates and coordinate sum $0$.  There is
%also an alternative description of affine permutations, due originally
%to Lusztig \cite{Lusztig83}, that is common in the combinatorics
%literature (see, e.g., \cite{Eriksson98}).
%
%\begin{defn}[Affine permutations as bijections]\label{rem:affS-comb}
%  Let $w\colon \Z \to \Z$ be a bijection from the integers to the
%  integers.  We call the map $w$ an \emph{affine permutation} of
%  \emph{order} $n$ if it satisfies three conditions:
%  \begin{enumerate}
%    \item  $w(i+n)=n+w(i)$ for all $i \in \Z$,
%    \item  $w(i) = w(j) \mod n$ if and only if $i = j \mod n$, and   
%    \item $w(1) + w(2) + \cdots + w(n) = 1 + 2 + \cdots + n$.
%  \end{enumerate}
%  By the first condition, such periodic maps can be described in
%  \emph{one line notation} by listing the $n$ values
%  $[w(1),w(2),\ldots,w(n)] \in \Z^n$.  It is easy to check whether
%  such a list of $n$ integers satisfies the second and third
%  conditions.
%\end{defn}
%
%\begin{rem}[Isomorphisms]
%  The isomorphisms between the combinatorial and geometric
%  descriptions of $\affS_n$ go as follows.  Since every integer can be
%  uniquely written as a number in $[n]$ plus a multiple of $n$, the
%  one line notation $[w(1),w(2),\ldots,w(n)]$ can be uniquely written
%  as $u + t$, where $u$ has all entries in $[n]$ and $t = n \cdot
%  \lambda$ for some vector $\lambda \in \Z^n$.  By the second
%  condition, $u$ is the one line notation of a permutation $\pi$ in
%  $\Sym_n$, while by third condition, $\lambda$ has coordinate sum
%  $0$.  The bijection $w$ with this one line notation is sent by the
%  isomorphism to the element $t_\lambda u_\pi \in \affS_n$.
%\end{rem}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Partitions, cycles, and root arrangements}
%
%In this section, we describe the move-sets and fixed spaces of
%permutations in terms of their cycle structure.  For this purpose, it
%is helpful to work with the language of set partitions.
%
%\begin{defn}[Partitions]\label{def:partitions}
%  A \emph{partition} $P$ of a set $A$ is a collection of pairwise
%  disjoint nonempty subsets whose union is $A$.  The subsets in $P$
%  are called \emph{blocks}, and the \emph{size} $|P|$ of $P$ is the
%  number of blocks it contains.  Concretely, a partition of size $k$
%  is a collection $P = \{B_1, B_2, \ldots, B_k\}$ with $\emptyset \neq
%  B_i \subset A$ for all $i$, $B_i \cap B_j = \emptyset$ for $i \neq
%  j$, and $B_1 \cup \cdots \cup B_k = A$.
%\end{defn}
%
%\begin{defn}[Cycles and partitions]\label{rem:cyc-part}
%  For every permutation $\pi \in \Sym_n$, there is a partition
%  $P(\pi)$ with $i$ and $j$ in the same block of $P(\pi)$ if and only
%  if $i$ and $j$ are in the same orbit under the action of $\pi$.
%  Concretely, $i$ and $j$ are in the same block of $P(\pi)$ if and
%  only if there is an integer $\ell$ so that $\pi^\ell(i) = j$.  When
%  $\pi$ is written in \emph{cycle notation}, each block of $P(\pi)$
%  corresponds to the set of numbers contained one cycle of $\pi$.  For
%  example, the permutation $\pi$ in $\Sym_6$ with one line notation
%  $[4, 5, 1, 3, 2, 6]$ has cycle notation $\pi = (1,4,3)(2,5)(6)$ and
%  associated partition $P(\pi) = \{\{1,3,4\},\{2,5\},\{6\}\}$.  Let
%  $\cyc(\pi)$ be the set of cycles in $\pi$, so that $\size{\cyc(\pi)}
%  = \size{P(\pi)}$ is the number of cycles.
%\end{defn}
%
% 
%\begin{defn}[Partition lattice]\label{def:part_n}
%  Let $P$ and $Q$ be two partitions of the same set $A$. We say that
%  $P$ \emph{refines} $Q$ and write $P \leq Q$ if every block of $P$ is
%  contained in some block of $Q$.  We denote by $\prt_A$ the set of
%  all partitions of a fixed finite set $A$, ordered by
%  refinement. This poset is a bounded, graded lattice, and two
%  partitions are at the same height if and only if they have the same
%  size.  The partition $P_\one = \{A\}$ with only one block is the
%  unique maximum element.  The partition $P_\zero$ with $\size{A}$
%  blocks, each containing a single element, is the unique minimum
%  element.  When $A = [n]$, we write $\prt_n$ instead of $\prt_{[n]}$.
%\end{defn}
%
%The next few definitions describe the spaces that turn out to be equal
%to the fixed spaces and move-sets of permutations.
%
%\begin{defn}[Special vectors]\label{def:special}
%  Let $\R^n$ be a euclidean vector space and let $e_1, e_2,\ldots,
%  e_n$ be a fixed ordered orthonormal basis. For each subset
%  $\emptyset \neq B \subset [n]$, let $\one_B$ be the sum of the basis
%  vectors indexed by the numbers in $B$, i.e., $\one_B = \sum_{i\in B}
