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\def\Z{\mathbb{Z}}
\def\R{\mathbb{R}}
\def\N{\mathbb{N}}
\def\simp{\Delta}
\def\wa{\widetilde{W}}
\def\ac{\mathcal{A}}%_{\circ}}
\def\Q{\check{Q}}
\def\bijQtoW{\mathsf{crt}}
\def\m{m}
\def\Ell{\mathscr{L}}
\def\I{\mathcal{I}}
\def\J{\mathcal{J}}
\def\HH{\mathcal{H}}
\def\A{\mathcal{A}}
\def\alc{\mathsf{Alc}^m_{\Phi}}
\def\park{\mathsf{Park}^{(m)}(\Phi)}
\def\pta{\Theta}
\def\ptt{\Gamma}
\def\A{\mathcal{A}}
\def\L{\mathcal{L}}
\def\krew{\mathsf{Krew}^{(\m)}_{\Phi}}
\def\nar{\mathsf{Nar}^{(\m)}_{\Phi}}
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\def\fix{\mathsf{Fix}}
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\def\T{\mathcal{T}}
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\def\Pl{\check{P}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  Title page
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title[Strange Expectations II]
  {Strange Expectations in Affine Weyl Groups}


%\email{stuck127@umn.edu}
%\email{thiel.marko@gmail.com}
%\email{nathan.f.williams@gmail.com}


%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[E.N.~Stucky, M.~Thiel, N.~Williams]{Eric Nathan Stucky\thanks{\href{mailto:stuck127@umn.edu}{stuck127@umn.edu}}\addressmark{1} \and Marko Thiel\thanks{\href{mailto:thiel.marko@gmail.com}{thiel.marko@gmail.com}}\addressmark{2}, \and Nathan Williams\thanks{\href{mailto:nathan.f.williams@gmail.com}{nathan.f.williams@gmail.com}}\addressmark{3}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}University of Minnesota \\ \addressmark{2}Marshall Wace LLP \\ \addressmark{3}University of Texas at Dallas}

\received{\today}

%\revised{}

\abstract{
We previously generalized P.~Johnson's proof of D.~Armstrong's conjecture for the expected number of boxes in a
simultaneous core to simply-laced type.  After recalling combinatorial core-like models for coroot lattices in the classical types, our main result is a generalization of this theorem to all affine Weyl groups.}

%\resume{
%We previously generalized P.~Johnson's proof of D.~Armstrong's conjecture for the expected number of boxes in a simultaneous core to simply-laced type.  After recalling combinatorial core-like models for coroot lattices in the classical types, our main result is a generalization of this theorem to all affine Weyl groups.}
\keywords{root system, affine Weyl group, integer partition, core, expected value}
\usepackage[backend=bibtex]{biblatex}
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\begin{document}
\maketitle



%======%
\section{Introduction}
\label{sec:introduction}


%------%
\subsection{Motivation}


An \defn{$a$-core} is an integer partition with no hook of length $a$.  An \defn{$(a,b)$-core} is a partition that is simultaneously an $a$-core and a $b$-core.  For $a$ and $b$ relatively prime, it turns out that there are only finitely many $(a,b)$-cores: \[\big|\core(a,b)\big| = \frac{1}{a+b}\binom{a+b}{b}.\]
For $\lambda$ a partition, write $\lambda^\intercal$ for its conjugate and $\size(\lambda)$ for the number of its boxes.  The starting point for a number of recent investigations has been Armstrong's conjecture on the average number of boxes in an $(a,b)$-core, and in a self-conjugate $(a,b)$-core~\cite{armstrong2011conjecture,armstrong2014results}.

\begin{theorem}[{\cite{johnson2015lattice}}]\label{thm:arm}  For $\gcd(a,b)=1$,
\begin{equation*}
\Expt{\lambda \in \core(a,b)}{\size(\lambda)} = \frac{(a-1)(b-1)(a+b+1)}{24} = \Expt{\substack{\lambda \in \core(a,b) \\ \lambda = \lambda^\intercal}}{\size(\lambda)}.
\end{equation*}
\end{theorem}
Both equalities in~\Cref{thm:arm} were proven by Johnson using weighted Ehrhart theory~\cite{johnson2015lattice}.  In~\cite{thiel2017strange}, we generalized Armstrong's conjecture and Johnson's proof (of the first equality) to all \emph{simply-laced} affine Weyl groups.  In this extended abstract, we complete the generalization to all affine Weyl groups. 


%------%
\subsection{Combinatorial Models of Coroot Lattices}


The set of $a$-cores under the action of the affine symmetric group $\widetilde{\mathfrak{S}}_a$ is a well-studied combinatorial model for the coroot lattice $\chk{\Q}_{a}$ of type $A_{a-1}$. Indeed, for all affine Weyl groups $\wa=\wa(X_n)$, there is a well-known $\wa$-equivariant map from the group to the coroot lattice $\wa = W \ltimes \chk{\Q}_{X_n} \to \chk{\Q}_{X_n}$
\begin{align*}
\waf = w \cdot t_{-q} &\mapsto \waf^{-1}(0)=t_{q}\cdot w^{-1}(0)=t_{q}(0)=q,
\end{align*}
which restricts to a $\wa$-equivariant bijection on the cosets $W\backslash\wa$.\footnote{Where $\widetilde{v} \in \wa$ acts on the right for $\waf \in W\backslash\wa$ by $\waf \cdot \widetilde{v}$, and on the left for $q \in \chk{\Q}_{X_n}$ by $\widetilde{v}^{-1}(q)$.}  Thus, combinatorial models for $\chk{\Q}_{X_n}$ also give models for $W\backslash \wa$, representatives usually taken to be dominant affine elements.  In type $A_{a-1}$, these correspondences give $\widetilde{\mathfrak{S}}_a$-equivariant bijections
\begin{equation}
\begin{alignedat}{2}
\core(a) &\leftrightarrow \chk{\Q}_{a} &&\leftrightarrow \mathfrak{S}_{a}\backslash\widetilde{\mathfrak{S}}_a \\
 \lambda &\leftrightarrow q_\lambda &&\leftrightarrow \waf_\lambda.
\end{alignedat}
\label{eq:core_to_coroot}
\end{equation}

It is an easy exercise to produce similar combinatorial models for the quotients $W\backslash \widetilde{W}$ of other classical types ($X_n \in \{A_n,B_n,C_n,D_n\}$) by embedding $\chk{\Q}_{X_n}$ into an appropriate type $A$ coroot lattice. These, as well as a model for $X_n=G_2$, are illustrated for rank two root systems in~\Cref{fig:a2}.
%--in~\Cref{sec:models} we use this strategy to produce a new model for $\chk{\Q}_{G_2}$ using $3$-cores.

