Connecting marginally large tableaux and rigged configurations via crystals
Ben Salisbury
Travis Scrimshaw
We show that the bijection from rigged configurations to tensor products of Kirillov-Reshetikhin crystals extends to a crystal isomorphism between the $B(\infty)$ models given by rigged configurations and marginally large tableaux.
a3mlt.tex

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B2-21-RC.tex

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%
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B2-21.tex
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%
\end{tikzpicture}
RC_to_MLT.tex
\documentclass[reqno]{amsart}
\usepackage{amssymb,setspace,tikz,xcolor,mathrsfs,listings,multicol,mathabx,bm}
\usepackage{rotating}
\usepackage[vcentermath]{youngtab}
\usepackage{hyperref}
\usetikzlibrary{arrows,matrix}
\usepackage[margin=1.25in]{geometry}
\onehalfspacing
\tikzset{tab/.style={matrix of math nodes,column sep=-.35, row sep=-.35,text height=7pt,text width=7pt,align=center,inner sep=2,font=\footnotesize}}

\newcommand{\bon}{\overline{1}}
\newcommand{\btw}{\overline{2}}
\newcommand{\bth}{\overline{3}}
\newcommand{\bfo}{\overline{4}}
\newcommand{\bi}{\overline\imath}
\newcommand{\bk}{\overline{k}}
\newcommand{\brn}{\overline{n}}
\newcommand{\cc}{\mathrm{cc}}
\newcommand{\g}{\mathfrak{g}}
\newcommand{\HH}{\mathcal{H}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\inner}[2]{\left\langle #1, #2 \right\rangle}
\newcommand{\innernoscale}[2]{\langle #1, #2 \rangle} % inner product but does not scale the brackets
\newcommand{\iso}{\cong}
\newcommand{\longdoublearrow}{\relbar\joinrel\twoheadrightarrow}
\newcommand{\longhookarrow}{\lhook\joinrel\longrightarrow}
\renewcommand{\mid}{:}
\newcommand{\p}{{\bm p}}
\newcommand{\proj}{\operatorname{proj}}
\newcommand{\qedbox}{\rule{2mm}{2mm}}
\renewcommand{\qedsymbol}{\qedbox}	
\newcommand{\RC}{\operatorname{RC}} % rigged configurations
\newcommand{\hwRC}{\RC^*} % highest weight rigged configurations
\newcommand{\sgn}{\!\operatorname{-sgn}}
\newcommand{\TT}{\mathcal{T}} % B(\infty) of marginally large tableaux
\newcommand{\VRC}{\operatorname{VRC}} % valid rigged configurations
\newcommand{\EVRC}{\operatorname{EVRC}} % extra valid rigged configurations
\newcommand{\seg}{\operatorname{seg}} % seg statistic
\newcommand{\rem}{\operatorname{rem}} % rem statistic
\newcommand{\rpt}{\operatorname{rpt}} % rpt statistic
\newcommand{\diff}{\operatorname{diff}} % diff statistic
\newcommand{\wt}{\mathrm{wt}}
\newcommand{\ZZ}{\mathbf{Z}}
\newcommand{\qbinom}[3]{\genfrac{[}{]}{0pt}{}{#1}{#2}_{#3}}
\newcommand{\absval}[1]{\left\lvert #1 \right\rvert}
\newcommand{\virtual}[1]{\widehat{#1}}
\newcommand{\folding}{\searrow}

%% For typesetting code listings                                                
\usepackage{listings}
\lstdefinelanguage{Sage}[]{Python}
{morekeywords={False,sage,True},sensitive=true}
\lstset{
  frame=single,
  showtabs=False,
  showspaces=False,
  showstringspaces=False,
  commentstyle={\ttfamily\color{dgreencolor}},
  keywordstyle={\ttfamily\color{dbluecolor}\bfseries},
  stringstyle={\ttfamily\color{dgraycolor}\bfseries},
  language=Sage,
  basicstyle={\footnotesize\ttfamily},
  aboveskip=0.75em,
  belowskip=0.75em,
  xleftmargin=.15in,
}
\definecolor{dblackcolor}{rgb}{0.0,0.0,0.0}
\definecolor{dbluecolor}{rgb}{0.01,0.02,0.7}
\definecolor{dgreencolor}{rgb}{0.2,0.4,0.0}
\definecolor{dgraycolor}{rgb}{0.30,0.3,0.30}
\newcommand{\dblue}{\color{dbluecolor}\bf}
\newcommand{\dred}{\color{dredcolor}\bf}
\newcommand{\dblack}{\color{dblackcolor}\bf}

\usepackage{xparse}

\makeatletter
% \specialmergetwolists{<coupler>}{<list1>}{<list2>}{<return macro>}
% \specialmergetwolists*{<coupler>}{<listcmd1>}{<listcmd2>}{<return macro>}
\protected\def\specialmergetwolists{%
  \begingroup
  \@ifstar{\def\cnta{1}\@specialmergetwolists}
    {\def\cnta{0}\@specialmergetwolists}%
}
\def\@specialmergetwolists#1#2#3#4{%
  \def\tempa##1##2{%
    \edef##2{%
      \ifnum\cnta=\@ne\else\expandafter\@firstoftwo\fi
      \unexpanded\expandafter{##1}%
    }%
  }%
  \tempa{#2}\tempb\tempa{#3}\tempa
  \def\cnta{0}\def#4{}%
  \foreach \x in \tempb{%
    \xdef\cnta{\the\numexpr\cnta+1}%
    \gdef\cntb{0}%
    \foreach \y in \tempa{%
      \xdef\cntb{\the\numexpr\cntb+1}%
      \ifnum\cntb=\cnta\relax
        \xdef#4{#4\ifx#4\empty\else,\fi\x#1\y}%
        \breakforeach
      \fi
    }%
  }%
  \endgroup
}
\makeatother

\usepackage{xparse}
\DeclareDocumentCommand\rpp{ m m g }{
	\foreach \x [count=\s from 1] in {#1}{
	        {\ifnum\s=1
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	   \foreach \y in {0, ..., \x} {\draw (\y,-\s)--(\y,-\s-1);}
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\theoremstyle{plain}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{cor}[thm]{Corollary}
\theoremstyle{definition}
\newtheorem{dfn}[thm]{Definition}
\newtheorem{ex}[thm]{Example}
\newtheorem{remark}[thm]{Remark}
\numberwithin{equation}{section}
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%\setcounter{section}{-1}

\begin{document}
\title[Marginally large tableaux and rigged configurations]{Connecting marginally large tableaux and rigged configurations via crystals}
\author{Ben Salisbury}
\address{Department of Mathematics, Central Michigan University, Mt. Pleasant, MI 48859}
\email{ben.salisbury@cmich.edu}
\urladdr{http://people.cst.cmich.edu/salis1bt/}
\author{Travis Scrimshaw}
\address{Department of Mathematics, University of California, Davis, CA 95616}
\email{tscrim@ucdavis.edu}
\urladdr{http://www.math.ucdavis.edu/~scrimsha/}
\thanks{T.S.\ was partially supported by NSF grant OCI-1147247.}
\keywords{crystal, rigged configuration, marginally large tableaux, segments}
\subjclass[2010]{05E10, 17B37}

\begin{abstract}
We show that the bijection from rigged configurations to tensor products of Kirillov-Reshetikhin crystals extends to a crystal isomorphism between the $B(\infty)$ models given by rigged configurations and marginally large tableaux.
\end{abstract}

\maketitle

\section{Introduction}

In~\cite{KKR86,KR86}, Kerov, Kirillov, and Reshetikhin described a recursive bijection between classically highest-weight rigged configurations in type $A_n^{(1)}$ and standard Young tableaux, showing the Kostka polynomial can be expressed as a fermionic formula. This was then extended to Littlewood-Richardson tableaux and classically highest weight elements in a tensor product of Kirillov-Reshetikhin (KR) crystals in~\cite{KSS1999} for, again, type $A_n^{(1)}$. A similar bijection $\Phi$ between rigged configurations and tensor products of the KR crystal $B^{1,1}$ corresponding to the vector representation was extended to all non-exceptional affine types in~\cite{OSS03}, type $E_6^{(1)}$ in~\cite{OS12}, and $D_4^{(3)}$ in~\cite{Scrimshaw15}.

Following~\cite{KSS1999}, it was conjectured that the bijection $\Phi$ can be further extended to a tensor product of general KR crystals with the major step being the algorithm for $B^{1,1}$. This has been proven in a variety of cases~\cite{OSS03III,OSS13,S05,SS2006,SS15,Scrimshaw15}. Despite this bijection's recursive definition, it is conjectured (see for instance~\cite{SS15}) that $\Phi$ sends a combinatorial statistic called cocharge~\cite{OSS03} to the algebraic statistic called energy~\cite{HKOTY99}, proving the so-called $X = M$ conjecture of~\cite{HKOTT02,HKOTY99}. Additionally, the bijection $\Phi$ is conjectured to translate the combinatorial $R$-matrix~\cite{KKMMNN91} into the identity on rigged configurations.

The description of $\Phi$ on classically highest-weight elements led to a description of classical crystal operators in simply-laced types in~\cite{S06} and non-simply-laced finite types in~\cite{SS15}. It was shown for type $A_n^{(1)}$ in~\cite{DS06} and $D_n^{(1)}$ in~\cite{Sakamoto13} that $\Phi$ is a classical crystal isomorphism. Using virtual crystals~\cite{OSS03II}, it can be shown $\Phi$ is a classical crystal isomorphism in non-exceptional affine types~\cite{SS15}.

Rigged configurations were also extended beyond the context of highest weight classical crystals to $B(\infty)$ in~\cite{SS2015}. There is also a similar extension of the Kashiwara-Nakashima tableaux~\cite{KN94} and Kang-Misra tableaux~\cite{KM94} (for type $G_2$), which are used to describe KR crystals~\cite{FOS09,KMOY07,Yamane98}, to the marginally large tableaux model of $B(\infty)$~\cite{HL08,HL12}. The goal of this paper is to connect with a crystal isomorphism these two models by using the bijection $\Phi$. In particular, the crystal isomorphism we obtain from Corollary~\ref{cor:bijection} is given combinatorially, in the sense that the description does not use the Kashiwara operators.

This paper is organized as follows. In Section~\ref{sec:crystals}, we give a background on crystals. In Section~\ref{sec:tableaux}, we describe the tableaux model for highest weight crystals and marginally large tableaux. In Section~\ref{sec:rigcon}, we give background on rigged configurations and describe the bijection $\Phi$. In Section~\ref{sec:isomorphism}, we construct our isomorphism between the rigged configuration model and marginally large tableaux model for $B(\infty)$. In Section~\ref{sec:statistics}, we describe certain statistics on highest weight crystals and $B(\infty)$.

\section{Crystals}
\label{sec:crystals}

Let $\g$ be a finite-dimensional simple Lie algebra with index set $I$, Cartan matrix $A = (A_{ab})_{a,b\in I}$, weight lattice $P$, root lattice $Q$, fundamental weights $\{\Lambda_a \mid a \in I\}$, simple roots $\{\alpha_a \mid a\in I\}$, and simple coroots $\{h_a \mid a\in I\}$.  There is a canonical pairing $\langle\ ,\ \rangle\colon P^\vee \times P \longrightarrow \ZZ$ defined by $\langle h_a, \alpha_b \rangle = A_{ab}$, where $P^{\vee} = \bigoplus_{a\in I} \ZZ h_a$ is the dual weight lattice.  The quantum group associated to $\g$ is denoted $U_q(\g)$, though we will not need the details concerning $U_q(\g)$.  The interested reader is encouraged to see \cite{HK02,Lusztig93} for more details.