%  e_i$.  We call $\one_B$ a \emph{special vector}.  When $B = \{i\}$
%  only contains a single element, $\one_{\{i\}} = e_i$ is a basis
%  vector.  At the other extreme, $\one_{[n]} = \one = (1, \ldots, 1)$
%  is the all-$1$s vector.
%\end{defn}
%
%\begin{defn}[$F$-spaces]\label{def:f-spaces}
%  For each partition $P \in \prt_n$, we define the \emph{$F$-space}
%  $F_P \subset \R^n$ to be the set of all vectors whose coordinates
%  are constant on the blocks of $P$.  In other words, a vector $v =
%  (v_1, \ldots, v_n)$ is in $F_P$ if and only if $v_i = v_j$ for all
%  $i, j$ belonging to the same block $B \in P$.
%\end{defn}  
%
%\begin{remark}[$F$-spaces]
%  When $P = \{B_1, B_2, \ldots, B_k\}$, we have $F_P$ is equal to the
%  span of $\one_{B_1}$, $\one_{B_2}$, \ldots, $\one_{B_k}$, and these
%  special vectors form a basis for $F_P$.  In particular, $\dim(F_P) =
%  \size{P}$. At the extremes, $F_{P_\zero} = \spn(\{e_1,e_2,\ldots
%  e_n\}) = \R^n$ is the entire space and $F_{P_\one} = \spn(\{\one\})
%  = \R\one$ is a line.  Also note that $F_P \subset F_Q$ if and only
%  if $Q \leq P$ in $\prt_n$.  Thus every $F$-space contains the line
%  $F_{P_\one} = \R\one$.
%\end{remark}
%
%\begin{defn}[$M$-spaces]\label{def:m-spaces}
%  For each partition $P \in \prt_n$, we define the \emph{$M$-space}
%  $M_P \subset \R^n$ to be the orthogonal complement of the
%  corresponding $F$-space $F_P$.  In symbols, $M_P = (F_P)^\perp$ and
%  $F_P \oplus M_P = \R^n$.  In terms of coordinates, $M_P$ is the set
%  of vectors $v$ such that, for each block $B$ of $P$, the sum of the
%  coordinates of $v$ indexed by elements of $B$ is equal to $0$.
%\end{defn}
%
%\begin{remark}[$M$-spaces]\label{rem:m-spaces}
%  Since $\dim(F_P) = \size{P}$, we have $\dim(M_P) =
%  n-\size{P}$. Since $F_{P_\zero} = \R^n$, $M_{P_\zero} = \{0\}$ is
%  the trivial subspace.  At the other extreme, since $F_{P_\one} =
%  \R\one$ is a line, $M_{P_\one} = (F_\one)^\perp$ is the
%  codimension-$1$ subspace that we call $V$.  Taking orthogonal
%  complements reverses containment, so $M_P \subset M_Q$ in $\R^n$ if
%  and only if $P \leq Q$ in $\prt_n$.  Because every $F$-space
%  contains the line $\R\one = F_{P_\one}$, every $M$-space is
%  contained in the subspace $V = M_{P_\one}$.
%\end{remark}
%
%\begin{prop}[Permutations and partitions]\label{prop:perm-part}
%  For each $\pi \in \Sym_n$ with partition $P(\pi) \in \prt_n$, the
%  elliptic isometry $u_\pi$ acting on $\R^n$ has $\fix(u_\pi) =
%  F_{P(\pi)}$ and $\mov(u_\pi) = M_{P(\pi)}$.
%\end{prop}
%
%\begin{proof}
%  That a point $v$ is fixed under the action of $u_\pi$ if and only if
%  $v$ is in $F_{P(\pi)}$ is clear from the definition of the action,
%  which proves the assertion about the fixed space.  Since move-sets
%  and fixed spaces of elliptic isometries are orthogonal complements
%  \cite[Lemma~3.6]{BradyMcCammond-15}, as are $F_{P(\pi)}$ and
%  $M_{P(\pi)}$ (Definition~\ref{def:m-spaces}), the assertion about
%  move-sets follows from the assertion about the fixed space.
%\end{proof}
%
%\begin{remark}[Root arrangement]\label{rem:arr-affS}
%  Proposition~\ref{prop:perm-part} shows that the move-set of every
%  permutation $\pi \in \Sym_n$ is an $M$-space.  Moreover, it is easy
%  to see that every partition $P$ of $[n]$ is induced as $P = P(\pi)$
%  for some permutation $\pi$.  Thus, the move-sets of permutations are
%  exactly the $M$-spaces.  It follows from Lemma~\ref{lem:red-ell}
%  that these spaces are also exactly the set of root spaces for the
%  root system $\Phi_n$, and so the arrangement of $M$-spaces is
%  exactly the root space arrangement
%  (Definition~\ref{def:move-set-arr}).
%\end{remark}
%
%The root arrangement for $\Phi_n$ should not be confused with the
%better known braid arrangement.
%
%\begin{rem}[Braid arrangement]\label{rem:braid-arr}
%  The \emph{braid arrangement} in $\R^n$ is the collection of
%  $\binom{n}{2}$ hyperplanes that are orthogonal to a root in
%  $\Phi_n$.  Concretely, there is a hyperplane $H_{ij}$ for all
%  distinct $i,j\in [n]$ that contains the vectors $v$ with $v_i =
%  v_j$.  The result of taking all intersections of hyperplanes in the
%  braid arrangement is a subspace arrangement that is equal to the
%  collection of $F$-spaces $\{F_P \mid P \in \prt_n\}$.  By
%  Remark~\ref{rem:arr-affS}, the spaces in the root arrangement
%  $\arr(\Phi_n)$ are the $M$-spaces, i.e., the orthogonal complements
%  of the $F$-spaces.  The hyperplanes in $\arr(\Phi_n)$ are the