% PICTURES FOR A_2, C_2, AND G_2
\import{}{fpsaccorepictures.tex} 

Under the correspondence between $a$-cores and $\chk{\Q}_{a}$ of~\Cref{eq:core_to_coroot}, the set of $(a,b)$-cores are exactly those coroot points that sit inside of a certain $b$-fold dilation of the fundamental alcove called the $b$-Sommers region (see \Cref{def:sommers}).  The natural generalization of $\core(a,b)$ to any affine Weyl group is the intersection of the coroot lattice $\chk{\Q}_{X_n}$ with the $b$-Sommers region, so that $\core(a,b) = \core(A_{a-1},b)$, where
\begin{equation}
\label{eq:Wcores}
    \core(X_n,b) := \chk{\Q}_{X_n} \cap \Sommers(b).
\end{equation}





%------%
\subsection{Size Statistics}


%\subsubsection{Simply-Laced}

Under the bijections of~\Cref{eq:core_to_coroot}, we noticed in~\cite{thiel2017strange} that the number of boxes in $\lambda$ could be computed from the coroot $q_\lambda$ as described above, or the inversion set of $\widetilde{w}_\lambda^{-1}$, where $\mathsf{inv}(\widetilde{w})=\widetilde{\Phi}^+ \cap \widetilde{w}(-\widetilde{\Phi}^+)$. More precisely:

\begin{proposition}[{\cite[Proposition 6.4 \& Corollary 6.7]{thiel2017strange}}] Let $\lambda$ be an $a$-core and $\chk{\rho}$ be the sum of the fundamental coweights in type $A_{a-1}$. Then
$$\size(\lambda) = \sum\limits_{\alpha+k\delta\in\mathsf{inv}(\waf^{-1}_\lambda)}k = \left\langle \frac{a}{2}q_{\lambda} - \chk{\rho},q_\lambda \right\rangle.$$
\end{proposition}
It was natural to consider the corresponding statistic in \emph{any} affine Weyl group $\widetilde{W}(X_n)$ acting on $V$, restricting to a certain finite set of coroots $\core(X_n,b)$ (defined below in~\Cref{eq:Wcores}, in analogy with simultaneous $(a,b)$-cores) .  For simply-laced Weyl groups, our result mirrored~\Cref{thm:arm}.

\begin{theorem}[{\cite[Theorem 1.10]{thiel2017strange}}]
\label{thm:us}
Let $X_n$ be a simply-laced Cartan type with Coxeter number $h$, and let $b$ be coprime to $h$.  Then
\begin{equation*} \Expt{q \in \core(X_n,b)}{\size(q)} = \frac{n(b-1)(h+b+1)}{24}. \end{equation*}
\end{theorem}

% TODO: Can these be removed?
When applied to $X_n=A_{a-1}$ (so that $n=a-1$ and $h=a$), our result gives a proof of the left equality of \Cref{thm:arm} for the expected size of simultaneous $(a,b)$-cores.  But since self-conjugate cores are a combinatorial model for coroots in the \emph{non-simply-laced} type $C_n$, we were unable to similarly specialize \Cref{thm:us} to conclude the right equality of \Cref{thm:arm} for the expected size of a self-conjugate simultaneous core.  

Our mistake was to take \Cref{thm:us} as evidence that we had determined the ``correct'' generalization of the number of boxes of an $a$-core to all affine Weyl groups---the trouble is that we were unable to apply the Ehrhart-theoretic techniques of~\Cref{sec:expectation} using this definition outside of simply-laced type.  To be able to apply these techniques, it turns out that we must modify the above definitions to incorporate different root lengths.


\subsection{Expected Size}
%\subsubsection{Non-Simply-Laced Types}
Normalize root systems so that the highest root has length $2$, and write $r$ for the ratio of the length of a long to a short root.  We define a new statistic on coroots that recovers our old definition of $\size$ in simply-laced type, but disagrees in non-simply-laced type.   For $\waf=w\cdot t_{-q} \in W\backslash \widetilde{W}$, define (the second equality is proven in~\Cref{thm:size_eq})
\begin{equation}
    \chk{\size}(\waf) := \left(\sum_{\substack{\alpha+k\delta \in \mathsf{inv}(\waf^{-1})\\\alpha \text{ long}}}k\right)+r\left(\sum_{\substack{\alpha+k\delta \in \mathsf{inv}(\waf^{-1})\\\alpha \text{ short}}}k\right) = \left\langle \frac{h}{2}q - \chk{\rho},q \right\rangle.
\end{equation}

We interpret $\chk{\size}$ as statistics on the combinatorial models of~\Cref{sec:models}, showing (for example) that $\chk{\size}$ in type $C_n$ corresponds to the number of boxes in the corresponding self-conjugate $2n$-core (see~\Cref{fig:a2}).


%------%


Following the same strategy as in~\cite{thiel2017strange}, we find an affine Weyl group element that maps $\Sommers(b)$ to a $b$-fold dilation of the fundamental alcove (correctly modifying the $\chk{\size}$ statistic), and then apply Ehrhart theory to compute the expected value of $\chk{\size}$ on $\core(X_n,b)$.

\begin{restatable}{theorem}{mainthm}
\label{thm:main_thm}
For $X_n$ an irreducible rank $n$ Cartan type with root system $\Phi$,
\begin{align*}
%\max{\lambda \in \chk{\Lambda} \cap b\mathcal{A}}{\chk{\size}_b(\lambda)} &=    \frac{r \chk{g}}{h}\frac{n(b^2-1) (h+1)}{24} &&\overset{\gcd(h,b)=1}{=} \max{\lambda \in \core(\widetilde{W},b)}{\chk{\size}(\lambda)} , \text{ and}\\
\Expt{\lambda \in \core(X_n,b)}{\chk{\size}(\lambda)}=\frac{r \chk{g}}{h}\frac{n(b-1)(h+b+1)}{24},
\end{align*}
where $h$ is the Coxeter number of $X_n$, $\chk{g}$ is the dual Coxeter number for $\chk{\Phi}$, and $r$ is the ratio of the length of a long root to the length of a short root in $\Phi$.
\end{restatable}

The extra factor of $\frac{r\chk{g}}{h}$ is \emph{invisible} in the simply-laced case, where $\chk{\Phi}=\Phi$, $\chk{g}=h$, and $r=1$.  As an immediate application of~\Cref{thm:main_thm}, we conclude both equalities in~\Cref{thm:arm} by specializing to these types.    Interestingly, although the expected number of boxes in a simultaneous core and in a self-conjugate simultaneous core happen to be the same, the formulas have quite different interpretations: the factor of $a-1$ corresponds to the dimension $n$ for ordinary simultaneous cores, but to $\chk{g}$ in the self-conjugate case.   



%======%
\section{Affine Weyl Groups}


Let $\Phi=\Phi^+ \sqcup \Phi^-$ be an irreducible crystallographic root system of Cartan type $X_n$ with ambient space $V$.  Let $n$ be its rank, $h$ its Coxeter number, $\Delta$ be its set of simple roots, and $\chk{\Q}$ be its coroot lattice. Write $\chk{\omega_i}$ for the fundamental coweights, and also set $\chk{\omega}_0:=0$. Normalize the inner product on $V$ so that $\langle \beta,\beta \rangle=2$ for $\beta$ a long root and define $r:=\frac{\langle \beta,\beta \rangle}{\langle \alpha,\alpha \rangle}$ for $\beta$ a long root and $\alpha$ a short root.