An \emph{abstract $U_q(\g)$-crystal} is a nonempty set $B$ together with maps
\[
e_a, f_a \colon B \longrightarrow B\sqcup\{0\}, \ \ \
\varepsilon_a, \varphi_a\colon B \longrightarrow \ZZ\sqcup\{-\infty\}, \ \ \
\wt \colon B \longrightarrow P,
\]
subject to the conditions
\begin{enumerate}
\item $f_a b = b'$ if and only if $b = e_a b'$ for $b,b'\in B$ and $a \in I$;
\item if $f_a b \neq 0$, then $\wt(f_a b) = \wt(b) - \alpha_a$ for all $a\in I$; and
\item $\varphi_a(b) - \varepsilon_a(b) = \langle h_a, \wt(b) \rangle$ for all $b\in B$ and $a \in I$.
\end{enumerate}
The maps $\{e_a \mid a \in I\}$ are called the \emph{Kashiwara raising operators} and the maps $\{f_a \mid a\in I\}$ are called the \emph{Kashiwara lowering operators}.

\begin{ex}
For a dominant integral weight $\lambda$, the crystal basis 
\[
B(\lambda) = \{ f_{a_k}\cdots f_{a_1}u_{\lambda} \mid a_1,\ldots,a_k \in I,\ k\in\ZZ_{\ge0} \} \setminus \{0\}
\] 
of an irreducible, highest weight $U_q(\g)$-module $V(\lambda)$ is an abstract $U_q(\g)$-crystal. (See~\cite{HK02,K91} for details.)  The crystal $B(\lambda)$ is characterized by the following properties.
\begin{enumerate}
\item The element $u_\lambda \in B(\lambda)$ is the unique element such that $\wt(u_\lambda) = \lambda$.
\item For all $a \in I$, $e_a u_{\lambda} = 0$.
\item For all $a \in I$, $f_a^{\inner{h_a}{\lambda} + 1} u_{\lambda} = 0$. 
\end{enumerate}
\end{ex}

\begin{ex}
The crystal basis
\[
B(\infty) = \{f_{a_k}\cdots f_{a_1} u_{\infty} \mid a_1,\dots,a_k\in I,\ k\in\ZZ_{\ge0}\}
\] 
of the negative half $U_q^-(\g)$ of the quantum group (equivalently a Verma module of highest weight $0$) is a $U_q(\g)$-crystal.  (See~\cite{HK02,K91} for details.)  Some important properties of $B(\infty)$ are the following.
\begin{enumerate}
\item The element $u_\infty \in B(\infty)$ is the unique element such that $\wt(u_\infty) = 0$.
\item For all $a\in I$, $e_au_\infty =0$.
\item For any sequence $(a_1,\dots,a_k)$ from $I$, $f_{a_k}\cdots f_{a_1}u_\infty \neq 0$.
\end{enumerate}
\end{ex}

%An abstract $U_q(\g)$-crystal is said to be \emph{upper regular} if, for all $b\in B$,
%\[
%\varepsilon_i(b) = \max\{ k \in\ZZ_{\ge0} \mid e_i^k b \neq 0 \}.
%\]
%Similarly, an abstract $U_q(\g)$-crystal is said to be \emph{lower regular} if, for all $b\in B$,
%\[
%\varphi_i(b) = \max\{ k \in \ZZ_{\ge0} \mid f_i^kb \neq 0\}.
%\]
%If $B$ is both upper regular and lower regular, then we say $B$ is \emph{regular}.  In this latter case, we may depict the entire $i$-string through $b\in B$ diagrammatically as
%\[
%e_i^{\varepsilon_i(b)}b \overset{i}{\longrightarrow}
%e_i^{\varepsilon_i(b)-1}b \overset{i}{\longrightarrow}
%\cdots \overset{i}{\longrightarrow}
%e_ib \overset{i}{\longrightarrow}
%b \overset{i}{\longrightarrow}
%f_ib \overset{i}{\longrightarrow}
%\cdots \overset{i}{\longrightarrow}
%f_i^{\varphi_i(b)-1}b \overset{i}{\longrightarrow}
%f_i^{\varphi_i(b)}b.
%\]
%Note that $B(\lambda)$ is a regular $U_q(\g)$-crystal, but $B(\infty)$ is only upper regular.

Let $B_1$ and $B_2$ be two abstract $U_q(\g)$-crystals.  A \emph{crystal morphism} $\psi\colon B_1 \longrightarrow B_2$ is a map $B_1\sqcup\{0\} \longrightarrow B_2 \sqcup \{0\}$ such that
\begin{enumerate}
\item $\psi(0) = 0$;
\item if $b \in B_1$ and $\psi(b) \in B_2$, then $\wt(\psi(b)) = \wt(b)$, $\varepsilon_a(\psi(b)) = \varepsilon_a(b)$, and $\varphi_a\psi(b)) = \varphi_a(b)$;
\item for $b \in B_1$, we have $\psi(e_a b) = e_a \psi(b)$ provided $\psi(e_ab) \neq 0$ and $e_a\psi(b) \neq 0$;
\item for $b\in B_1$, we have $\psi(f_a b) = f_a \psi(b)$ provided $\psi(f_ab) \neq 0$ and $f_i\psi(b) \neq 0$.
\end{enumerate}
A morphism $\psi$ is called \emph{strict} if $\psi$ commutes with $e_a$ and $f_a$ for all $a \in I$.  Moreover, a morphism $\psi\colon B_1 \longrightarrow B_2$ is called an \emph{embedding} if the induced map $B_1 \sqcup\{0\} \longrightarrow B_2 \sqcup \{0\}$ is injective.

Again let $B_1$ and $B_2$ be abstract $U_q(\g)$-crystals.  The tensor product $B_2 \otimes B_1$ is defined to be the Cartesian product $B_2\times B_1$ equipped with crystal operations defined by
\begin{align*}
e_a(b_2 \otimes b_1) &= \begin{cases}
e_a b_2 \otimes b_1 & \text{if } \varepsilon_a(b_2) > \varphi_a(b_1) \\
b_2 \otimes e_a b_1 & \text{if } \varepsilon_a(b_2) \le \varphi_a(b_1),
\end{cases} \\
f_a(b_2 \otimes b_1) &= \begin{cases}
f_a b_2 \otimes b_1 & \text{if } \varepsilon_a(b_2) \ge \varphi_a(b_1) \\
b_2 \otimes f_a b_1 & \text{if } \varepsilon_a(b_2) < \varphi_a(b_1),
\end{cases} \\ 
\varepsilon_a(b_2 \otimes b_1) &= \max\big( \varepsilon_a(b_2), \varepsilon_a(b_1) - \inner{h_a}{\wt(b_2)} \bigr) \\
\varphi_a(b_2 \otimes b_1) &= \max\big( \varphi_a(b_1), \varphi_a(b_2) + \inner{h_a}{\wt(b_1)} \bigr) \\
\wt(b_2 \otimes b_1) &= \wt(b_2) + \wt(b_1).
\end{align*}

\begin{remark}
Our convention for tensor products is opposite the convention given by Kashiwara in~\cite{K91}.
\end{remark}

%More generally if $B_1, \dotsc, B_t$ are crystals, to compute the action of the Kashiwara operators on the tensor product $B = B_t \otimes \cdots \otimes B_2 \otimes B_1$, we use the \emph{signature rule}.  Indeed, for $a \in I$ and $b = b_t \otimes \cdots \otimes b_2 \otimes b_1$ in $B$, write
%\[
%\underbrace{-\cdots-}_{\varphi_a(b_t)}\
%\underbrace{+\cdots+}_{\varepsilon_a(b_t)}\
%\cdots\
%\underbrace{-\cdots-}_{\varphi_a(b_1)}\
%\underbrace{+\cdots+}_{\varepsilon_a(b_1)}\ .
%\]
%From the above sequence, successively delete any $(+,-)$-pair to obtain a sequence
%\[
%a\sgn(b) :=
%\underbrace{-\cdots-}_{\varphi_a(b)}\
%\underbrace{+\cdots+}_{\varepsilon_a(b)}
%\ .
%\]
%Suppose $1 \leq j_-, j_+ \leq t$ are such that $b_{j_-}$ contributes the leftmost $-$ in $a\sgn(b)$ and $b_{j_+}$ contributes the rightmost $+$ in $a\sgn(b)$.  Then 
%\begin{align*}
%e_a b &= b_t \otimes \cdots \otimes b_{j_-+1} \otimes e_ab_{j_-} \otimes b_{j_--1} \otimes \cdots \otimes b_1, \\
%f_a b &= b_t \otimes \cdots \otimes b_{j_++1} \otimes f_ab_{j_+} \otimes b_{j_+-1} \otimes \cdots \otimes b_1.
%\end{align*}

We say an abstract $U_q(\g)$-crystal is simply a \emph{$U_q(\g)$-crystal} if it is crystal isomorphic to the crystal basis of an integrable $U_q(\g)$-module.







\section{Tableaux model}
\label{sec:tableaux}

Let $\g$ be of finite classical type. We review the Kashiwara-Nakashima tableaux and Kang-Misra tableaux, which we call \emph{classical tableaux}, model for highest weight crystals $B(\lambda)$ and the marginally large tableaux model for $B(\infty)$.

\subsection{Fundamental crystals and classical tableaux}

Recall that a tableau is called semistandard over an alphabet $J = \{ j_1 \prec j_2 \prec \cdots \prec j_p \}$ if entries are weakly increasing in rows and strictly increasing in columns, with respect to $\prec$.  Let $J(X_n)$ be the alphabet for the semistandard tableaux of type $X_n$.  Then
\begin{equation}\label{eq:alphabets}
\begin{aligned}
J(A_n) &= \{ 1 \prec 2 \prec \cdots \prec n+1 \} ,\\
J(B_n) &= \{ 1 \prec \cdots \prec n \prec 0 \prec \overline n \prec \cdots \prec \overline1 \},\\
J(C_n) &= \{ 1 \prec \cdots \prec n \prec \overline n \prec \cdots \prec \overline1 \},\\
J(D_{n+1}) &= \{ 1 \prec \cdots \prec n \prec n+1,\overline{n+1}, \prec \overline{n} \prec \cdots \prec \overline1 \}, \\
J(G_2) &= \{ 1 \prec 2 \prec 3 \prec 0 \prec \overline{3} \prec \overline{2} \prec \overline{1} \}.
\end{aligned}
\end{equation}