%  $M$-spaces $M_P$ where $P$ is a partition of size $2$, and there are
%  exactly $(2^n-2)/2 = 2^{n-1}-1$ of these.  Unlike the $F$-spaces,
%  the $M$-spaces are \emph{not} closed under intersection.  For
%  example, in $\R^4$, the intersection $M_{\{\{1,2\},\{3,4\}\}} \cap
%  M_{\{\{1,4\},\{2,3\}\}}$ is the line consisting of vectors of the
%  form $(a,-a,a,-a)$ for $a\in \R$, and this is not an $M$-space.
%\end{rem}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Null partitions}\label{sec:null-parts}
%
%In this section, we assemble the final combinatorial and
%linear-algebraic objects necessary to reinterpret reflection length.
%
%\begin{defn}[$L$-maps]\label{def:l-maps}
%  Let $v$ be a vector in $\R^n$.  For each nonempty subset $B \subset
%  [n]$, we define the real number $v_B$ to be the sum of the
%  coordinates of $v$ in the positions indexed by the set $B$.
%  Equivalently, $v_B = \langle v, \one_B\rangle$, where $\one_B$ is
%  the special vector defined in Definition~\ref{def:special}.  For
%  each partition $P \in \prt_n$ with $\size{P} = k$, we define a
%  linear transformation $L_P\colon \R^n \to \R^k$ as follows.  When $P
%  = \{B_1,\ldots, B_k\}$ we assign the names $e_{B_i}$ to an
%  orthonormal basis of $\R^k$ and we define $L_P(v) = \sum_{i=1}^k
%  v_{B_i} e_{B_i}$.  We call the maps of this form \emph{$L$-maps}.
%\end{defn}
%
%If we impose an order on the blocks of $P$, then in coordinates
%$L_P(v) = (v_{B_1},v_{B_2},\ldots, v_{B_k})$, but note that $L_P$ is a
%well defined map independent of such an ordering.  For example, if $v
%= (-5,2,1,0,4,-2)$ and $P= \{\{1,3\},\{2,4,6\},\{5\}\}$ with the
%blocks in that order, then $L_P(v) = (-4,0,4)$.
%
%\begin{prop}[$M$-space membership]\label{prop:m-l}
%  Let $P \in \prt_n$ be a partition of $[n]$.  Then $\ker L_P = M_P$.
%\end{prop}
%
%\begin{proof}
%  Let $P = \{B_1, \ldots, B_k\}$.  The vector $v$ is in the space
%  $M_P$ if and only if it is orthogonal to every vector in $F_P$.  Since
%  $\{\one_{B_1}, \ldots, \one_{B_k}\}$ is a basis of $F_P$, it is
%  sufficient to test whether $v$ is orthogonal to each $\one_{B_i}$,
%  and this is equivalent to being in the kernel of map $L_P$.
%\end{proof}
%
%\begin{defn}[Null partitions]\label{def:null-parts}
%  Let $v$ be a vector in $V = M_\one$. A nonempty subset $B\subset
%  [n]$ is called a \emph{null block} of $v$ if the sum $v_B$
%  (Definition~\ref{def:l-maps}) of the entries of $v$ indexed by $B$
%  is equal to $0$, or equivalently if $v$ is orthogonal to the special
%  vector $\one_B$ (Definition~\ref{def:special}).  A partition $P \in
%  \prt_n$ is called a \emph{null partition} of $v$ if every block of
%  $P$ is a null block of $v$.  Equivalently, $P$ is a null partition
%  of $v$ if and only if $v$ is in the kernel of $L_P$
%  (Definition~\ref{def:l-maps}) and if and only if $v \in M_P$
%  (Proposition~\ref{prop:m-l}).  We write $\nprt(v)$ for the
%  collection of all null partitions of $v$.  This collection is not
%  empty since $v\in V$ implies $\langle v, \one \rangle = 0$, which
%  means $P_\one \in \nprt(v)$.  The \emph{nullity} of $v$ is the
%  maximal size of a null partition of $v$.  In symbols, $\nll(v) =
%  \max\{ \size{P} \mid P \in \nprt(v)\}$.
%\end{defn}
%
%\begin{example}[Null partitions]\label{ex:null-parts}
%  Consider $v = (-5,2,1,0,4,-2)$ in $V = M_\one \subset \R^6$.  The
%  partition $P = \{\{1,3,5\},\{4\},\{2,6\}\}$ is a null partition of
%  $v$ and its blocks are null blocks of $v$.  The set $\{1,2,3\}$ is
%  not a null block of $v$ because $\langle v, \one_{\{1,2,3\}}\rangle
%  = -2 \neq 0$.
%\end{example}
%
%\begin{prop}[Nullity and dimension]\label{prop:null-dim}
%  For each $v \in V$, $\nll(v) + \dim_{\Phi_n}(\{v\}) = n$.
%\end{prop}
%
%\begin{proof}
%  By Definition~\ref{def:root-dim}, $\dim_{\Phi_n}(\{v\})$ is the
%  minimal dimension of a root space in $\arr(\Phi_n)$ that contains
%  $v$.  By Remark~\ref{rem:arr-affS}, root spaces ares $M$-spaces.
%  For each $P \in \prt_n$, $P \in \nprt(v)$ if and only if $v \in M_P$
%  (Definition~\ref{def:null-parts}) and $\size{P} + \dim(M_P) = n$
%  (Remark~\ref{rem:m-spaces}).  Thus, among the null partitions for
%  $v$, the ones that maximize $|P|$ simultaneously minimize
%  $\dim(M_P)$.  For a partition $P$ with these properties, the
%  equation $\size{P} + \dim(M_P) = n$ becomes $\nll(v) +
%  \dim_{\Phi_n}(\{v\}) = n$, as claimed.
%\end{proof}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Combinatorial formulas for statistics}
%
%In this section, we complete the work of
%Section~\ref{sec:aff-sym}. 