Recall that the corresponding Weyl group $W=W(X_n)$ is generated by the reflections $s_\alpha(x) := x - 2\frac{\langle \alpha,x\rangle}{\langle \alpha,\alpha\rangle}\alpha$ for $\alpha \in \Delta$.  There is a unique \defn{highest root} $\widetilde{\alpha} \in \Phi^+$ for which $\widetilde{\alpha}+\alpha \not \in \Phi^+$ for any $\alpha \in \Delta$. The corresponding affine Weyl group $\wa= W \ltimes \chk{\Q}$ is generated by the $s_\alpha$ for $\alpha \in \Delta$ along with the additional affine simple reflection $s_{\widetilde{\alpha},1}:=x - (\langle \widetilde{\alpha},x\rangle-1)\widetilde{\alpha}.$  More generally, 
  reflections act by $s_{\alpha,k}(x)=x-(\langle\alpha,x\rangle-k)\chk{\alpha} = s_\alpha(x)+k\chk{\alpha}.$

Recall that $\wa$ acts on the affine roots $\widetilde{\Phi} = \{\alpha+k\delta : \alpha \in \Phi,k\in \mathbb{Z}\}$ by 
%\begin{equation}
%\label{eq:rootaction}
$\waf \cdot (\alpha + k \delta) = w(\alpha) + (k-\langle \alpha,q\rangle)\delta$
when $\waf = w \cdot t_q$ for $w \in W$ and $q \in \chk{\Q}$~%\cite{dyer2011reflection}. 
% TODO: Is it okay to omit this citation? If it is included, the last page is two lines too long. I wasn't able to find any easy cosmetic changes that fix it otherwise.
Given a reduced word $\mathsf{\widetilde{w}}=a_{1} a_2 \cdots a_{\ell}$ for  $\waf \in \wa$ (with the $a_i$ simple reflections), we associate the affine roots in the inversion set of $\waf$ to the letters $a_{i}$ in $\mathsf{\widetilde{w}}$ by $(a_1\cdots a_{i-1})(\alpha_{a_i})=\beta_i+k_i\delta$.  We call the ordered sequence $\inv(\mathsf{\widetilde{w}})=\beta_1+k_1\delta,\beta_2+k_2\delta,\ldots,\beta_\ell+k_\ell\delta$ the \defn{inversion sequence} corresponding to the word $\mathsf{\widetilde{w}}$.  (These record the affine hyperplanes that separate $w(\mathcal{A})$ from the fundamental alcove $\mathcal{A}$.)


\begin{comment}
Let $W$ be a Weyl group acting in the reflection representation on the $n$-dimensional real vector space $V$ with \defn{simple roots} $\Delta$ and \defn{roots} $\Phi=\Phi^+ \sqcup \Phi^-$.  $W$ is generated by the reflections $s_\alpha(x) := x - 2\frac{\langle \alpha,x\rangle}{\langle \alpha,\alpha\rangle}\alpha$ for $\alpha \in \Delta$.  There is a unique \defn{highest root} $\widetilde{\alpha} \in \Phi^+$ for which $\widetilde{\alpha}+\alpha \not \in \Phi^+$ for any $\alpha \in \Delta$.  We normalize the highest root (and, hence, all long roots) to have length $\sqrt{2}$. The corresponding affine Weyl group $\widetilde{W}$ is generated by the $s_\alpha$ for $\alpha \in \Delta$ along with the additional affine simple reflection $s_{\widetilde{\alpha},1}:=x - (\langle \widetilde{\alpha},x\rangle-1)\widetilde{\alpha}.$  More generally, 
  reflections act by \begin{equation}
\label{eq:reflection}
s_{\alpha,k}(x)=x-(\langle\alpha,x\rangle-k)\chk{\alpha} = s_\alpha(x)+k\chk{\alpha}.
\end{equation}

Let $\Phi$ be an irreducible crystallographic root system with ambient space $V$.  Let $n$ be its rank and $h$ its Coxeter number.  Normalize the inner product on $V$ so that $\langle \beta,\beta \rangle=2$ for $\beta$ a long root and define $r:=\frac{\langle \beta,\beta \rangle}{\langle \alpha,\alpha \rangle}$ for $\beta$ a long root and $\alpha$ a short root.  Write $\chk{\Lambda\!}$ for the coweight lattice, $\chk{\rho}$ for the sum of the fundamental coweights, and $\widetilde{\alpha}$ for the highest root.  
Write $\chk{\omega_i}$ for the fundamental coweights, with $\chk{\omega}_0:=0$.   Recall that $\wa = W \ltimes \chk{\Q}$ acts on the affine roots $\widetilde{\Phi} = \{\alpha+k\delta : \alpha \in \Phi,k\in \mathbb{Z}\}$ by 
\begin{equation}
\label{eq:rootaction}
\waf \cdot (\alpha + k \delta) = w(\alpha) + (k-\langle \alpha,q\rangle)\delta
\end{equation}
when $\waf = w \cdot t_q$ for $w \in W$ and $q \in \chk{\Q}$~\cite{dyer2011reflection}.  We normalize so that long roots have length $2$ and write $r$ for the ratio of a long root to a short root.
% TODO: IS THIS THE CORRECT DIRECTION, OR IS IT $t_q w$
\end{comment}


\begin{example}
\label{ex:a2inv}
%=s_0s_1s_2s_0s_1s_0=
We compute the inversion sequence for the reduced word $s_0s_1s_2s_1s_0s_1$ for the element $\waf=s_1s_2t_{-2\chk{\alpha}_2} \in \widetilde{A}_2$ (see also~\Cref{fig:abacus}):
\begin{align*}
    \inv(s_0s_1s_2s_1s_0s_1) &= -\widetilde{\alpha}+\delta,-\alpha_2+\delta,-\widetilde{\alpha}+2\delta,-\alpha_1+\delta,-\widetilde{\alpha}+3\delta,-\alpha_2+2\delta.
\end{align*}

\end{example}

\begin{theorem}
\label{thm:coset_to_coroot}
The $\wa$-equivariant map $\wa = W \ltimes \chk{\Q} \to \chk{\Q}$ defined by% (reflecting the semidirect product structure $\wa = $)
\begin{align*}
\waf = w \cdot t_{-q} \mapsto \waf^{-1}(0)=t_{q}\cdot w^{-1}(0)=t_{q}(0)=q
\end{align*}
restricts to a $\wa$-equivariant bijection on the cosets $W\backslash\wa$.
%$\waf = w \cdot t_{-q}$, we compute the corresponding coroot as %$\waf^{-1}(0)=t_{q}\cdot w^{-1}(0)=t_{q}(0)=q \in \chk{Q}$.
\end{theorem}
% REMARK: I agree that this is the correct use of inverses, in order to make W act on the RIGHT (e.g.\ so that the transpositions act in "reading order")

%We will always take our coset representatives of $W \backslash \wa$ to be the dominant affine permutations.




%======%
\section{Combinatorial Models for Coroot Lattices}
\label{sec:models}

We explain the $a$-core model for the type $A_{a-1}$ coroot lattice $\chk{\Q}_a$.  Similar combinatorial models for affine Weyl groups of classical type can be produced using coroot lattice embeddings, as in~\cite{hanusa2012abacus}.  We illustrate this is type $G_2$, interpreting the statistics $\chk{\size}_i$ and $\chk{\size}$ on the model.% on these models. % This recovers and extends some results in~\cite{hanusa2012abacus}.


\subsection{Partitions, Abaci, and the Coroot Lattice in Type \texorpdfstring{$A$}{A}}
An integer partition $\lambda$ (in English notation) can be characterized by its \defn{boundary word}---the bi-infinite sequence of $\bullet$s and $\circ$s (with $\bullet$s representing steps up and $\circ$s representing steps right) that begins with an infinite sequence of only $\bullet$s and ends with an infinite sequence of only $\circ$s, encoding the boundary of $\lambda$ when read from bottom left to top right.  Partitioning this sequence into consecutive subsequences of length $a$ and stacking them vertically gives an \defn{$a$-abacus} representation of $\lambda$. Finally, an $a$-abacus is called \defn{balanced} if we can draw a horizontal line between two rows with as many $\circ$s above the line as $\bullet$s below; every partition has a unique representation as a balanced $a$-abacus. 