\begin{figure}[t]
\[
\begin{array}{|rl|}\hline
A_n: & 
\begin{tikzpicture}[scale=1.45,baseline=-4]
\node (1) at (0,0) {$\young(1)$};
\node (2) at (1.5,0) {$\young(2)$};
\node (d) at (3.0,0) {$\cdots$};
\node (n-1) at (4.5,0) {$\young(n)$};
\node (n) at (6,0) {$\boxed{n+1}$};
\draw[->] (1) to node[above]{\tiny$1$} (2);
\draw[->] (2) to node[above]{\tiny$2$} (d);
\draw[->] (d) to node[above]{\tiny$n-1$} (n-1);
\draw[->] (n-1) to node[above]{\tiny$n$} (n);
\end{tikzpicture}\\
B_n: &
\begin{tikzpicture}[baseline=-4]
\node (1) at (0,0) {$\young(1)$};
\node (d1) at (1.5,0) {$\cdots$};
\node (n) at (3,0) {$\young(n)$};
\node (0) at (4.5,0) {$\young(0)$};
\node (bn) at (6,0) {$\young(\brn)$};
\node (d2) at (7.5,0) {$\cdots$};
\node (b1) at (9,0) {$\young(\bon)$};
\draw[->] (1) to node[above]{\tiny$1$} (d1);
\draw[->] (d1) to node[above]{\tiny$n-1$} (n);
\draw[->] (n) to node[above]{\tiny$n$} (0);
\draw[->] (0) to node[above]{\tiny$n$} (bn);
\draw[->] (bn) to node[above]{\tiny$n-1$} (d2);
\draw[->] (d2) to node[above]{\tiny$1$} (b1);
\end{tikzpicture} 
\\[10pt]
C_n: &
\begin{tikzpicture}[baseline=-4]
\node (1) at (0,0) {$\young(1)$};
\node (d1) at (1.8,0) {$\cdots$};
\node (n) at (3.6,0) {$\young(n)$};
\node (bn) at (5.4,0) {$\young(\brn)$};
\node (d2) at (7.2,0) {$\cdots$};
\node (b1) at (9,0) {$\young(\bon)$};
\draw[->] (1) to node[above]{\tiny$1$} (d1);
\draw[->] (d1) to node[above]{\tiny$n-1$} (n);
\draw[->] (n) to node[above]{\tiny$n$} (bn);
\draw[->] (bn) to node[above]{\tiny$n-1$} (d2);
\draw[->] (d2) to node[above]{\tiny$2$} (b1);
\end{tikzpicture}
\\[10pt]
D_{n+1}: & 
\begin{tikzpicture}[baseline=-4]
\node (1) at (0,0) {$\young(1)$};
\node (d1) at (1.5,0) {$\cdots$};
\node (n-1) at (3,0) {$\young(n)$};
\node (n) at (4.5,.75) {$\boxed{n+1}$};
\node (bn) at (4.5,-.75) {$\boxed{\overline{n+1}}$};
\node (bn-1) at (6,0) {$\young(\brn)$};
\node (d2) at (7.5,0) {$\cdots$};
\node (b1) at (9,0) {$\young(\bon)$};
\draw[->] (1) to node[above]{\tiny$1$} (d1);
\draw[->] (d1) to node[above]{\tiny$n-1$} (n-1);
\draw[->] (n-1) to node[above,sloped]{\tiny$n$} (n);
\draw[->] (n-1) to node[below,sloped]{\tiny$n+1$} (bn);
\draw[->] (n) to node[above,sloped]{\tiny$n+1$} (bn-1);
\draw[->] (bn) to node[below,sloped]{\tiny$n$} (bn-1);
\draw[->] (bn-1) to node[above]{\tiny$n-1$} (d2);
\draw[->] (d2) to node[above]{\tiny$1$} (b1);
\end{tikzpicture}
\\[30pt]
G_2: & 
\begin{tikzpicture}[baseline=-4]
\node (1) at (0,0) {$\young(1)$};
\node (2) at (1.5,0) {$\young(2)$};
\node (3) at (3,0) {$\young(3)$};
\node (0) at (4.5,0) {$\young(0)$};
\node (b3) at (6,0) {$\young(\bth)$};
\node (b2) at (7.5,0) {$\young(\btw)$};
\node (b1) at (9,0) {$\young(\bon)$};
\path[->,font=\tiny]
 (1) edge node[above]{$1$} (2)
 (2) edge node[above]{$2$} (3)
 (3) edge node[above]{$1$} (0)
 (0) edge node[above]{$1$} (b3)
 (b3) edge node[above]{$2$} (b2)
 (b2) edge node[above]{$1$} (b1);
\end{tikzpicture}
\\\hline
\end{array}
\]
\caption{The fundamental crystals $\TT(\Lambda_1)$ when the underlying Lie algebra is of finite type.}\label{fig:fundcrystal}
\end{figure}

For our purposes, we need only define highest weight crystals for specific fundamental weights.  Namely, define a subset $\widehat P^+$ of $P^+$ by
\[
\widehat P^+ = \begin{cases}
 \ZZ\Lambda_1 \oplus \cdots \oplus \ZZ\Lambda_{n-1} \oplus \ZZ\Lambda_n & \text{ if } \g = A_n, C_n, \\
 \ZZ\Lambda_1 \oplus \cdots \oplus \ZZ\Lambda_{n-1} \oplus \ZZ(2\Lambda_n) & \text{ if } \g = B_n, \\
 \ZZ\Lambda_1 \oplus \cdots \oplus \ZZ\Lambda_{n-1} \oplus \ZZ(\Lambda_n+\Lambda_{n+1}) & \text{ if } \g = D_{n+1}, \\
 \ZZ\Lambda_1 \oplus \ZZ\Lambda_2 & \text{ if } \g = G_2.
\end{cases}
\]
The reason $\widehat P^+$ suffices is due to the constructions we will use in what follows.  In particular, these weights suffice to define the marginally large tableaux model $\TT(\infty)$ of the crystal $B(\infty)$, and thus will be sufficient for us to define our crystal morphism from the rigged configuration model for $B(\infty)$ to $\TT(\infty)$.  These weights also ensure that we will not have any ``spin columns'' in types $B_n$ and $D_{n+1}$.

Recall the fundamental crystals $\TT(\Lambda_1)$ in Figure~\ref{fig:fundcrystal}. Next consider some $\lambda \in \widehat P^+$; we wish to model an element in $B(\lambda)$. It is from these fundamental crystals that the more general crystals will be defined.  For $\lambda \in \widehat P^+$ defined by
\[
\lambda = \begin{cases}
 c_1\Lambda_1 + \cdots + c_{n-1}\Lambda_{n-1} + c_n\Lambda_n & \text{ if } \g = A_n, C_n, \\
 c_1\Lambda_1 + \cdots + c_{n-1}\Lambda_{n-1} + c_n(2\Lambda_n) & \text{ if } \g = B_n, \\
 c_1\Lambda_1 + \cdots + c_{n-1}\Lambda_{n-1} + c_n(\Lambda_n+\Lambda_{n+1}) & \text{ if } \g = D_{n+1}, \\
 c_1\Lambda_1 + c_2\Lambda_2  & \text{ if } \g = G_2,
\end{cases}
\]
let $Y_\lambda$ be the Young diagram with $c_i$ columns of height $i$.  Define $T_\lambda$ to be the unique tableau of shape $Y_\lambda$ such that all entries in the $j$th row of $T_\lambda$ are $j$.   We may embed $T_\lambda$ into $\TT(\Lambda_1)^{\otimes|\lambda|}$, where $|\lambda|$ is the number of boxes in $Y_\lambda$, by reading the tableaux entries from top-to-bottom starting with the right-most column.  Then $f_aT_\lambda$, for $a\in I$, is defined using the tensor product rule and the corresponding fundamental crystal above.  Now let $\TT(\lambda)$ be the set generated by $f_a$ $(a \in I)$ and $T_\lambda$.  This is the set of \emph{classical tableaux} of shape $\lambda$.  The description of type $A_n, B_n, C_n, D_n$ tableaux is due to Kashiwara and Nakashima~\cite{KN94} and type $G_2$ tableaux is due to Kang and Misra \cite{KM94}.
The resulting set is a crystal of semistandard tableaux (with respect to $J(X_n)$) satisfying certain filling conditions.  The explicit description of these crystals may be found in \cite{HK02,KM94,KN94}.


\begin{figure}[ht]
\input{./B2-21.tex}
\caption{The crystal graph $\TT(\Lambda_1+2\Lambda_2)$ of type $B_2$, created using Sage.}
\end{figure}



\subsection{Marginally large tableaux}

Following~\cite{Cliff98}, a semistandard tableau is called \emph{large} if the difference of the number of boxes in the $i$-th row containing the element $i$ and the total number of boxes in the $(i+1)$-th row is positive.  Additionally, following \cite{HL08}, a large (semistandard) tableau is called \emph{marginally large} if the aforementioned difference exactly 1.  Such tableaux are defined for simple Lie algebras $\g$ of type $A_n$, $B_n$, $C_n$, $D_{n+1}$, and $G_2$ in~\cite{HL08}, and of type $E_6$, $E_7$, $E_8$, and $F_4$ in~\cite{HL12}.  The alphabets over which each tableaux from \cite{HL08} are defined are given in Equation \eqref{eq:alphabets}.
%\begin{table}[ht]
%\doublespacing
%\[
%\begin{array}{|c|c|}\hline
%X_n & J(X_n) \\\hline
%A_n & \{1 \prec \cdots \prec n \prec n+1 \} \\\hline
%B_n & \{1 \prec \cdots \prec n \prec 0 \prec \overline n \prec \cdots \prec \overline 1 \} \\\hline
%C_n & \{1 \prec \cdots \prec n \prec \overline n \prec \cdots \prec \overline 1 \} \\\hline
%D_{n+1} & \left\{1 \prec \cdots \prec  n+1, \overline{n+1} \prec \cdots \prec \overline 1 \right\} \\\hline
%G_2 & \{1 \prec 2 \prec 3 \prec 0 \prec \overline3 \prec \overline2 \prec \overline1 \} \\\hline 
%\end{array}
%\]
%\caption{Alphabets for the tableaux in each type}
%\label{table:alphabets}
%\end{table}

The set of marginally large tableaux may be generated through successive application of the Kashiwara lowering operators $f_a$ ($a \in I$) to a specified highest weight vector.  It is in this way that the set of marginally large tableaux work as a combinatorial model for $B(\infty)$.  In certain types, additional conditions are required to precisely define the model, so we give the list of conditions for each type-by-type.

\begin{dfn}\label{ABCDG-tab}
For $X_n = A_n, B_n, C_n, D_{n+1}, G_2$, define the set $\TT(\infty)$ as follows.  (By convention, we assume $n=2$ when $X_n = G_2$.)
\begin{enumerate}
\item The highest weight vector is the unique tableau which consists of $n+1-i$ $i$-colored boxes in the $i$th row from the top (when written using the English convention).
\item Each element is marginally large, semistandard with respect to $J(X_n)$, and consists of exactly $n$ rows.
\end{enumerate}
We also have the following additional type-specific requirements.
\begin{itemize}
\item $X_n = B_n$ $(n\ge 2)$
\begin{enumerate}
\item Elements in the $i$th row are $\preceq \overline\imath$.
\item A $0$-box may occur at most once in a given row.
\end{enumerate}
\item $X_n = C_n$ $(n\ge 2)$
\begin{enumerate}
\item Elements in the $i$th row are $\preceq \overline\imath$.
\end{enumerate}
\item $X_n = D_{n+1}$ $(n\ge 3)$
\begin{enumerate}
\item Elements in the $i$th row are $\preceq \overline\imath$.
\item Both $n+1$ and $\overline{n+1}$ may not appear simultaneously in a single row.
\end{enumerate}
\item $X_n = G_2$
\begin{enumerate}
\item Elements in the second row are $\preceq 3$.
\item A $0$-box may occur at most once in a given row.
\end{enumerate}
\end{itemize}
\end{dfn}

The crystal operators are defined in the same way as in $\TT(\lambda)$.  Namely, read entries of a tableau $T \in \TT(\infty)$ from top-to-bottom in columns starting with the right-most column to obtain an element of $\TT(\Lambda_1)^{\otimes N}$, where $N$ is the number of boxes in $T$.  Then apply the tensor product rule to obtain $f_aT$ and $e_aT$, $a\in I$. 



\begin{thm}[{\cite{HL08}}] %HL12
Let $\g$ be a finite simple Lie algebra of type $X_n$.  Then $\TT(\infty) \cong B(\infty)$ as $U_q(\g)$-crystals.
\end{thm}

\begin{ex}
Consider type $A_3$.  The top part of the crystal graph $\TT(\infty)$ is shown in Figure~\ref{A3mlt} down to depth 3.  The notation used at the vertices is condensed so that all place holding $i$-boxes in the $i$th row are removed.  If the resulting reduction yields a row with no boxes, then that row appears with one box containing $*$.  The graph in the figure is modified output from Sage \cite{combinat,sage}.
\begin{figure}[ht]
\input{a3mlt}
\caption{The top part of $\TT(\infty)$ for type $A_3$.}
\label{A3mlt}
\end{figure}
\end{ex}

%Since $B(\infty)$ is a direct limit of the highest weight crystals $B(\lambda)$, there is a natural embedding $B(\infty) \longrightarrow B(\lambda)$ for each $\lambda \in P^+$.  As such, we may define a function $\TT(\infty) \longrightarrow \TT(\lambda)$, where $\TT(\lambda)$ is the tableaux model for $B(\lambda)$ defined in \cite{KN94}.  Moreover, for each $T$ in $\TT(\infty)$, there is a minimal $\lambda$ (with respect to usual partial order on $P^+$) such that $T$ is nonzero when projected to $\TT(\lambda)$.  Denote this weight by $\lambda_T$.  