%For any affine permutation $w$, we give combinatorial expressions for the elliptic dimension $e(w)$ and the differential dimension $d(w)$. Thus by Theorem \ref{main:dim}, we
%obtain a combinatorial expression for the reflection length $\lr(w)$.
%
First, we briefly recall the normal form of an element in the affine symmetric group $\affS_n$. (See \cite{LMPS-arxiv} for full details.) If $w \in \affS_n$, we can write $w = t_{\lambda} u_{\pi}$, where $\pi \in \Sym_n$ is a permutation and $\lambda$ is a coroot, which we can express as a vector in $\Z^n$ whose entries sum to zero. For example, the pair $\lambda = (-2, -1,3,1,1,-2,0)$ and $\pi = (1,5,7)(2,4)(3)(6)$ determines a unique element $w = t_{\lambda}u_{\pi}$ in $\affS_7$. (Note we write $\pi$ in cycle notation.) We will fix this element as an example throughout this section.

Now we consider the elliptic dimension.  Our result generalizes
D\'enes's result that the reflection length of a permutation $\pi$ is
$n - \size{\cyc(\pi)}$.

\begin{prop}[Elliptic dimension]\label{prop:e-affS}
  Let $w \in \affS_n$ be an affine permutation, with normal form $w =
  t_\lambda u_\pi$ (so that $u_\pi$ is the elliptic part of $w$).
  Then $e(w) = n - \size{\cyc(\pi)}$.
\end{prop}