An integer partition $\lambda$ is an $a$-core if and only if its $a$-abacus representation is \defn{flush}---that is, if each of the vertical ``runners'' of the abacus consists of an infinite sequence of only $\bullet$s followed by an infinite sequence of only $\circ$s. A flush, balanced $a$-abacus can be encoded as the $a$-tuple of signed distances from the lowest $\bullet$ in each runner to the line witnessing the balanced condition---the balanced condition ensures that these distances sum to zero.  See~\Cref{fig:abacus} for an illustration.

\begin{figure}[htbp]
\[
\text{Coroot point } q=(0,2,-2), \text{ abacus }  \raisebox{-.5\height}{\begin{tikzpicture}[scale=1,bull/.style={circle,draw=black,fill=black,thick,scale=.6},circ/.style={circle,draw=black,fill=white,thin,scale=.6}]
\draw[->,black,thin](-.25,-2.25)--(-.25,-5.5);
\draw[->,black,thin](.25,-2.25)--(.25,-5.5);
\draw[->,black,thin](.75,-2.25)--(.75,-5.5);
\node at (-.25,-2.5)[bull]{};
\node at (.25,-2.5)[bull]{};  
\node at (.75,-2.5)[bull]{};

\node at (-.25,-3)[bull]{};
\node at (.25,-3)[bull]{};  
\node at (.75,-3)[circ]{};

\node at (-.25,-3.5)[bull]{};
\node at (.25,-3.5)[bull]{};  
\node at (.75,-3.5)[circ]{};

\node at (-.25,-4)[circ]{};
\node at (.25,-4)[bull]{};  
\node at (.75,-4)[circ]{};

\node at (-.25,-4.5)[circ]{};
\node at (.25,-4.5)[bull]{};  
\node at (.75,-4.5)[circ]{};

\node at (-.25,-5)[circ]{};
\node at (.25,-5)[circ]{};  
\node at (.75,-5)[circ]{};
\draw[-,blue,thick](-.5,-3.75)--(1,-3.75);
\end{tikzpicture}}
  \text{ and } \text{$3$-core } \lambda=\raisebox{-.5\height}{\begin{tikzpicture}[scale=1,bull/.style={circle,draw=black,fill=black,thick,scale=.5},circ/.style={circle,draw=black,fill=white,thin,scale=.5}] \node (y) at (0,0) {\young(01201,201,1,0)};
  \node at (-1.15,-1.6)[bull]{};
  \node at (-1.15,-1.15)[bull]{};
  \node at (-.9,-.95)[circ]{};
  \node at (-.45,0)[circ]{};
  \node at (0,0)[circ]{};
  \node at (.45,.45)[circ]{};
  \node at (.9,.45)[circ]{};
  \node at (1.35,.9)[circ]{};
  \node at (1.8,.9)[circ]{};
  \node at (-.70,-.70)[bull]{};
  \node at (-.70,-.25)[bull]{};
  \node at (.25,.25)[bull]{};
  \node at (1.15,.7)[bull]{};
  \end{tikzpicture}}\]%\young(01201,201,1,0)}\]
\caption{An example of the bijection between $a$-core, abaci, and ${\protect \chk{\Q}_{A_{a-1}}}$ (for $a=3$).}
\label{fig:abacus}
\end{figure}

%\medskip

%\subsection{The Coroot Lattice}
In type $A_{a-1}$, the simple roots are $\alpha_i:=e_i-e_{i+1}$ for $1\leq i < a$, the highest root is $\widetilde{\alpha}:=e_1-e_a$, and the coroot lattice\footnote{For safety---even though roots and coroots can be identified in type $A$---we already throw in the distinguishing check.} is
%\begin{equation}
$\chk{\Q}_{a} = \chk{\Q}_{A_{a-1}} := \left\{q=(q_1,q_2\ldots,q_a) \in \mathbb{Z}^a : \sum_{i=1}^a q_i = 0 \right\}.$
%\end{equation}
By the discussion above, $\chk{\Q}_{a}$ is in bijection with the set of $a$-cores $\core(a)$.    For $q \in \chk{\Q}_a$, we write $\lambda_q$ for the corresponding $a$-core; for $\lambda \in \core(a)$, we write $q_\lambda$ for the corresponding coroot.  The action of the affine symmetric group $\widetilde{\mathfrak{S}}_a=\wa(A_{a-1})$ on $\chk{\Q}_a$ is generated by the usual simple reflections along with an additional affine simple reflection: 
\begin{align*}
    s_i(q_1,\ldots,q_i,q_{i+1},\ldots,q_a) &= (q_1,\ldots,q_{i+1},q_{i},\ldots,q_a), \text{ and}\\
    s_0(q_1,\ldots,q_a) &= (q_a+1,\ldots,q_1-1).
\end{align*}
%Rephrasing in terms of the $a$-abacus, the action of $s_i$ interchanges the $i$th and $i{+}1$st runners for $1 \leq i < a$.  The action of $s_0$ interchanges the first and last runners, shifting the last runner down by one and the first runner up by one.  
It is elegant to rephrase this action of $\widetilde{\mathfrak{S}}_a$ in the language of $a$-cores.  We think of an $a$-core as an order ideal in $\mathbb{N} \times \mathbb{N}$, where each $(i,j) \in \mathbb{N} \times \mathbb{N}$ is indexed by its content $(i-j) \mod a$.  For $0 \leq i < a$, the action of a simple reflection $s_i$ is then to add or remove all possible boxes indexed by $i$.   The following theorem now follows immediately.

\begin{theorem}
The map $\lambda \to q_\lambda$ is an $\widetilde{\mathfrak{S}}_a$-equivariant bijection between $\core(a)$ and $\chk{\Q}_a$, with inverse given by $q \to \lambda_q$.
\label{thm:sa_eq}
\end{theorem}


%------%
\subsection{The statistic size in type \texorpdfstring{$A$}{A}}
\label{sec:size}


We interpret the size statistic for the number of boxes in an $a$-core in terms of the affine symmetric group $\widetilde{\mathfrak{S}}_a$).  Recall that in type $A_{a-1}$ (up to the usual normalization that the sum of the entries ought to be zero, which we will safely ignore), we have
\begin{align*}
    \chk{\omega}_i = \sum_{j=1}^i e_i = (\underbrace{1,1,\ldots,1}_{i \text{ ones}},0,0,\ldots,0) \text{ and }    \chk{\rho} = \sum_{i=1}^{a-1} \chk{\omega}_i = (a-1,a-2,\ldots,1,0),
\end{align*}
where we use the convention that $\omega_0=0$.  For $\lambda$ a partition, write $\lambda^\intercal$ for its conjugate, $\size_i(\lambda)$ for the the number of boxes in $\lambda$ with content $i \mod a$, and $\size(\lambda)$ for the total number of its boxes.  For $q=(q_1,\ldots,q_a) \in \chk{\Q}_a$, write $q^\intercal:=(-q_a,\ldots,-q_1)$ and define \begin{align*}
    \chk{\size}_i(q):=\left\langle \frac{1}{2} q-\chk{\omega}_i,q\right\rangle \text{ and }    \chk{\size}(q)=\sum_{i=1}^{a-1} \chk{\size}_i(q):=\left\langle \frac{a}{2} q-\chk{\rho},q\right\rangle.
\end{align*}