Following \cite{HL12}, we call a column of any tableau $T$ a \emph{basic column} if it has height $r$ and is filled with $(1, \ldots, r)$. From~\cite{HL08}, consider the set
\[
\TT^L := \left\{T \in \bigcup_{\lambda \in \widehat P_+} \TT(\lambda) \mid T \text{ is large} \right\}
\]
We may partition $\TT^L$ into equivalence classes by saying $T \sim T'$ if they differ only by basic columns.  Note that $f_a T \neq 0$ for all $T \in \TT^L$. If $f_a T$ is large, then for all $S \in [T]$ such that $f_a S$ is large, we have $f_a S \in [f_a T]$. In other words, the crystal operators essentially preserve equivalence classes. Moreover, if $f_a S \notin [f_a T]$, then $f_a S$ differs from a unique element in $[f_a T]$ only by adding a single basic column of height $a$. Additionally, every equivalence class has exactly one marginally large tableaux. The details of these statements can be found in~\cite{HL08}.







\section{Rigged configurations}
\label{sec:rigcon}

\subsection{Crystal structure}

%{\color{blue} Change this entire section to assume all tensor products of KR cystals are of the form $B = \bigotimes_{i=1}^N B^{r_i, 1}$.}

We first need to consider an affine type $\widetilde{\g}$ whose classical subalgebra is $\g$. However we do not do so in the usual fasion by taking the untwisted affine algebra, but instead consider those given by Table~\ref{table:affine}.
\begin{table}[ht]
\doublespacing
\[
\begin{array}{|c|c|c|c|c|c|}\hline
\g & A_n & B_n & C_n & D_{n+1} & G_2 \\\hline
\widetilde{\g} & A_n^{(1)} & D_{n+1}^{(2)} & A_{2n-1}^{(2)} & D_{n+1}^{(1)} & D_4^{(3)} \\\hline
\end{array}
\]
\singlespacing
\caption{The association of affine type $\widetilde{\g}$ with a classical type $\g$ used here.}
\label{table:affine}
\end{table}

Set $\HH = I \times \ZZ_{>0}$.  Consider a multiplicity array
\[
L = \big(L^{(a)} \in \ZZ_{\ge0} \mid a \in I \big)
\]
and a dominant integral weight $\lambda$ for $\g$.  We call a sequence of partitions $\nu = \{ \nu^{(a)} \mid a \in I \}$ an $(L, \lambda)$-\emph{configuration} if 
\begin{equation}\label{LL-config}
\sum_{(a,i)\in\HH} i m_i^{(a)} \alpha_a = \sum_{a \in I} L^{(a)} \Lambda_a - \lambda,
\end{equation}
where $m_i^{(a)}$ is the number of parts of length $i$ in the partition $\nu^{(a)}$ and $\{\alpha_a \mid a \in I\}$ are the simple roots for $\g$.  The set of all such $(L,\lambda)$-configurations is denoted $C(L,\lambda)$.
%Let $c_a$ and $c_a^{\vee}$ denote the Kac and dual Kac labels respectively given in Table Aff~1-3~\cite{kac90} of type $\widetilde{\g}$, and let
%\[
%t_a = \max\left(\frac{c_a}{c_a^{\vee}}, c_0^{\vee}\right) \quad\quad t_a^{\vee} = \max\left(\frac{c_a^{\vee}}{c_a}, c_0\right), \quad\quad a \in I.
%\]
To an element $\nu \in C(L,\lambda)$, define the \emph{vacancy numbers} of $\nu$ to be 
\begin{equation}
\label{eq:vacancy}
p_i^{(a)}(\nu) = p_i^{(a)} = L^{(a)} - \sum_{(b,j) \in \HH_0} A_{ab} \min(i, j) m_j^{(b)}.
\end{equation}

Recall that a partition is a multiset of integers (typically sorted in weakly decreasing order).  More generally, a \emph{rigged partition} is a multiset of pairs of integers $(i, x)$ such that $i > 0$ (typically sorted under weakly decreasing lexicographic order).  Each $(i,x)$ is called a \emph{string}, while $i$ is called the length or size of the string and $x$ is the \emph{rigging}, \emph{label}, or \emph{quantum number} of the string.  Finally, a \emph{rigged configuration} is a pair $(\nu, J)$ where $\nu \in C(L,\lambda)$ and $J = \big( J_i^{(a)} \big)_{(a, i) \in \HH}$, where each $J_i^{(a)}$ is a weakly decreasing sequence of riggings of strings of length $i$ in $\nu^{(a)}$. We call a rigged configuration \emph{valid} if every label $x \in J_i^{(a)}$ satisfies the inequality $p_i^{(a)} \geq x$ for all $(a, i) \in \HH$. We say a rigged configuration is \emph{highest weight} if $x \geq 0$ for all labels $x$. Define the \emph{colabel} or \emph{coquantum number} of a string $(i,x)$ to be $p_i^{(a)} - x$.  
For brevity, we will often denote the $a$th part of $(\nu,J)$ by $(\nu,J)^{(a)}$ (as opposed to $(\nu^{(a)},J^{(a)})$).

%\begin{ex}\label{ex:runningrig}
%Rigged configurations we will be depicted as sequences of partitions with parts labeled on the left by the corresponding vacancy number and labeled on the right by the corresponding rigging.  For example, 
%\[
%\begin{tikzpicture}[scale=.35,anchor=top]
% \rpp{2,2}{-1,-1}{-1,-1}
% \begin{scope}[xshift=6cm]
% \rpp{2,2,1}{1,1,1}{1,1,1}
% \end{scope}
% \begin{scope}[xshift=12cm]
% \rpp{2,2,1,1}{0,-2,0,0}{0,0,0,0}
% \end{scope}
% \begin{scope}[xshift=18cm]
% \rpp{2,1}{0,0}{0,0}
% \end{scope} 
% \begin{scope}[xshift=24cm]
% \rpp{2,1}{0,0}{0,0}
% \end{scope}
%\end{tikzpicture}
%\]
%is a rigged configuration with $\g = D_5$ and $L$ is given by $L_2^{(1)} = L_1^{(2)} = L_1^{(3)} = 1$ with all other $L^{(a)} = 0$. % and $\lambda = \Lambda_2 + \Lambda_3$.
%See the appendix on how to construct this example in Sage.
%\end{ex}

%Denote $\hwRC(L, \lambda)$ the set of valid highest weight rigged configurations $(\nu, J)$ such that $\nu \in C(L, \lambda)$. In~\cite{S06}, an abstract $U_q(\g)$-crystal structure was given to rigged configurations in simply-laced types, and it was extended to all finite types in~\cite{SS15}. We first recall the Kashiwara operators.
%
%\begin{dfn}
%\label{def:rc_crystal_ops}
%Let $\g$ be a Lie algebra of finite type and $L$ a multiplicity array. Let $(\nu, J)$ be a valid rigged configuration. Fix $a\in I$ and let $x$ be the smallest label of $(\nu,J)^{(a)}$.  
%\begin{enumerate}
%\item If $x \geq 0$, then set $e_a(\nu,J) = 0$. Let $\ell$ be the minimal length of all strings in $(\nu,J)^{(a)}$ which have label $x$.  Otherwise, the rigged configuration $e_a(\nu,J)$ is obtained by replacing the string $(\ell, x)$ with the string $(\ell-1, x+1)$ and changing all other labels so that all colabels remain fixed.
%\item If $x > 0$, then add the string $(1,-1)$ to $(\nu,J)^{(a)}$.  Otherwise, let $\ell$ be the maximal length of all strings in $(\nu,J)^{(a)}$ which have label $x$.   Replace the string $(\ell, x)$ by the string $(\ell+1, x-1)$ and change all other labels so that all colabels remain fixed.  If the result is a valid rigged configuration, then it is $f_a(\nu, J)$ .  Otherwise $f_a(\nu, J) = 0$.
%\end{enumerate}
%\end{dfn}
%
%Let $\RC(L, \lambda)$ denote the set generated by $\hwRC(L, \lambda)$ by the Kashiwara operators. For $(\nu,J) \in \RC(L, \lambda)$, if $f_a$ adds a box to a string of length $\ell$ in $(\nu,J)^{(a)}$, then the vacancy numbers are changed using the formula
%\begin{equation}\label{f-vacancy}
%p_i^{(b)} = \begin{cases}
% p_i^{(b)} & \text{if } i \le \ell,\\
% p_i^{(b)} - (\alpha_a | \alpha_b) & \text{if } i > \ell.
%\end{cases}
%\end{equation}
%On the other hand, if $e_a$ removes a box from a string of length $\ell$, then the vacancy numbers must be changed using
%\begin{equation}\label{e-vacancy}
%p_i^{(b)} = \begin{cases}
% p_i^{(b)} & \text{if } i < \ell,\\
% p_i^{(b)} + (\alpha_a|\alpha_b) & \text{if } i \geq \ell.
%\end{cases}
%\end{equation}
%
%Let $\RC(L)$ be the closure under the Kashiwara operators of the set $\hwRC(L) = \bigcup_{\lambda\in P^+} \hwRC(L, \lambda)$. Lastly, the weight map $\wt \colon \RC(L) \longrightarrow P$ is defined as
%\begin{equation}\label{RC_weight}
%\wt(\nu,J) = \sum_{(a,i) \in \HH} i\big( L^{(a)}\Lambda_a - m_i^{(a)}\alpha_a\big).
%\end{equation}
%
%\begin{ex}
%Let $(\nu,J)$ be the rigged configuration from Example~\ref{ex:runningrig}.  Then
%\begin{align*}
%e_3(\nu,J) &= 
%\begin{tikzpicture}[scale=.35,baseline=-20]
% \rpp{2,2}{-1,-1}{-1,-1}
% \begin{scope}[xshift=6cm]
% \rpp{2,2,1}{0,0,1}{0,0,1}
% \end{scope}
% \begin{scope}[xshift=12cm]
% \rpp{2,1,1,1}{2,0,0,-1}{2,0,0,0}
% \end{scope}
% \begin{scope}[xshift=18cm]
% \rpp{2,1}{-1,0}{-1,0}
% \end{scope} 
% \begin{scope}[xshift=24cm]
% \rpp{2,1}{-1,0}{-1,0}
% \end{scope}
%\end{tikzpicture}
%\intertext{and}
%f_2(\nu,J) &= 
%\begin{tikzpicture}[scale=.35,baseline=-20]
% \rpp{2,2}{0,0}{0,0}
% \begin{scope}[xshift=6cm]
% \rpp{2,2,1,1}{-1,-1,-1,-1}{-1,-1,-1,-1}
% \end{scope}
% \begin{scope}[xshift=12cm]
% \rpp{2,2,1,1}{1,-1,1,1}{1,1,1,1}
% \end{scope}
% \begin{scope}[xshift=18cm]
% \rpp{2,1}{0,0}{0,0}
% \end{scope} 
% \begin{scope}[xshift=24cm]
% \rpp{2,1}{0,0}{0,0}
% \end{scope}
%\end{tikzpicture}.
%\end{align*}
%\end{ex}



\begin{dfn}
Let $(\nu_\emptyset,J_\emptyset)$ be the rigged configuration with empty partition and empty riggings and let $L$ be the multiplicity array of all zeros.
Define $\RC(\infty)$ to be the graph generated by $(\nu_\emptyset,J_\emptyset)$, $e_a$, and $f_a$, for $a \in I$, where $e_a$ and $f_a$ acts on elements $(\nu,J)$ in $\RC(\infty)$ as follows. Fix $a \in I$ and let $x$ be the smallest label of $(\nu,J)^{(a)}$.
\begin{itemize}
\item[$e_a$:] If $x \geq 0$, then set $e_a(\nu,J) = 0$. Otherwise, let $\ell$ be the minimal length of all strings in $(\nu,J)^{(a)}$ which have label $x$.  The rigged configuration $e_a(\nu,J)$ is obtained by replacing the string $(\ell, x)$ with the string $(\ell-1, x+1)$ and changing all other labels so that all colabels remain fixed.
\item[$f_a$:] If $x > 0$, then add the string $(1,-1)$ to $(\nu,J)^{(a)}$.  Otherwise, let $\ell$ be the maximal length of all strings in $(\nu,J)^{(a)}$ which have label $x$.   Replace the string $(\ell, x)$ by the string $(\ell+1, x-1)$ and change all other labels so that all colabels remain fixed.
\end{itemize}
The remaining crystal structure on $\RC(\infty)$ is given by
\begin{subequations}\label{epphiwt}
\begin{align}
\label{Binf_ep} \varepsilon_a(\nu,J) &= \max\{ k \in \ZZ_{\ge0} \mid  e_a^k(\nu,J) \neq 0 \}, \\ 
\label{Binf_phi} \varphi_a(\nu,J) &= \varepsilon_a(\nu,J) + \langle h_a,\wt(\nu,J)\rangle, \\ 
\label{Binf_wt} \wt(\nu,J) &= -\sum_{(a,i)\in\HH} im_i^{(a)}\alpha_a = -\sum_{a\in I} |\nu^{(a)}|\alpha_a.
\end{align}
\end{subequations}
\end{dfn}