The elliptic dimension of our running example is thus $e(w) =7 - 4 = 3$.

%\begin{proof}
% By Remark~\ref{rem:computing-e}, $e(w) = \dim(\mov(u_\pi))$.  By
% Proposition~\ref{prop:perm-part}, this is equal to
% $\dim(M_{P(\pi)})$.  As noted in Remark~\ref{rem:m-spaces}, this is
% equal to $n - \size{P(\pi)} = n - \size{\cyc(\pi)}$, as claimed.
%\end{proof}
%

In Remark \ref{rem:computing-d}, we have that $d(w) = \dim_{\Phi/U}(\lambda/U)$. In the more combinatorial setting of the affine symmetric group, this relative dimension boils down to a statement about partitioning the vector $\lambda$. We use the term \emph{nullity}, denoted $\nu(\lambda)$, to describe the maximal size of a partition of the entries of $\lambda$ whose blocks all sum to zero. For example, $\nu((-2, -1,3,1,1,-2,0)) = 3$ since the partition $\{\{ -2, -1,3\}, \{-2,1,1\}, \{0\}\}$ has all its blocks sum to zero, and no partition into four or more parts has all its parts sum to zero. (Note that the partition $\{\{-2,-2,1,3\},\{-1,1\},\{0\}\}$ is also maximally refined in this way.) The \emph{relative nullity}, denoted $\nu(\lambda/\pi)$, is the nullity of the vector $\lambda'$ obtained by first coarsening $\lambda$ according to the cycle structure of $\pi$. With our running example, $\pi=(1,5,7)(2,4)(3)(6)$, so $\lambda' = (\lambda_1+\lambda_5+\lambda_7, \lambda_2+\lambda_4, \lambda_3,\lambda_6) = (-1,0,3,-2)$. This vector has nullity 2, so the relative nullity is $\nu(\lambda/\pi) = 2$.

%
%\begin{defn}[Relative nullity]\label{def:rel-null}
%  Let $w = t_\lambda u_\pi$ be an affine permutation in $\affS_n$ with
%  $\pi \in \Sym_n$ and $\lambda$ in the $\Z$-span of $\Phi_n$ and let
%  $k = \size{P(\pi)} = \cyc(\pi)$ be the number of cycles of the
%  permutation $\pi$ (Definition~\ref{def:perms}).  We call the nullity
%  of the point $L_{P(\pi)}(\lambda)$ in the arrangement $\arr(\Phi_k)$
%  the \emph{relative nullity of $\lambda$ modulo $\pi$} and we denote
%  this number by $\nll(\lambda/\pi)$.  In combinatorial language, the
%  relative nullity is the maximum size of a null partition $P$ of
%  $\lambda = w(0)$ such that every cycle of $\pi = p(w)$ is contained
%  in a single part of $P$.
%\end{defn}

We now state the combinatorial result for differential dimension.

\begin{prop}[Differential dimension]\label{prop:d-affS}
  An affine permutation $w \in \affS_n$ with normal form $w =
  t_\lambda u_\pi$ has $d(w) = |\cyc(\pi)|-\nll(\lambda/\pi)$.
% or $d(w) = n - e(w)-\nll(\lambda/\pi)$
\end{prop}

Thus in our example, the differential dimension is $d(w) = 4-2 = 2$.