\def\numz{{\textcolor{blue}{0}}}
\def\numo{{\textcolor{green}{1}}}
\def\numt{{\textcolor{red}{2}}}
\begin{example}
\label{ex:a22}
Continuing~\Cref{ex:a2inv}, the element $\waf=s_1s_2t_{-2\chk{\alpha}_2} \in \widetilde{A}_2$ corresponds to the coroot $q=2\chk{\alpha}_2=(0,2,-2)$ and the $3$-core $\lambda=\raisebox{\height}{\young(\numz\numo\numt\numz\numo,\numt\numz\numo,\numo,\numz)}$.  This $\lambda$ has four boxes with content $\numz \mod 3$, four with content $\numo \mod 3$, and two with content $\numt \mod 3$.  We compute (again, safely ignoring normalization)

\begin{align*}
    \chk{\size}_\numz(q) &= \left\langle \frac{1}{2}(0,2,-2),(0,2,-2)\right\rangle = 4 = \size_\numz(\lambda),\\
    \chk{\size}_\numo(q) &= \left\langle \frac{1}{2}(0,2,-2)-\textcolor{green}{(1,0,0)},(0,2,-2)\right\rangle = 4=\size_\numo(\lambda),\\
     \chk{\size}_\numt(q) &= \left\langle \frac{1}{2}(0,2,-2)-\textcolor{red}{(1,1,0)},(0,2,-2)\right\rangle = 2=\size_\numt(\lambda).
\end{align*}
\end{example}

\begin{proposition}
\label{prop:ip_content}
For $q \in \chk{\Q}_a$, $\lambda_{q^\intercal}=\lambda_q^\intercal$ and $\chk{\size}(q)=\size(\lambda_q))$.  Furthermore, for any $0\leq i\leq a-1$, $\size_i(\lambda)=\chk{\size}_i(q_\lambda)$.
\end{proposition}

\begin{proof}
We compute directly that the number of boxes in the $a$-core $\lambda_q$ is given by \[\sum_{\bullet \in \lambda_q} \# \left\{\circ \text{ to the left of } \bullet\right\} = \sum_{1 \leq i < j \leq n} \frac{(x_i-x_j-1)(x_i-x_j)}{2} = \left\langle \frac{a}{2} q-\chk{\rho}, q \right\rangle,\]
where the first equality comes from counting the number of inversions coming from $\bullet$s and $\circ$s between runner $i$ and runner $j$, while the second equality comes from rearranging and using $\chk{\rho}=(a,a-1,\ldots,1)-\frac{a+1}{2}(1,1,\ldots,1).$  The statement about conjugation follows by observing that the boundary path of a partition and its conjugate are related by reversing and interchanging $\bullet \leftrightarrow \circ$.  The remaining statement about $\size_i$ will follow from the more general~\Cref{thm:sizer} below.
\end{proof}


%------%
\subsection{The statistic size in general type}
\label{sec:sizegen}
%TODO: Connecting sentences between Prop 3.6 – Thm 3.8 ??

We now turn to the general definition of the size statistic for affine Weyl groups.  In fact, we generalize the refined statistic $\size_i$ (keeping track of the number of boxes with content $i \mod a$ in type $A$) for both models of $X_n$-cores: first for $W\backslash\widetilde{W}$, and then for $\chk{\Q}_{X_n}$.
%We will interpret this statistic on the combinatorial models for coroot lattices $\chk{\Q}_{X_n}$ in the sequel.

\begin{definition}
\label{def:sizewords}
Fix $\waf \in \wa$ and a reduced word $\mathsf{\widetilde{w}}=a_1 \cdots a_{\ell}$ for $\waf$, with inversion sequence $\inv(\mathsf{\widetilde{w}})=\beta_1+k_1\delta,\beta_2+k_2\delta,\ldots,\beta_\ell+k_\ell\delta.$   For any $i \in \{0,1,\ldots,n\}$ with corresponding simple reflection $s_i$ and simple root $\alpha_i$, define \begin{align*}  \chk{\size}_i(\mathsf{\widetilde{w}})&=\frac{2}{\left\langle \alpha_i, \alpha_i\right\rangle}  \sum_{\substack{1\leq j \leq \ell \\ a_j=s_i}} k_j .\end{align*}
\end{definition}

\begin{example}
\label{ex:a2}
Continuing~\Cref{ex:a2inv,ex:a22}, we recall that the inversion sequence for $\wa=\widetilde{A}_2$ and $\mathsf{\widetilde{w}} = s_{\textcolor{blue}{0}}s_{\textcolor{green}{1}}s_{\textcolor{red}{2}}s_{\textcolor{green}{1}}s_{\textcolor{blue}{0}}s_{\textcolor{green}{1}}$ is
\[\inv(\mathsf{\widetilde{w}})=-\widetilde{\alpha}+\textcolor{blue}{1}\cdot\delta,-\alpha_2+\textcolor{green}{1}\cdot\delta,-\widetilde{\alpha}+\textcolor{red}{2}\cdot\delta,-\alpha_1+\textcolor{green}{1}\cdot\delta,-\widetilde{\alpha}+\textcolor{blue}{3}\cdot\delta,-\alpha_2+\textcolor{green}{2}\cdot\delta,\]
and we compute $\size_{\textcolor{blue}{0}}(\mathsf{\widetilde{w}})=\textcolor{blue}{1}+\textcolor{blue}{3}=4$, 
    $\size_{\textcolor{green}{1}}(\mathsf{\widetilde{w}})=\textcolor{green}{1}+\textcolor{green}{1}+\textcolor{green}{2}=4$, and 
    $\size_{\textcolor{red}{2}}(\mathsf{\widetilde{w}})=\textcolor{red}{2}.$
\end{example}

\begin{proposition}
For any two reduced words $\mathsf{\widetilde{w}}$ and $\mathsf{\widetilde{w}}'$ representing the same group element $\widetilde{w}\in\widetilde{W}$, we have $\chk{\size}_i(\mathsf{\widetilde{w}})=\chk{\size}_i(\mathsf{\widetilde{w}}')$.
\end{proposition}
% TODO: Eliminate?

\begin{definition}
\label{def:sizelattice}
%$\widetilde{\alpha}=\sum_{i=1}^n c_i \alpha_i$ and
For $q \in \chk{\Q}_{X_n}$, define
$\chk{\size_i}(q)=\Big\langle \frac{c_i}{2}q-\chk{\omega_i},q\Big\rangle,$ where the highest root $\widetilde{\alpha}$ is expressed in terms of the simple roots as $\widetilde{\alpha}=\sum_{i=1}^n c_i \alpha_i$ (so that $\left\langle \chk{\omega}_i,\widetilde{\alpha}\right\rangle =c_i$), with $c_0:=1$.
\end{definition}

\begin{theorem}
\label{thm:sizer}
Under the bijection of~\Cref{thm:coset_to_coroot} sending a dominant affine element $\waf = w t_{-q} \in W \backslash \wa$ to the coroot $q \in \chk{\Q}_{X_n}$, we have $    \chk{\size}_i(\waf) =\chk{\size_i}(q).$
\end{theorem}

\begin{proof}
Let $j \neq 0$ and let $i \in \{0,1,\ldots,n\}$.  We compute $\chk{\size}_i(s_j(q))$:
\begin{align*}
    \chk{\size}_i(s_j(q))
    &= \left\langle \frac{c_i}{2} s_j(q)-\chk{\omega}_i,s_j(q)\right\rangle \\
    &= \left\langle \frac{c_i}{2} \left[q-\langle \chk{\alpha}_j,q\rangle\alpha_j\right]-\chk{\omega}_i,\left[q-\langle \chk{\alpha}_j,q\rangle\alpha_j\right]\right\rangle \\
    &=\left\langle \frac{c_i}{2} q-\chk{\omega}_i,q\right\rangle+\left\langle \chk{\alpha}_j, q\right\rangle \cdot \left\langle \chk{\omega}_i,\alpha_j\right\rangle\\
    &=\chk{\size}_i(q)+\begin{cases}\big\langle \chk{\alpha}_j, q\big\rangle & \text{if } i=j \\ 0 & \text{if } i\neq j\end{cases}.
\end{align*}
    Similarly, we compute $\chk{\size}_i(s_0(q)) = \chk{\size}_i(q)+\begin{cases}1-\left\langle \widetilde{\alpha}, q\right\rangle & \text{if } i=0 \\ 0 & \text{if } i\neq 0\end{cases}$.
    