It is worth noting that, in this case, the definition of the vacancy numbers reduces to
\begin{equation}
p_i^{(a)}(\nu) = p_i^{(a)} = -\sum_{(b,j) \in \HH_0} A_{ab} \min(i, j) m_j^{(b)}.
\end{equation}


\begin{ex}
Rigged configurations will be shown as a sequence of partitions where the vacancy numbers will be written on the left and the corresponding rigging on the right. Let $\g$ be of type $A_5$ and $(\nu,J) = f_5 f_4 f_5 f_2 f_1 f_2 f_3(\nu_{\emptyset}, J_{\emptyset})$ be the rigged configuration
\begin{align*}
(\nu,J) &= 
\begin{tikzpicture}[scale=.35,baseline=-18]
 \rpp{1}{-1}{-1}
\begin{scope}[xshift=5cm]
 \rpp{2}{-1}{-2}
\end{scope}
\begin{scope}[xshift=12cm]
 \rpp{1}{1}{0}
\end{scope}
\begin{scope}[xshift=16cm]
 \rpp{1}{-1}{0}
\end{scope}
\begin{scope}[xshift=21cm]
 \rpp{2}{-1}{-3}
\end{scope}
\end{tikzpicture}
\intertext{Then $\wt(\nu,J) = -\alpha_1 - 2\alpha_2 - \alpha_3 - \alpha_4 - 2\alpha_5$,}
e_2(\nu,J) &= 
\begin{tikzpicture}[scale=.35,baseline=-18]
 \rpp{1}{-1}{-1}
\begin{scope}[xshift=5cm]
 \rpp{1}{0}{0}
\end{scope}
\begin{scope}[xshift=12cm]
 \rpp{1}{1}{0}
\end{scope}
\begin{scope}[xshift=16cm]
 \rpp{1}{-1}{0}
\end{scope}
\begin{scope}[xshift=21cm]
 \rpp{2}{-1}{-3}
\end{scope}
\end{tikzpicture}
\intertext{and}
f_2(\nu,J) &= 
\begin{tikzpicture}[scale=.35,baseline=-18]
 \rpp{1}{-1}{-1}
\begin{scope}[xshift=5cm]
 \rpp{3}{-2}{-4}
\end{scope}
\begin{scope}[xshift=12cm]
 \rpp{1}{1}{0}
\end{scope}
\begin{scope}[xshift=16cm]
 \rpp{1}{-1}{0}
\end{scope}
\begin{scope}[xshift=21cm]
 \rpp{2}{-1}{-3}
\end{scope}
\end{tikzpicture}
\end{align*}
\end{ex}


\begin{thm}[\cite{SS2015}]
\label{thm:RCinf_nsl}
The map defined by $(\nu_\emptyset,J_\emptyset) \mapsto u_\infty$, where $u_\infty$ is the highest weight element of $B(\infty)$, is a $U_q(\g)$-crystal isomorphism $\RC(\infty) \cong B(\infty)$.
\end{thm}

We can extend the crystal structure on rigged configurations to model $B(\lambda)$ as follows. Consider a multiplicity array $L$ such that $\lambda = \sum_{a \in I} L^{(a)} \Lambda_a$. We first we modify the definition of the weight to be $\wt'(\nu, J) = \wt(\nu, J) + \lambda$. Next, modify the crystal operators by saying $f_a(\nu, J) = 0$ if $\varphi_a(\nu, J) = 0$. Equivalently, we say $f_a(\nu, J) = 0$ if the result under $f_a$ above is not a valid rigged configuration. Let $\RC(\lambda)$ denote the closure of $(\nu_{\emptyset}, J_{\emptyset})$ under these modified crystal operators. This arises from the natural projection of $B(\infty) \longrightarrow B(\lambda)$.

\begin{thm}[\cite{SS15}]
Let $\g$ be of finite type. Then
$
\RC(\lambda) \iso B(\lambda).
$
\end{thm}

\begin{figure}[ht]
\input{./B2-21-RC.tex}
\caption{The crystal graph $\RC(\Lambda_1+2\Lambda_2)$ of type $B_2$, created using Sage.}
\end{figure}


\subsection{Bijection with tableau model for highest weight crystals}

Kirillov-Reshetikhin (KR) crystals are crystals $B^{r,s}$ associated to certain finite-dimensional $U_q([\widetilde{\g}, \widetilde{\g}])$-modules $W^{r,s}$, where $r$ is a node in the Dynkin diagram and $s$ is a positive integer. 
As $U_q(\g)$-crystals, KR crystals have the direct sum decompositions
\[
B^{r,s} = B(s\Lambda_r) \oplus \bigoplus_{\lambda} B(\lambda),
\]
where the sum is over $\lambda \in P^+$ satisfying certain conditions~\cite{Chari01, FOS09, HKOTT02,HKOTY99, Hernandez06,Hernandez10, Nakajima03}. We note that we only work with the classical crystal structure of $B^{r,s}$, and as such, we simply consider $B^{r,s}$ to be a $U_q(\g)$-crystal.
%Next we only describe Kirillov-Reshetikhin crystals $B^{r,s}$ where $r \in I$ and $s \in \ZZ_{>0}$ of type $\widetilde{\g}$ as a direct sum of classical crystals. For $\g$ of type $A_n$, we have $B^{r,s} = B(s\Lambda_r)$. For $\g$ of type $B_n$, we have $B^{r,s} = \bigoplus_{\lambda} B(\lambda)$ where we take all partition that fit inside an $r \times s$ box. For $\g$ of type $C_n$ or $D_{n+1}$, we have $B^{r,s} = \bigoplus_{\lambda} B(\lambda)$ where $\lambda$ is formed by removing vertical dominoes from an $r \times s$ box. {\color{blue} Note: we don't need the spinor cases here since all of our shapes are partitions, as long as we're careful with our wording.}

In~\cite{OSS03}, a bijection $\Phi$ from classically highest weight elements in a tensor product of KR crystals of the form $(B^{1,1})^{\otimes N}$ for all non-exceptional affine types was described.  A similar bijection in type $E_6^{(1)}$ and $D_4^{(3)}$ was given in~\cite{OS12} and~\cite{Scrimshaw15}, respectively. For simplicity, if $B = \bigotimes_{i=1}^N B^{r_i,1}$ we write $\RC(B) = \RC(\lambda)$, where $\lambda = \sum_{a \in I} c_a \Lambda_a$ with $c_a$ equal to the number of factors $B^{a,1}$ occurring in $B$.

\begin{remark}
\label{remark:KR_crystal difference}
We define
\[
\widetilde{B}^{r,1} = \begin{cases}
B^{n,2} & \g = B_n \text{ and } r = n, \\
B^{n,1} \otimes B^{n+1,1} & \g = D_{n+1} \text{ and } r = n, \\
B^{r,1} & \text{otherwise.}
\end{cases}
\]
Because we will only use $\widetilde{B}^{r,s}$ for the remainder of this paper and to ease the burden of notation, we will simply write $B^{r,s}$. We also note that this allows us to not consider any special modifications to $\Phi$ as in~\cite{SS15}.
\end{remark}


The bijection $\Phi$ is given by applying the basic algorithm given in~\cite{OSS03}
\[
\delta \colon \RC(B^{1,1} \otimes B^*) \longrightarrow \RC(B^*)
\] 
as many times as necessary, where $B^* = \bigotimes_{i=1}^N B^{r_i,1}$. The algorithm $\delta$ is given by traversing the crystal graph $\TT(\Lambda_1)$ (of type $\g$) starting at $\boxed{1} \in \TT(\Lambda_1)$, where for each edge $a$ we remove a box from a singular string from $\nu^{(a)}$ of strictly longer length than the previously selected string after removal. For $\widetilde{\g}$ of type $D_{n+1}^{(1)}$, we choose the smaller singular string between $\nu^{(n)}$ and $\nu^{(n+1)}$ when we are at $\boxed{n-1} \in \TT(\Lambda_1)$.  If we cannot find a singular string or there are no outgoing arrows when we are at $\boxed{r} \in \TT(\Lambda_1)$, then we say $\delta$ returns $r$ and we make all changed strings singular.

We also have the following modification for $\widetilde{\g}$ of type $D_{n+1}^{(2)}$.  Let $\ell^{(a)}$ denote the original length of the selected strings in $\nu^{(a)}$ (for $a < r$, we have $\ell^{(a)} = 1$). We say a string $(i, x)$ is \emph{quasi-singular} if $x = p_i^{(n)} - 1$ and there does not exist a singular string of length $i$. For $\nu^{(n)}$, if the singular string has length $1$, we immediately return $\emptyset \in B^{1,1}$.  Otherwise we look for the smallest string of longer length than the previously which is either
\begin{itemize}
\item[(S)] singular,
\item[(Q)] quasi-singular.
\end{itemize}
If no such string exists, we return $n-1$ (as usual). If we are in case~(S), we remove 2 boxes from the singular string and proceed from $\boxed{\overline{n-1}} \in \TT(\Lambda_1)$.  If we are in case~(Q), we remove a box from the quasi-singular string and look for a larger singular string in $\nu^{(n)}$. If no such string exists, we return $0$. Otherwise we say we are in case~(Q,S) and remove a box from the found singular string. We then continue from $\boxed{\overline{n-1}} \in \TT(\Lambda_1)$. If we are at $\boxed{\bi} \in \TT(\Lambda_1)$ and the length of the previously selected string before removal equals $\ell^{(a)}$, we remove a second box from the string originally selected in $\nu^{(a)}$.

After all boxes are removed, we make all of the changed strings singular unless we are in case~(Q,S), in which case the (longer) selected singular string in $\nu^{(n)}$ is made quasi-singular.  

\begin{ex}
Consider $B = (B^{1,1})^{\otimes 5}$ of type $D_5^{(2)}$. Therefore applying $\delta$ each time, we have
\[
\begin{tikzpicture}[scale=.35]
\begin{scope}[yshift=1cm]
 \rpp{2,1,1}{0,0,0}{0,1,1}
\begin{scope}[xshift=5cm]
 \rpp{2,1}{0,0}{0,0}
\end{scope}
\begin{scope}[xshift=10cm]
 \rpp{2}{1}{1}
\end{scope}
\begin{scope}[xshift=15cm]
 \rpp{2}{0}{0}
\end{scope}
\end{scope} % yshift
%%%
\draw[->] (8,-3cm) -- (8,-6cm) node[midway,right] {$\bon$};
\begin{scope}[yshift=-6cm]
 \rpp{1,1}{0,0}{1,1}
\begin{scope}[xshift=5cm]
 \rpp{1}{0}{0}
\end{scope}
\begin{scope}[xshift=10cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\begin{scope}[xshift=15cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\end{scope} % yshift
%%%
\draw[->] (8,-9cm) -- (8,-12cm) node[midway,right] {$1$};
\begin{scope}[yshift=-12cm]
 \rpp{1,1}{0,0}{0,0}
\begin{scope}[xshift=5cm]
 \rpp{1}{0}{0}
\end{scope}
\begin{scope}[xshift=10cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\begin{scope}[xshift=15cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\end{scope} % yshift
%%%
\draw[->] (8,-15cm) -- (8,-18cm) node[midway,right] {$3$};
\begin{scope}[yshift=-18cm]
 \rpp{1}{0}{0}
\begin{scope}[xshift=5cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\begin{scope}[xshift=10cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\begin{scope}[xshift=15cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\end{scope} % yshift
%%%
\draw[->] (8,-21cm) -- (8,-24cm) node[midway,right]{$2$};
\begin{scope}[yshift=-24cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\begin{scope}[xshift=5cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\begin{scope}[xshift=10cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\begin{scope}[xshift=15cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\end{scope} % yshift
%%%
\draw[->] (8,-27cm) -- (8,-30cm) node[midway,right] {$1$};
\begin{scope}[yshift=-30cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\begin{scope}[xshift=5cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\begin{scope}[xshift=10cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\begin{scope}[xshift=15cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\end{scope} % yshift
%%%
\end{tikzpicture}
\]
(where the result from each application of $\delta$ is to the right of the arrow) and resulting in
\[
\young(\bon) \otimes \young(1) \otimes \young(3) \otimes \young(2) \otimes \young(1)\ .
\]
\end{ex}