%\begin{proof}
%  Let $U = \mov(u_\pi)$ be the move-set of the elliptic part $u_{\pi}$
%  of $w$, and let $k = \size{P(\pi)} = \size{\cyc{\pi}}$.  By
%  Remark~\ref{rem:computing-d}, the differential dimension $d(w)$ is
%  equal to the dimension of the point $\lambda/U$ in the subspace
%  arrangement $\arr(\Phi_n/U)$.  By Proposition~\ref{prop:perm-part},
%  the move-set $U$ is equal to the $M$-space $M_{P(\pi)}$, and by
%  Proposition~\ref{prop:m-l}, quotienting by the $M$-space
%  $M_{P(\pi)}$ is achieved by applying the $L$-map $L_{P(\pi)}$.  In
%  particular, the point $\lambda/U$ is the point $L_{P(\pi)}(\lambda)$
%  in $\R^k$.
%  
%  Let $\overline{\imath}$ denote the block of $P(\pi)$ containing $i$.
%  The image of the root $e_i-e_j$ in $\Phi_n$ under $L_{P(\pi)}$ is
%  $e_{\overline{\imath}}-e_{\overline{\jmath}}$, which is either equal
%  to $0$ (if $i$ and $j$ belong to the same cycle of $\pi$) or is a
%  root in $\Phi_k$.  Moreover, for each root in $\Phi_k$, we can find
%  a preimage in $\Phi_n$ by picking representatives.  It follows that
%  the quotient arrangement $\arr(\Phi_n/U)$ is equal to the root
%  arrangement $\arr(\Phi_k)$.  Thus $d(w)$ is the dimension of the
%  point $L_{P(\pi)}(\lambda)$ in the arrangement $\arr(\Phi_k)$.  By
%  Proposition~\ref{prop:null-dim} and Definition~\ref{def:rel-null},
%  we have
%  \[
%   \nll(\lambda/\pi) + \dim_{\Phi_k}(\lambda/U) = k.
%  \]
%  As $k=|\cyc(\pi)|$, we can rewrite this as
%  \[
%   d(w) = |\cyc(\pi)|-\nll(\lambda/\pi),
%  \]
%  as claimed.
%\end{proof}

Combining these propositions with the result Theorem \ref{main:dim}, that $\lr(w) = 2d(w) + e(w)$, gives us a combinatorial formula for the
reflection length of an element in an affine symmetric group. 

\begin{thm}[Formula]\label{thm:lr-affS}
  An affine permutation $w \in \affS_n$ with normal form $w =
  t_\lambda u_\pi$ has $\lr(w) = n - 2\cdot \nll(\lambda/\pi) +
  \size{\cyc(\pi)}$.
\end{thm}

For our running example, we compute the reflection length as $\lr(w) = 2\cdot 2 + 3 = 7$ or as $\lr(w) = 7 - 2\cdot 2 + 4 = 7$.


%\begin{proof}
%  By Theorem~\ref{main:dim} and Propositions~\ref{prop:e-affS}
%  and~\ref{prop:d-affS}, 
%  \[
%    \lr(w) = 2\cdot (\size{\cyc(\pi)} - \nll(\lambda/\pi)) + (n - \size{\cyc(\pi)}),
%  \] 
%  which simplifies to the expression in the statement of the theorem.
%\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Future work\label{sec:future}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Non-euclidean Coxeter groups}
%
%Is there a natural extension of Theorem~\ref{main:dim} to other
%infinite Coxeter groups?  Because reflection length is unbounded for
%all irreducible non-euclidean Coxeter groups of infinite type
%\cite{Duszenko12}, there cannot be a simple formula with a bounded
%number of summands representing bounded dimensions.  Thus any
%extension must necessarily have a different flavor.
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Identifying reduced factorizations}
%
%It is easy to characterize when reflection is minimal in spherical
%Coxeter groups: the product $r_1 \cdots r_k$ of reflections in a
%spherical Coxeter group has reflection length $k$ if and only if the
%associated roots are linearly independent.  Is there a similar
%criterion for affine Coxeter groups?  Because of the delicate behavior
%of some of our examples in the affine symmetric group, the criterion
%is likely to be complicated.
% 


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