\begin{comment}
\begin{align*}
    \chk{\size}_i(s_0(q))
    &= \left\langle \frac{c_i}{2} s_0(q)-\chk{\omega}_i,s_0(q)\right\rangle \\
    &= \left\langle \frac{c_i}{2} \left[q-(\langle \widetilde{\alpha},q\rangle-1)\widetilde{\alpha}\right]-\chk{\omega}_i,\left[q-(\langle \widetilde{\alpha},q\rangle-1)\widetilde{\alpha}\right]\right\rangle \\
    &=\left\langle \frac{c_i}{2} q-\chk{\omega}_i,q\right\rangle+\left(\left\langle \widetilde{\alpha}, q\right\rangle-1\right) \cdot \left(\left\langle \chk{\omega}_i,\widetilde{\alpha}\right\rangle -c_i\right)\\
    &=\chk{\size}_i(q)+\begin{cases}1-\left\langle \widetilde{\alpha}, q\right\rangle & \text{if } i=0 \\ 0 & \text{if } i\neq 0\end{cases},
\end{align*}
\end{comment}

We now argue by induction on the length of the dominant affine element $\waf$, with base case coming from the identity $e \leftrightarrow 0$ giving $\chk{\size}_i(e)=\chk{\size}_i(0)=0$.  Consider now $\waf=a_1a_2\cdots a_{\ell-1}=w \cdot t_{-q}$ with (right) ascent $s_j$ so that $\waf s_j$ is still dominant.  We can compare the root $-(\waf s_j)(\alpha_{j})=-\waf(-\alpha_{j})=\waf(\alpha_{j})$ with the previous computations to conclude the result by induction: for $j \neq 0$, we have
$    \waf (\alpha_j)
    = w(\alpha_j) + \langle \alpha_j,q\rangle \delta,$
while if $j=0$, then 
$    \waf (-\widetilde{\alpha}+\delta)
    = w(-\widetilde{\alpha}) + (1-\langle \widetilde{\alpha},q\rangle)\delta.$  The result follows by observing that $\chk{\size}_i(\waf)$ picks off the coefficient of $\delta$ for this last inversion, recording the correction factor of $r$ if $\alpha_j$ was a short root.
\end{proof}

Because $\frac{2}{\left\langle \alpha_i, \alpha_i\right\rangle} = 1$ if $\alpha_i$ is long, and is $r$ if $\alpha_i$ is short, and because $\sum_{i=0}^n c_i = h$ and $\sum_{i=0}^n \chk{\omega}_i = \chk{\rho}$, we conclude the following theorem.

\begin{theorem}
\label{thm:size_eq}
For $\waf \in W\backslash \wa$, define \[\chk{\size}(\waf) = \sum_{i=0}^n \chk{\size}_i(\waf) =  \left(\sum_{\substack{\alpha+k\delta \in \mathsf{inv}(\waf^{-1})\\\alpha \text{ long}}}k\right)+r\left(\sum_{\substack{\alpha+k\delta \in \mathsf{inv}(\waf^{-1})\\\alpha \text{ short}}}k\right).\]
For $q \in \chk{\Q}_{X_n}$, define $\chk{\size}(q) = \sum_{i=0}^n \chk{\size}_i(q) = \left\langle \frac{h}{2}q - \chk{\rho},q \right\rangle.$
Then if $\waf = w t_{-q} \in W \backslash \wa$ corresponds to the coroot $q \in \chk{\Q}_{X_n}$, we have $\chk{\size}(\waf)=\chk{\size}(q).$
\end{theorem}


\begin{comment}
%------%
\subsection{Type \texorpdfstring{$C$}{C}}
\label{sec:modelC}

In this section we recall the well-known fact that a combinatorial model for the coroot lattice of type $C_n$ is given by self-conjugate $2n$-cores, and show that the number of boxes in such a core is given by the statistic $\chk{\size}$ on the corresponding coroot point.  The simple roots for the root system of type $C_n$ are $\alpha_i:=\frac{1}{\sqrt{2}}(e_i-e_{i+1})$ for $1\leq i < n$ and $\alpha_n:=\sqrt{2}e_n$. Hence, the coroot lattice is $\chk{Q}_{C_n} = \sqrt{2}\mathbb{Z}^n.$ The action of the affine Weyl group \[\widetilde{C}_n = \langle s_0^C,s_1^C,s_2^c,\ldots,s_n^C : (s_0^Cs_1^C)^4=(s_{n-1}^Cs_n^C)^4=\underbrace{(s_is_{i+1})^3}_{0< i<n-1}=\underbrace{s_is_j}_{|i-j|>1}=e\rangle\] 
\vspace{-0.1in}
on $\chk{Q}_{C_n}$ is given explicitly by
\begin{align*}s_i^C(x_1,\ldots,x_i,x_{i+1},\ldots,x_n) &= (x_1,\ldots,x_{i+1},x_{i},\ldots,x_n) \text{ for } 1 \leq i < n,\\
s_n^C(x_1,\ldots,x_n) &= (x_1,\ldots,-x_n), \text{ and }\\
s_0^C(x_1,\ldots,x_n) &= (1-x_1,\ldots,x_n).
\end{align*}
We embed the type $C_n$ coroot lattice into the coroot lattice for $\widetilde{\mathfrak{S}}_{2n}$ by $\iota_C:\chk{Q}_C \hookrightarrow \chk{Q}_{2n}$
\[\iota_C(x_1,\ldots,x_n) = \frac{1}{\sqrt{2}}(x_1,\ldots,x_n,-x_n,\ldots,-x_1)\]
from which it is easy to mimic the action of $\widetilde{C}_n$ using $\widetilde{\mathfrak{S}}_{2n}$ by
$s_i^C \mapsto s_i s_{2n-i}$ for $1 \leq i < n$, 
$s_n^C \mapsto s_n$, and $s_0^C \mapsto s_0.$
It is straightforward to check that the simple reflections of $\widetilde{C}_n$ agree with these $\widetilde{\mathfrak{S}}_{2n}$ operators on the $\iota_C$-embedded coroot lattice. 

\begin{theorem}
\label{thm:cn_cores}
The map $\iota_C(q)$ is an $\widetilde{C}_n$-equivariant bijection between the $C_n$ coroot lattice and self-conjugate $(2n)$-cores.  For $q \in \chk{Q}_{C_n}$, $\chk{\size}_i(q)=\chk{size}_i(\iota_C(q))$.
\end{theorem}

\begin{example}
\label{ex:c2}
The inversion sequence for $\wa=\widetilde{C}_2$ and $\mathsf{\widetilde{w}}=s_0s_1s_2s_0s_1=s_2 t_{\chk{\alpha}_1}$ is
\[-\widetilde{\alpha}+\delta,-\alpha_1-\alpha_2+\delta,-\widetilde{\alpha}+2\delta,-\alpha_2+\delta,-\alpha_1-\alpha_2+2\delta\]
and since $\alpha_1$ is short and $r=2$, we have (by~\Cref{def:sizewords}) that $\chk{\size}_1(\mathsf{\widetilde{w}})=2\cdot(1+2)=6$.
On the other hand, $\waf= s_0s_1s_2s_0s_1$ corresponds to $q=\sqrt{2}(-1,1) \in \chk{Q}_{C_2}$.   Since $\chk{\omega}_1 = \sqrt{2}(1,0)$ and $c_1=2$, we compute (by~\Cref{def:sizelattice}) that
\[\chk{\size}_1(q) = \left\langle \sqrt{2}(-1,1)-\sqrt{2}(1,0),\sqrt{2}(-1,1)\right\rangle = 2\left\langle (-1,1)-(1,0),(-1,1)\right\rangle = 6.\]  The equality $\chk{\size}_1(\mathsf{\widetilde{w}})=\chk{\size}_1(q)$ illustrates~\Cref{thm:size_eq}.
Finally, the corresponding $4$-core \raisebox{\height}{\young(0121,101,21,1)} has $6$ boxes with content $1$---which illustrates~\Cref{thm:cn_cores}.
\end{example}