We can now extend this bijection to $\bigotimes_{i=1}^N B^{r_i,1}$ as given for types $A_n^{(1)}$~\cite{KSS1999}, $D_{n+1}^{(1)}$~\cite{S05}, $A_{2n-1}^{(2)}$~\cite{SS15}, $D_{n+1}^{(2)}$~\cite{OSS03II}, and $D_4^{(3)}$ \cite{Scrimshaw15} by considering the map 
\[
\mathrm{lt} \colon \RC(B^{r,1} \otimes B^*) \longrightarrow \RC(B^{1,1} \otimes B^{r-1,1} \otimes B^*)
\] 
for $r \geq 2$ which adds a singular string of length 1 to all $\nu^{(a)}$ for $a < r$ and then applying $\delta$. We can combine these two steps $\delta' := \delta \circ \mathrm{lt}$ where we just begin $\delta$ starting at $\boxed{r} \in \TT(\Lambda_1)$ (i.e., the first box we try to remove is in $\nu^{(r)}$). Unless otherwise noted, we will be using $\delta'$ in place of $\delta$.

\begin{ex}
Consider $B^{1,1} \otimes B^{2,1} \otimes B^{1,1}$ of type $A_5^{(2)}$. Therefore applying $\delta$ each time, we have
\[
\begin{tikzpicture}[scale=.35]
 \rpp{1,1}{0,0}{0,0}
\begin{scope}[xshift=6cm]
 \rpp{1,1}{1,0}{1,1}
\end{scope}
\begin{scope}[xshift=12cm]
 \rpp{1}{0}{0}
\end{scope}
%%%
\draw[->] (6,-4cm) -- (6,-7cm) node[midway,right] {$\bth$};
\begin{scope}[yshift=-7cm]
 \rpp{1}{0,0}{0}
\begin{scope}[xshift=6cm]
 \rpp{1}{1,0}{1}
\end{scope}
\begin{scope}[xshift=12cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\end{scope} % yshift
%%%
\draw[->] (6,-10cm) -- (6,-13cm) node[midway,right] {$3$};
\begin{scope}[yshift=-13cm]
 \rpp{1}{0}{0}
\begin{scope}[xshift=6cm]
 \rpp{1}{0}{0}
\end{scope}
\begin{scope}[xshift=12cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\end{scope} % yshift
%%%
\draw[->] (6,-16cm) -- (6,-19cm) node[midway,right] {$2$};
\begin{scope}[yshift=-19cm]
 \rpp{1}{0}{0}
\begin{scope}[xshift=6cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\begin{scope}[xshift=12cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\end{scope} % yshift
%%%
\draw[->] (6,-22cm) -- (6,-25cm) node[midway,right] {$1$};
\begin{scope}[yshift=-25cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\begin{scope}[xshift=6cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\begin{scope}[xshift=12cm]
 \draw (0.5,-1.5) node {$\emptyset$};
\end{scope}
\end{scope} % yshift
%%%
\end{tikzpicture}
\]
and resulting in
\[
\young(\bth) \otimes \young(3,2) \otimes \young(1)\ .
\]
\end{ex}

For $\delta^{-1}(b)$, in general we select the largest singular strings starting at $\boxed{b} \in \TT(\Lambda_1)$ and following arrows in reverse until we reach $\boxed{r} \in \TT(\Lambda_1)$.

\begin{remark}
We note that for $B(\Lambda_r) \subseteq B^{r,1}$, the classical tableaux representation is the same as the Kirillov--Reshetikhin tableaux representation of~\cite{S05,OSS13,SS15,Scrimshaw15}.
\end{remark}

We also have the following from~\cite[Prop.~6.4]{SS15} by using the results of~\cite{DS06,Sakamoto13,OSS03II,SS15}.

\begin{thm}
\label{thm:iso_columns}
Let $B = \bigotimes_{i=1}^N B^{r_i,1}$ be of type $\widetilde{\g}$. Then $\Phi$ is a $U_q(\g)$-crystal isomorphism, where $\widetilde{\g}$ and $\g$ are related via Table \ref{table:affine}.
\end{thm}

Given dominant integral weight $\lambda = \sum_{a\in I} c_a\Lambda_a \in \widehat{P}^+$, define
\[
B^{\otimes \lambda} = \bigotimes_{a \in I} (B^{a,1})^{\otimes c_a},
\]
where the factors are ordered to be weakly decreasing with respect to $a$.

\begin{ex}
If $\lambda = 2\Lambda_4 + 3\Lambda_1$ in type $A_{17}$, then $B^{\otimes \lambda} = (B^{4,1})^{\otimes 2} \otimes (B^{1,1})^{\otimes 3}$.
\end{ex}

Next we can restrict $\Phi$ to a (classical, or $U_q(\g)$-)crystal isomorphism between $\RC(\lambda)$ and $\TT(\lambda)$ as follows. We recall that there exists, by weight considerations, a unique copy
\[
B\left( \sum_{i=1}^N \mu_i \right) \subseteq \bigotimes_{i=1}^N B(\mu_i)
\]
generated by $u_{\mu_1} \otimes \cdots \otimes u_{\mu_N}$, where $u_{\mu_i} \in B(\mu_i)$ is the unique highest weight element. Then there exists a unique embedding
\begin{equation}
\label{eq:subcrystal}
\TT(\lambda) \subseteq \bigotimes_{a \in I} \TT(\Lambda_a)^{\otimes c_a} \subseteq B^{\otimes \lambda}.
\end{equation}
Because we have chosen the ordering of $B^{\otimes \lambda}$ to be in decreasing order, the natural embedding of $\TT(\lambda) \longhookarrow \TT(\Lambda_1)^{\otimes \absval{\lambda}}$ agrees with the natural classical embedding of $B^{\otimes \lambda} \longhookarrow \TT(\Lambda_1)^{\otimes \absval{\lambda}}$.
Moreover the highest weight element $T_{\lambda} \in \TT(\lambda)$ is given by (a tensor product of) basic columns. So in $\RC(B^{\otimes\lambda})$, considered as a $U_q(\g)$-crystal, the unique connected component generated by $(\nu_{\emptyset}, J_{\emptyset})$ is $\RC(\lambda)$. Therefore $\Phi(\nu_{\emptyset}, J_{\emptyset}) = T_{\lambda}$, and hence we have the following.

\begin{prop}
\label{prop:classical_bijection}
The crystal isomorphism $\Phi \colon \RC(B^{\otimes \lambda}) \longrightarrow B^{\otimes \lambda}$ restricts to a crystal isomorphism between $\RC(\lambda)$ and $\TT(\lambda)$.
\end{prop}






\section{The crystal isomorphism between $\RC(\infty)$ and $\TT(\infty)$}
\label{sec:isomorphism}

Let $\g$ be of type $A_n$, $B_n$, $C_n$, $D_{n+1}$, or $G_2$. By knowing to which $B(\lambda)$ a particular rigged configuration belongs (in fact, there are infinite such $\lambda$), we can extend the bijection between rigged configurations and tensor products of KR crystals to $B(\infty)$ by projecting down to $B(\lambda)$. We show that this implies the (induced) map given by lifting the isomorphism $\Phi \colon \RC(\lambda) \longrightarrow \TT(\lambda)$ from Proposition~\ref{prop:classical_bijection} is an isomorphism between $\RC(\infty)$ and $\TT(\infty)$ (and thus the unique isomorphism between $\RC(\infty)$ and $\TT(\infty)$ since the automorphism group of $B(\infty)$ is the trivial group). 

Consider
\[
%\RC^V = \left\{ (\nu,J) \in \bigsqcup_{\lambda \in P^+} \RC(\lambda) : (\nu,J) \text{ is valid} \right\}
\RC^V = \bigsqcup_{\lambda \in P^+} \RC(\lambda),
%\RC^V = \left\{ (\nu,J) \in \bigsqcup_{\lambda \in P^+} \bigl( T_\lambda \otimes \RC(\infty) \bigr) : (\nu,J) \text{ is valid} \right\}
\]
and denote $(\nu, J, \lambda) \in \RC^V$ as the element $(\nu, J) \in \RC(\lambda)$. Define an equivalence relation on $\RC^V$ by asserting
\begin{equation}
\label{eq:equivalence_class}
(\nu, J, \lambda) \sim (\nu', J', \lambda') \iff \nu = \nu' \text{ and } J = J'.
\end{equation}
Note the vacancy numbers will vary over the equivalence class.  Equivalently, we have defined a subset $W_{(\nu,J)} \subseteq P^+$ such that $(\nu, J) \in \RC(\lambda)$ for all $\lambda \in W_{(\nu,J)}$. We show that every equivalence class of large tableaux embeds into an equivalence class of $\RC^V$ and that $\Phi$ induces a bijection from $\RC^V / \sim$ to $\TT(\infty) = \TT^L / \sim$. Subsequently, we show that this induced bijection is the desired crystal isomorphism.

For a sequence of partitions $\nu = (\nu^{(a)})_{a \in I}$, define $\lambda_{\nu} \in P^+$ by
\[
\lambda_{\nu} := \sum_{\substack{a \in I \\ a < n}} \big( \lvert \nu^{(a)} \rvert + 1 \big) \Lambda_a + \lambda_{\nu}^{(n)},
\]
where
\[
\lambda_{\nu}^{(n)} := \begin{cases}
2 \big( \lvert \nu^{(n)} \rvert + 1 \big) \Lambda_n & \g = B_n,
\\ (\max\bigl( \lvert \nu^{(n)} \rvert, \lvert \nu^{(n+1)} \rvert \bigr) + 1 \big) (\Lambda_n + \Lambda_{n+1}) & \g = D_{n+1},
\\ \big( \lvert \nu^{(n)} \rvert + 1 \big) \Lambda_n & \text{otherwise,}
\end{cases}
\]
and
\[
\RC^{EV} = \{ (\nu, J, \lambda) \in \RC^V \mid \lambda \geq \lambda_{\nu} \}.
\]
Here, $\lambda \leq \mu$ means $\lambda = (\lambda_i : i \in I)$ and $\mu = (\mu_i \mid i \in I)$ with $\lambda_i \le \mu_i$ for all $i \in I$.
We can restrict the equivalence relation given in equation \eqref{eq:equivalence_class} to $\RC^{EV}$ (so that classes in $\RC^{EV}$ are subclasses of those in $\RC^V$).  Call a rigged configuration \emph{extra valid} if it belongs to $\RC^{EV}$, and call a rigged configuration \emph{marginally extra valid} if $(\nu, J) \in \RC(\lambda_{\nu})$. We note that for each equivalence class in $\RC^{EV}$, there is a unique marginally extra valid rigged configuration because it has the smallest possible vacancy numbers.

\begin{lemma}
If $(\nu, J) \in \RC(\infty)$, then $(\nu, J) \in \RC(\lambda_{\nu})$.
\end{lemma}

\begin{proof}
This clearly holds for $(\nu_{\emptyset}, J_{\emptyset}) \in \RC(0)$. We will now proceed by induction by applying $f_a$ for some $a \in I$. Suppose $(\nu, J) \in \RC(\lambda_{\nu})$, we will show that $(\nu',J') = f_a(\nu, J)$ is in $\RC(\lambda_{\nu'})$. We note that the only possibile failure will occur if $x' > p_{i+1}^{(a)}(\nu')$ for the string $(i, x)$ acted on by $f_a$ since all other colabels remain fixed. We have
\[
p_{i+1}^{(a)}(\nu') - p_{i+1}^{(a)}(\nu) = -1
\]
since $\lambda_{\nu'} - \lambda_{\nu} = \Lambda_a$.  But because $x' - x = -1$, we must have $x' \leq p_{i+1}^{(a)}(\nu')$. Therefore $(\nu', J') \in \RC(\lambda_{\nu'})$.  Since there is some path to $(\nu_{\emptyset}, J_{\emptyset})$, the proof follows by induction.
\end{proof}


\begin{lemma}
\label{lemma:embed_MLT}
Let $T \in \TT^L$.  Then $\Phi^{-1}(T) \in \RC^V$.  Moreover, if $t \in [T]$, then $\Phi^{-1}(t) \in [\Phi^{-1}(T)]$.
\end{lemma}

\begin{proof}
Fix a large tableau $T$. By the definition of $\Phi$ and $\RC^V$, we have $\Phi^{-1}(T) \in \RC^V$. Next we note a column in $T$ of height $r$ has the form
\begin{equation}\label{column}
\begin{tikzpicture}[baseline]
\matrix [matrix of math nodes,column sep=-.4, row sep=-.5,text height=10,text width=23,align=center,inner sep=3] 
 {
 	\node[draw,fill=gray!30]{1}; \\
  	\node[draw,fill=gray!30]{2}; \\
  	\node[draw,fill=gray!30]{\vdots}; \\ 
  	\node[draw,fill=gray!30]{r-1}; \\
	\node[draw]{x}; \\
 };
\end{tikzpicture}
\end{equation}
where $1 \le r \le n$. We are going to add a column of the form above in $B^{r,1}$. Suppose $B = \bigotimes_{i=1}^N B^{r_i, s_i}$ and $(\nu,J) \in \RC(B^{a-1,1} \otimes B)$ for $a < r$.  Applying $\delta^{-1}$ to the column in \eqref{column} will change $(\nu,J)$, and the order by which $(\nu,J)$ is affected is determined by reading the column from top to bottom. Indeed, applying $\delta^{-1}$ corresponding to the $a$-box of the column will add $1$ to the vacancy numbers of $\nu^{(a)}$ and subtract $1$ from the vacancy numbers of $\nu^{(a-1)}$ if $a > 1$.  Now suppose we are performing $\delta^{-1}$ corresponding to the $x$-box at the bottom of the column.  Then this application of $\delta^{-1}$ can only add boxes to $\nu^{(a)}$ for $a \geq r$, and can at most decrease the vacancy numbers in $\nu^{(r-1)}$ by $1$. In particular, if $x = r$, then the net result of adding this column is that the vacancy numbers of $\nu^{(r)}$ increased by $1$ and the vacancy numbers of $\nu^{(a)}$ for $a<r$ are left unchanged.

In applying $\Phi^{-1}$ to $T$, we are moving from right to left in $T$, so we are applying $\delta^{-1}$ to columns weakly increasing in height. Moreover, $t$ differs from $T$ by the, without loss of generality, addition of columns with $x = r$. Therefore from the above, we must have $\Phi^{-1}(t) \sim \Phi^{-1}(T)$ for any $t \in [T]$.
\end{proof}

\begin{lemma}
\label{lemma:embed_RC}
Let $(\nu, J) \in \RC^{EV}$. %The bijection $\Phi$ embeds an equivalence class $[(\nu_{EV}, J_{EV})] \in \RC^{EV}(\nu, J)$ into an equivalence class $[T] \in \TT^L$.
Then $\Phi(\nu,J) \in \TT^L$. Moreover if $(\nu',J') \in [(\nu,J)]$, then $\Phi(\nu',J') \in [\Phi(\nu,J)]$.
\end{lemma}

\begin{proof}
Fix some extra valid rigged configuration $(\nu, J)$. Suppose $\Phi(\nu, J) \notin \TT^L$. Therefore during the procedure of $\Phi$ in a column of height $r$ at height $a < r$, we return $x \geq a$, so we remove at least one box from $\nu^{(a)}$. Therefore we must remove at least
\[
1 + \inner{\alpha_a}{\lambda_{\nu}} = 1 + 1 + |\nu^{(a)}|
\]
boxes from $\nu^{(a)}$ since we must return at least $x$ by the semistandard condition. This is a contradiction, and so we must return $a$. Similarly for the left-most column of height $r$, we must return $r$. Hence $\Phi(\nu, J) \in \TT^L$.

Next we show $\Phi(\nu', J') \in [\Phi(\nu, J)]$. Consider $(\nu', J') \in \RC(\lambda_{\nu'})$ such that $\lambda_{\nu'} - \lambda_{\nu} = \Lambda_r$ for some $1 \leq r \leq n$. We will show that $\Phi(\nu', J')$ differs from $\Phi(\nu, J)$ by a basic column of height $r$. We note that
\begin{equation}
\label{eq:diff_vac_nums}
p_i^{(a)}(\nu') - p_i^{(a)}(\nu) = \delta_{ar},
\end{equation}
and therefore all columns of height at least $r+1$ are equal under $\Phi$. That is $\delta$ returns the same elements on both $(\nu', J')$ and $(\nu, J)$. Furthermore, once we've removed all such columns (there are the same number of columns in each), the results are equivalent such that the difference of the weights is still $\Lambda_r$. Hence Equation~\eqref{eq:diff_vac_nums} still holds.

Now we have one additional column of height $r$ in $\lambda_{\nu'}$. From Equation~\eqref{eq:diff_vac_nums}, we must have all strings of $\nu'^{(r)}$ being non-singular. Therefore $\delta$ returns $r$, and we increase all vacancy numbers of $\nu'^{(r-1)}$. Thus all strings of $\nu'^{(r-1)}$ are non-singular and iterating this, we see that we return a basic column of height $r$.

At this point, the resulting rigged configurations are equal (not just equivalent as they have the same weight), and hence the remaining result from $\Phi$ are equal. Since there exists a unique element of minimal weight in $[(\nu, J)]$, the claim follows by induction.
\end{proof}
%Since $(\nu, J)$ is extra valid, there does not exist any singular strings by definition. Note that splitting off the left column will increase all of the vacancy numbers for $\nu^{(r)}$, so the splitting does not create any singular strings. Thus when applying $\delta$ on $B^{r,1}$, we terminate immediately and return $r$. This then adds 1 to all of the vacancy numbers of $\nu^{(r-1)}$ and decreases 1 to all vacancy numbers from $\nu^{(r)}$ due to the new $L$ matrix. Thus there are no singular strings in $\nu^{(r-1)}$, and so by induction we get a column of height $(1, 2, \dotsc, r)$. Therefore increasing the vacancy numbers corresponds to adding basic columns. The operation of increasing vacancy numbers and adding basic columns corresponds to equivalence relation for rigged configurations and large tableaux respectively.

%Next we consider having removed all basic columns of height $r$, So the leftmost column is of the form $(1, \dotsc, r-1, k)$ for some letter $k$ between $r$ and $\overline{r}$ inclusive. Since we have not removed any columns of height $a < r$, there are no singular strings in $\nu^{(a)}$ for $a < r$. Hence $\delta$ will only affect $\nu^{(a)}$ for $a \geq r$ and will result in some $k$ between $r$ and $\overline{r}$. It is straightforward to see that in the resulting rigged configuration, there are no singular strings in $\nu^{(a)}$ for all $a < r$, so the remaining part of the column will be $1, 2, \dotsc, r-1$. Thus we have added 1 to all vacancy numbers of $\nu^{(r-1)}$ with potentially subtracting 1 as well. Hence $\nu^{(r-1)}$ contains no singular strings as in the previous cases. Therefore continuing to apply $\delta$ returns the remainder of the column.

%From the above, we can remove all leftmost columns of height $r$. Therefore the leftmost column has height $r-1$, and by induction, we have $\Phi(\nu, J) \in \bigl[\Phi(\nu', J')\bigr]$.
%\end{proof}

The following lemma is analogous to~\cite[Lemma~3.2]{HL08}, which shows that the crystal operators are well-defined on equivalence classes.

\begin{lemma}
\label{lemma:preserve_Kashiwara}
Fix $a\in I$.
\begin{enumerate}
\item If $(\nu,J) \in \RC^{EV}$, then $f_a(\nu,J) \neq 0$.
\item Given any element of $\RC(\infty)$, we can always find a representative $(\nu,J) \in \RC^{EV}$ such that $f_a(\nu,J)$ is a valid rigged configuration.
\item If $(\nu,J)$ and $(\nu',J')$ are in the same equivalence class in $\RC^V/\sim$, then $[f_a(\nu,J)] = [f_a(\nu',J')]$.
\item If $(\nu,J)$ is valid, then $e_a(\nu,J)$ is either valid or zero.
\item If $(\nu,J)$ and $(\nu',J')$ are in the same equivalence class in $\RC^V/\sim$, then either $[e_a(\nu,J)] = [e_a(\nu',J')]$ or both $e_a(\nu,J) = e_a(\nu',J') = 0$.
\end{enumerate}
\end{lemma}

\begin{proof}
These statements can be seen directly from the definitions.
\end{proof}

Thus we can define
\begin{subequations}
\label{eq:crystal_structure}
\begin{align}
e_a [(\nu, J)] & = [e_a(\nu, J)]
\\ f_a [(\nu, J)] & = [f_a(\nu, J)]
\\ \wt [(\nu, J)] & = \sum_{a \in I} \lvert \nu^{(a)} \rvert \Lambda_a,
\\ \varepsilon_a [(\nu, J)] & = \max\{ e_a^k(\nu, J) \neq 0 \mid k \in \ZZ_{>0} \},
\\ \varphi_a [(\nu ,J)] & = \varepsilon_a [(\nu, J)] + \inner{\wt [(\nu, J)]}{h_a},
\end{align}
\end{subequations}
for any $[(\nu, J)] \in \RC^V / \sim$ with appropriate representative $(\nu, J)$. Therefore a straightforward check shows the following.

\begin{prop}
Equation~\eqref{eq:crystal_structure} defines an abstract $U_q(\g)$-crystal structure on $\RC^V / \sim$.
\end{prop}

Define a map $\Psi\colon \RC(\infty) \longrightarrow \TT(\infty)$ using the sequence of maps
\[
\begin{array}{ccccccccc}
\RC(\infty) &\longdoublearrow& \RC(\lambda_\nu) &\longhookarrow& B^{\otimes \lambda_\nu} &\longdoublearrow& \TT(\lambda_\nu) &\longhookarrow& \TT(\infty), \\
(\nu,J) &\mapsto& (\nu,J) &\mapsto& (\nu,J) &\mapsto& \Phi(\nu,J) &\mapsto& \Phi(\nu,J).
\end{array}
\]
Conversely, for $T\in \TT(\infty)$, let $\lambda_T \in P^+$ partition shape of $T$.  There is a natural crystal embedding of a tableau $T$ into a tensor product of its columns in $B^{\otimes\lambda_T}$.  Denote the image of $T$ in $B^{\otimes\lambda_T}$ by $T^{\otimes\lambda_T
}$.  Now define a map $\Xi\colon \TT(\infty) \longrightarrow \RC(\infty)$ by the sequence of maps
\[
\begin{array}{ccccccccc}
\TT(\infty) &\longdoublearrow& \TT(\lambda_T) &\longhookarrow& B^{\otimes \lambda_T} &\longdoublearrow& \RC(\lambda_T)& \longhookarrow &\RC(\infty),\\
T &\mapsto& T &\mapsto& T^{\otimes\lambda_T} &\mapsto& \Phi^{-1}(T^{\otimes\lambda_T}) &\mapsto& \Phi^{-1}(T^{\otimes\lambda_T}).
\end{array}
\]

\begin{thm}
We have $\Xi\circ\Psi = \id_{\RC(\infty)}$ and $\Psi\circ\Xi = \id_{\TT(\infty)}$.
\end{thm}