\end{comment}


%------%
\subsection{Other Types}%Type \texorpdfstring{$G_2$}{G2}}
\label{sec:modelG}
It is easy to embed coroot lattices $\Q_{X_n}$ for the classical types into appropriate coroot lattices of type $A$. In so doing we find combinatorial models for their coroot lattices. and in so doing give combinatorial interpretations of $\chk{\size}$.  We illustrate this method here for the exceptional type $G_2$---we do not know of a similar method for finding reasonable combinatorial models for the remaining types $F_4$, $E_6, E_7, $ or $E_8$.  


As usual, we wish to think of $G_2$ as acting on the orthogonal complement of $\mathbb{R}(1,1,1)$ in $\mathbb{R}^3$.  The simple roots for $G_2$ can be taken to be $\alpha_1:=(1,-1,0) \text{ and } \alpha_2 := \frac{1}{3}(-1,2,-1).$  With these conventions, the type $G_2$ coroot lattice $\chk{\Q}_{G_2}$ coincides with the coroot lattice for $\widetilde{\mathfrak{S}}_{3}$.  The map $q \mapsto \lambda_q$ therefore gives a bijection between $\chk{\Q}_{G_2}$ and $3$-cores.  It remains to determine the action of $\widetilde{G}_2$ on 3-cores.  The action of the affine Weyl group $\widetilde{G}_2 = \langle s_0^G,s_1^G,s_2^G: (s_0^Gs_1^G)^3=(s_1^Gs_2^G)^6=(s_0s_2)^2=e\rangle$ on $\chk{\Q}_{G_2}$ is given explicitly by %$s_1^G(q_1,q_2,q_3)  = (q_2,q_1,q_3),$
%$s_2^G(q_1,q_2,q_3)  = (-q_3,-q_2,-q_1),$ and
%$s_0^G(q_1,q_2,q_3)  = (q_3+1,q_2,q_1-1).$
\begin{align*}s_1^G(q_1,q_2,q_3)  &= (q_2,q_1,q_3),\\
s_2^G(q_1,q_2,q_3)  &= (-q_3,-q_2,-q_1), \text{ and }\\
s_0^G(q_1,q_2,q_3)  &= (q_3+1,q_2,q_1-1).
\end{align*}
We emulate the action of $\widetilde{G}_2$ by $s_1^G(\lambda_q) = s_1(\lambda_q), s_2^G(\lambda_q) = \lambda_q^\intercal, \text{ and } s_0^G(\lambda_q) = s_0(\lambda_q).$  For $\lambda$ a $3$-core, define $\chk{\size}_G(\lambda):=\lambda_0+\lambda_1+4\lambda_2$.  We obtain a combinatorial model for $\chk{\Q}_{G_2}$.

\begin{theorem}
The map $q\mapsto \lambda_q$ is a $\widetilde{G}_2$-equivariant bijection between the $G_2$ coroot lattice and $3$-cores: $s_1^G$ acts on a 3-core by adding or removing all boxes of content $1$, $s_0^G$ acts similarly on boxes of content $0$, and $s_2^G$ acts by conjugation.  For $q \in \chk{\Q}_{G_2}$, $\chk{\size}_G(\lambda_q)=\chk{\size}(q)$.
\end{theorem}

\begin{comment}
\subsection{Other types}
It is easy to perform similar embeddings of the coroot lattices $\Q_{X_n}$ for the other classical types into an appropriate coroot lattice of type $A$; we do not know of a similar method for finding reasonable combinatorial models for the coroot lattices in the remaining types $F_4$, $E_6, E_7, $ or $E_8$.
\end{comment}


%======%
\section{Expected Size of Simultaneous Cores}
\label{sec:sommers}



Johnson {\cite{johnson2015lattice}} showed that $(a,b)$-cores satisfy certain inequalities; when considering them as elements of $\chk{\Q}_{A_{a-1}}$, these place them inside a particular simplex that had previously been considered by Sommers \cite{sommers2005b}. Recall that $\Phi$ is a root system with an irreducible Cartan type $X_n$, Coxeter number $h$. For $1 \leq i < h$, write $\Phi_{i}$ to denote the set of positive roots of height $i$.

%TODO: Include this? It's basically just recognizing that the boundary word can be partitioned into either $a$ or $b$ runners, and that the flush condition for $b$-abaci translates to the statement that any white bead translates to another white bead $b$ strands later; by coprimality, repeating this eventually finds white beads in all of the columns. But it might be somewhat awkward to state.

%The idea is that the coroots $q \in \chk{\Q}_a$ correspond to $a$-cores.  Throwing in the very explicit additional conditions that $\langle q,\alpha \rangle \geq -\bq$ for $\alpha \in \Phi_{\br}$ and $\langle q,\alpha \rangle \leq \bq+1$ for $\alpha \in \Phi_{a-\br}$ must give the $b$-core condition.  Translating, if $q=(q_1,\ldots,q_a)$, then we have $q_{i+\br}-q_i \leq \bq$ and $q_i-q_{i+a-\br} \leq \bq+1$...\Nathan{finish?}

\begin{definition}
\label{def:sommers}
For $b$ coprime to $h$, write $\rat=\bq h+\br$ with $t,r\in\Z_{\geq0}$ and $0<\br<h$.  We define the \defn{$b$-Sommers region} 
\begin{align*}
 \Sommers_{X_n}(b):=\left\{x\in V: \begin{array}{rl} \langle x,\alpha\rangle\geq-\bq & \text{for all }\alpha\in \Phi_{\br}\text{ and }\\
 \langle x,\alpha\rangle\leq \bq+1 &\text{for all }\alpha\in \Phi_{h-\br} \end{array} \right\}.
\end{align*}
\end{definition}

As in~\Cref{eq:Wcores}, a natural generalization of $\core(a,b)$ to any affine Weyl group is the intersection of the coroot lattice $\chk{\Q}_{X_n}$ with $\Sommers_{X_n}(b)$, so that $\core(a,b) = \core(A_{a-1},b)$.


%------%
\subsection{The Sommers Region and the Fundamental Alcove}

We would like to perform the $\chk{\size}$-weighted enumeration of $\core(X_n,b)$ using Ehrhart theory. Unfortunately, the family $\{\Sommers_{X_n}(b): \gcd(b,h)=1\}$ does not consist of dilations of a fixed polytope---but this difficulty can be circumvented.
Define, for any $x\in V$, the statistic
\begin{equation}\sizeb(x):=\frac{h}{2}\left(\left\|x - \frac{b \chk{\rho}}{h}\right\|^2 - \left\|\frac{\chk{\rho}}{h}\right\|^2 \right). \end{equation}
Notice that when $q\in\chk{\Q}_{X_n}$ and $b=1$, we have $\size^{(1)}(q)=\chk{\size}(q)$ (cf.\ \Cref{thm:size_eq}).
%this statistic agrees with the one given in \Cref{thm:size_eq}:

We recall from~\cite[\S 4]{thiel2017strange} that there is a unique element $\wf\in \wa$ such that $\frac{b}{h}\chk{\rho} = \wf(\frac{\chk{\rho}}{h})$, and that left-multiplication by this element maps $\Sommers_{X_n}(b)$ onto the $b$-fold dilation of the fundamental alcove $\ac$.

\begin{theorem}
\label{thm:size_after_wb}
For $b$ coprime to $h$, the following equality of multisets holds:
\[\Big\{\chk{\size}(q) : q \in \core(X_n,b)\Big\} = \Big\{\sizeb(q) : q \in \rat\ac \cap \chk{\Q}_{X_n}\Big\}.\]
\end{theorem}
\begin{proof}
We first note that since $\wf$ maps $\Sommers_{X_n}(b)$ onto $b\ac$, and also $\wf\in\wa$ and thus is a $\chk{\Q}_{X_n}$-preserving bijection, it restricts to a bijection $\core(\wa,b)\to \rat\ac \cap \chk{\Q}$.  Write $\wf = t_{q_0} w$; then $\wf = t_{\frac{b\chk{\rho}}{h}} w t_{-\frac{\chk{\rho}}{h}}$. Since $\|\cdot\|$ is $W$-invariant, for $q\in \core(\wa,b)$:
\begin{align*}
\sizeb(\wf(q)) &= \frac{h}{2}\left(\left\| t_{\frac{b}{h}\chk{\rho}} w t_{\frac{1}{h}\chk{\rho}} (q) - \frac{b \chk{\rho}}{h}\right\|^2 - \left\|-\frac{\chk{\rho}}{h}\right\|^2 \right) \\
&= \frac{h}{2}\left(\left\|w\left(q - \frac{\chk{\rho}}{h}\right)\right\|^2 - \left\|\frac{\chk{\rho}}{h}\right\|^2 \right) = \chk{\size}(q).\qedhere
\end{align*}
\end{proof}



%======%
\subsection{Proof Sketch}
\label{sec:expectation}

We are now ready to sketch a proof of \Cref{thm:main_thm}:

\mainthm*

We do this by computing the left-hand side explicitly for each type $X_n$, which by \cref{thm:size_after_wb} is
\begin{align*}
    \Expt{q \in \core(X_n,b)}{\chk{\size}(q)} 
    &= \frac{1}{|\core(X_n,b)|}\sum_{q\in \core(X_n,b)} \chk{\size}(q) = \frac{1}{|b\ac\cap\chk{\Q}_{X_n}|}\sum_{q\in b\ac\cap\chk{\Q}_{X_n}} \sizeb(q).
\end{align*} 
The denominator was explicitly (and uniformly) calculated by Haiman \cite{haiman1994conjectures}. To compute the sum, we first record the vertices of the fundamental alcove $\ac$: they are \(\Gamma:=\{0\}\cup\left\{\frac{\chk{\omega}_i}{c_i} : 1 \leq i \leq n\right\}\), where %$\chk{\omega}_i$ are the fundamental coweights
and $c_i$ are defined by $\widetilde{\alpha}=\sum_{i=1}^nc_i\alpha_i$.  As in \cite{thiel2017strange}, we proceed by translating the problem to the coweight lattice.  Define the \defn{extended affine Weyl group} of type $X_n$ by $\wex:=W\ltimes \cowl$, and write the group of automorphisms for $b\ac$ as $b\cycl:=\{\waf\in\wex:\waf(b\ac)=b\ac\}$. These groups are isomorphic for all $b$, and in particular have constant order $f$.

\begin{proposition}[{\cite[Theorem 2.5 \& Lemma 6.11]{thiel2017strange}}]
For any $b$ coprime to $h$:
	%\begin{enumerate}[(a)]
    \begin{enumerate}[label=(\alph*)]
	\item The action of $b\cycl$ on $b\ac\cap\cowl$ is free.
	\item Each $b\cycl$ orbit of $b\ac\cap\cowl$ contains exactly one element of $b\ac\cap\chk{\Q}_{X_n}=\core(X_n,b)$.
	\item For any $\waf\in b\cycl$ any any $\omega\in\cowl$, $\chk{\size}_b(\omega)=\chk{\size}_b(\waf\cdot\omega)$
	\end{enumerate}
\label{prop:barational}
\end{proposition}
%REMARK: Part (c) isn't actually stated in the previous paper because of the incorrect definition of size, although the proof is identical--- it even uses $\wf=t_{\frac{b}{h}\chk{\rho}}wt_{-\frac{b}{h}\chk{\rho}}$.

Using this, we finish the translation from $\chk{\Q}_{X_n}$ to $\cowl$:
\[
\Expt{q \in \core(X_n,b)}{\chk{\size}(q)} = \frac{1}{|b\ac\cap\chk{\Q}_{X_n}|}\cdot \frac{1}{f}\sum_{q\in b\ac\cap\cowl} \sizeb(q).
\]

Let us now recall the relevant Ehrhart-theoretic tools. For any degree-$r$ polynomial $F:\R^n\to\R$, its \defn{weighted lattice point enumerator} is
$ \ac_F(b) := \sum\limits_{q\in b\ac\cap\cowl} F(q)$ over $\chk{\Lambda}_{X_n}$. This $\ac_F(b)$ is a quasipolynomial in $b$, of degree $n+r$ and period $c:=\lcm(c_1,\dots,c_n)$, where the $c_i$ are again the denominators of the vertices of $\ac$. As $\size^{(b)}$ changes with $b$, Ehrhart theory appears to be inapplicable---however, a judicious rewriting shows that this is not the case.

\begin{proposition}
	The weighted lattice point enumerator $\ac_{\sizeb}(b)$
    is a quasipolynomial in $b$ of degree $n+2$ and period       $c_{X_n}:=\lcm(c_1,\ldots,c_n)$.
\label{prop:quasipoly}
\end{proposition}

\begin{proof}
    Notice that $\chk{\size}_b(x)=\frac{h}{2}\left\|x\right\|^2-b\langle x,\chk{\rho}\rangle+(b^2-1)\frac{\left\|\check{\rho}\right\|^2}{2h}.$
   Thus we find that
   \[\ac_{\sizeb}(b)={\textstyle\frac{h}{2}}\ac_{\|\cdot\|^2}(b)-b\ac_{\langle\cdot,\chk{\rho}\rangle}(b)+(b^2-1)\ac_{\frac{\left\|\chk{\rho}\right\|^2}{2h}}(b)\]
   is a quasipolynomial in $b$ of degree $n+2$ and period $c_{X_n}$.
\end{proof}

Therefore, to complete the proof of \Cref{thm:main_thm}, we may compute the quasipolynomial on for all components that contain a residue $b\mod c_{X_n}$ that is coprime to $h$. For the exceptional types, this is already a finite and computationally feasible calculation. For classical types, we use Ehrhart reciprocity to see that $-e_i$ is a root of $\ac_{\sizeb}(b)$ for all exponents $e_i$; when working through the details, this reduces the problem to interpolating a factor that is at worst quadratic (in type $D_n$). Thus, by explicitly computing the lattice points contained in several dilates of $\ac$, we complete the missing factor and recover the desired formula.


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\acknowledgements{We thank Benjamin Cotton for help drawing~\Cref{fig:a2}. The second author was partially supported by Simons grant 585380.}



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