\begin{proof}
Given a marginally large tableaux $T$ of shape $\lambda_T$, begin by projecting down to $B(\lambda_T)$. This preserves the tableaux $T$ and consider the natural embedding $T^{\prime}$ in $\bigotimes_i B^{r_i,1}$ given by Equation~\eqref{eq:subcrystal}. Next take $\Phi(T^{\prime})$, recall from Theorem~\ref{thm:iso_columns} that $\Phi$ is a bijection, and lift the resulting rigged configuration to $\RC(\infty)$. Last, we note that this procedure is well-defined over the equivalence class of large tableaux by Lemma~\ref{lemma:embed_MLT} and Lemma~\ref{lemma:embed_RC}, so this gives us the desired bijection.
\end{proof}

\begin{cor}
\label{cor:bijection}
The bijection $\Psi$ is a crystal isomorphism.
\end{cor}

\begin{proof}
This follows from the fact that $\Phi$ is a (classical) crystal isomorphism. Indeed, the map $\Psi$ is well-defined as a crystal morphism by Lemma~\ref{lemma:preserve_Kashiwara}. Let $(\nu,J) \in \RC(\infty)$ and $a\in I$. Next, denote the $e_a$ operator in the crystal $X$ by $e_a^X$. Then
\[
e_a^{\TT(\infty)}\Psi(\nu,J) = e_a^{\TT(\infty)}\Phi(\nu,J) = e_a^{\TT(\lambda_\nu)}\Phi(\nu,J) = e_a^{B^{\otimes\lambda_\nu}}(\nu,J) = e_a^{\RC(\lambda_\nu)}(\nu,J).
\]
Since $(\nu,J)$ is nonzero in $\RC(\lambda_\nu)$ by definition of $\lambda_\nu$, we have $e_a^{\RC(\lambda_\nu)}(\nu,J) = e_a^{\RC(\infty)}(\nu,J)$.  Thus $\Psi$ commutes with $e_a$.  Showing that $\Psi$ commutes with $f_a$ is similar.
\end{proof}

\begin{remark}
Consider some $T \in B(\infty)$, we can explicitly describe the image of $T$ when projecting under $p_{\lambda}$ to $B(\lambda)$ for all $\lambda \in P^+$. From this, we define an equivalence class $[T] = \{ p_{\lambda}(T) \mid p_{\lambda} \neq 0,\ \lambda \in P^+ \}$. We can check that $[T]$ corresponds to the class of all valid rigged configurations under $\Phi$.
\end{remark}



\begin{ex}
\label{ex:full_bijection}
Let $\g$ be of type $A_4$ and
\[
T = \begin{tikzpicture}[baseline]
\matrix [tab] 
 {
 	\node[draw,fill=gray!30]{1}; &  
	\node[draw,fill=gray!30]{1}; &
	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw]{3}; & 
	\node[draw]{3}; \\
  	\node[draw,fill=gray!30]{2}; &  
	\node[draw,fill=gray!30]{2}; &  
	\node[draw,fill=gray!30]{2}; & 
	\node[draw,fill=gray!30]{2}; &
	\node[draw,fill=gray!30]{2}; & 
	\node[draw,fill=gray!30]{2}; & 
	\node[draw]{5}; \\
  	\node[draw,fill=gray!30]{3}; & 
  	\node[draw,fill=gray!30]{3}; &
	\node[draw]{4}; &
	\node[draw]{5}; &
	\node[draw]{5}; \\
  	\node[draw,fill=gray!30]{4}; \\
 };
\end{tikzpicture}.
\]
We first project onto $B(\lambda) \subseteq B^{\otimes \lambda}$ with $\lambda = \Lambda_4 + 4 \Lambda_3 + 2 \Lambda_2 + 3 \Lambda_1$ (which results in the same tableaux). Next we apply $\Phi \colon B^{\otimes \lambda} \longrightarrow \RC(B^{\otimes \lambda})$, and have the following:
\[
\begin{tikzpicture}[scale=.35]
 \rpp{1}{0}{0}
\begin{scope}[xshift=6cm]
 \rpp{1}{-1}{-1}
\end{scope}
\begin{scope}[xshift=12cm]
 \draw (0,-1.5) node {$\emptyset$};
\end{scope}
\begin{scope}[xshift=19cm]
 \draw (0,-1.5) node {$\emptyset$};
\end{scope}
\begin{scope}[xshift=25.5cm,yshift=-1.3cm]
\matrix [tab] 
 {
	\node[draw]{3}; \\
 };
\end{scope}
%%%
\begin{scope}[yshift=-3cm]
 \rpp{2}{0}{0}
\begin{scope}[xshift=6cm]
 \rpp{2}{-2}{-2}
\end{scope}
\begin{scope}[xshift=12cm]
 \draw (0,-1.5) node {$\emptyset$};
\end{scope}
\begin{scope}[xshift=19cm]
 \draw (0,-1.5) node {$\emptyset$};
\end{scope}
\begin{scope}[xshift=26cm,yshift=-1.3cm]
\matrix [tab] 
 {
	\node[draw]{3}; & 
	\node[draw]{3}; \\
 };
\end{scope}
\end{scope} % yshift
%%%
\begin{scope}[yshift=-6cm]
 \rpp{2}{0}{1}
\begin{scope}[xshift=6cm]
 \rpp{2}{-2}{-2}
\end{scope}
\begin{scope}[xshift=12cm]
 \draw (0,-1.5) node {$\emptyset$};
\end{scope}
\begin{scope}[xshift=19cm]
 \draw (0,-1.5) node {$\emptyset$};
\end{scope}
\begin{scope}[xshift=26.5cm,yshift=-1.3cm]
\matrix [tab] 
 {
	\node[draw,fill=gray!30]{1}; & 
	\node[draw]{3}; & 
	\node[draw]{3}; \\
 };
\end{scope}
\end{scope} % yshift
%%%
\begin{scope}[yshift=-9cm]
 \rpp{2}{0}{2}
\begin{scope}[xshift=6cm]
 \rpp{2,1}{-2,-1}{-2,-1}
\end{scope}
\begin{scope}[xshift=12cm]
 \rpp{1}{1}{1}
\end{scope}
\begin{scope}[xshift=19cm]
 \rpp{1}{-1}{-1}
\end{scope}
\begin{scope}[xshift=27cm,yshift=-1.5cm]
\matrix [tab] 
 {
 	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw]{3}; & 
	\node[draw]{3}; \\
	\node[draw]{5}; \\
 };
\end{scope}
\end{scope} % yshift
%%%
\begin{scope}[yshift=-12cm]
 \rpp{2}{0}{2}
\begin{scope}[xshift=6cm]
 \rpp{2,1}{-2,-1}{-1,0}
\end{scope}
\begin{scope}[xshift=12cm]
 \rpp{1}{1}{1}
\end{scope}
\begin{scope}[xshift=19cm]
 \rpp{1}{-1}{-1}
\end{scope}
\begin{scope}[xshift=27.5cm,yshift=-1.5cm]
\matrix [tab] 
 {
 	\node[draw,fill=gray!30]{1}; & 
 	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw]{3}; & 
	\node[draw]{3}; \\
	\node[draw,fill=gray!30]{2}; & 
	\node[draw]{5}; \\
 };
\end{scope}
\end{scope} % yshift
%%%
\begin{scope}[yshift=-16cm]
 \rpp{2}{0}{2}
\begin{scope}[xshift=6cm]
 \rpp{2,1}{-2,-1}{0,0}
\end{scope}
\begin{scope}[xshift=12cm]
 \rpp{2}{2}{2}
\end{scope}
\begin{scope}[xshift=19cm]
 \rpp{2}{-2}{-2}
\end{scope}
\begin{scope}[xshift=28cm,yshift=-1.8cm]
\matrix [tab] 
 {
	\node[draw,fill=gray!30]{1}; & 
 	\node[draw,fill=gray!30]{1}; & 
 	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw]{3}; & 
	\node[draw]{3}; \\
	\node[draw,fill=gray!30]{2}; & 
	\node[draw,fill=gray!30]{2}; & 
	\node[draw]{5}; \\
	\node[draw]{5}; \\
 };
\end{scope}
\end{scope} % yshift
%%%
\begin{scope}[yshift=-20cm]
 \rpp{2}{0}{2}
\begin{scope}[xshift=6cm]
 \rpp{2,1}{-2,-1}{0,0}
\end{scope}
\begin{scope}[xshift=12cm]
 \rpp{3}{2}{2}
\end{scope}
\begin{scope}[xshift=19cm]
 \rpp{3}{-3}{-3}
\end{scope}
\begin{scope}[xshift=28.5cm,yshift=-1.8cm]
\matrix [tab] 
 {
	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
 	\node[draw,fill=gray!30]{1}; & 
 	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw]{3}; & 
	\node[draw]{3}; \\
	\node[draw,fill=gray!30]{2}; & 
	\node[draw,fill=gray!30]{2}; & 
	\node[draw,fill=gray!30]{2}; & 
	\node[draw]{5}; \\
	\node[draw]{5}; & 
	\node[draw]{5}; \\
 };
\end{scope}
\end{scope} % yshift
%%%
\begin{scope}[yshift=-24cm]
 \rpp{2}{0}{2}
\begin{scope}[xshift=6cm]
 \rpp{2,1}{-2,-1}{0,0}
\end{scope}
\begin{scope}[xshift=12cm]
 \rpp{4}{1}{1}
\end{scope}
\begin{scope}[xshift=19cm]
 \rpp{3}{-3}{-3}
\end{scope}
\begin{scope}[xshift=29cm,yshift=-1.8cm]
\matrix [tab] 
 {
	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
 	\node[draw,fill=gray!30]{1}; & 
 	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw]{3}; & 
	\node[draw]{3}; \\
	\node[draw,fill=gray!30]{2}; & 
	\node[draw,fill=gray!30]{2}; & 
	\node[draw,fill=gray!30]{2}; & 
	\node[draw,fill=gray!30]{2}; & 
	\node[draw]{5}; \\
	\node[draw]{4}; & 
	\node[draw]{5}; & 
	\node[draw]{5}; \\
 };
\end{scope}
\end{scope} % yshift
%%%
\begin{scope}[yshift=-29cm]
 \rpp{2}{0}{2}
\begin{scope}[xshift=6cm]
 \rpp{2,1}{-2,-1}{0,0}
\end{scope}
\begin{scope}[xshift=12cm]
 \rpp{4}{1}{2}
\end{scope}
\begin{scope}[xshift=19cm]
 \rpp{3}{-3}{-2}
\end{scope}
\begin{scope}[xshift=30cm,yshift=-1.8cm]
\matrix [tab] 
 {
	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
 	\node[draw,fill=gray!30]{1}; & 
 	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw]{3}; & 
	\node[draw]{3}; \\
	\node[draw,fill=gray!30]{2}; & 
	\node[draw,fill=gray!30]{2}; & 
	\node[draw,fill=gray!30]{2}; & 
	\node[draw,fill=gray!30]{2}; & 
	\node[draw,fill=gray!30]{2}; & 
	\node[draw,fill=gray!30]{2}; & 
	\node[draw]{5}; \\
	\node[draw,fill=gray!30]{3}; & 
	\node[draw,fill=gray!30]{3}; & 
	\node[draw]{4}; & 
	\node[draw]{5}; & 
	\node[draw]{5}; \\
	\node[draw,fill=gray!30]{4}; \\
 };
\end{scope}
\end{scope} % yshift
\end{tikzpicture}
\]
By mapping back the rigged configuration into $\RC(\infty)$, we obtain
\[
\Xi(T) = 
\begin{tikzpicture}[scale=.35,baseline=-18]
 \rpp{2}{0}{-1}
\begin{scope}[xshift=6cm]
 \rpp{2,1}{-2,-1}{-2,-2}
\end{scope}
\begin{scope}[xshift=12cm]
 \rpp{4}{1}{-2}
\end{scope}
\begin{scope}[xshift=19cm]
 \rpp{3}{-3}{-3}
\end{scope}
\end{tikzpicture}.
\]
In Sage, we can reproduce the example using
\begin{lstlisting}
sage: B = crystals.infinity.Tableaux("A4")
sage: t = B(rows=[[1,1,1,1,1,1,1,1,3,3],\
....:             [2,2,2,2,2,2,5],[3,3,4,5,5],[4]])
sage: RC = crystals.infinity.RiggedConfigurations("A4")
sage: defn = {B.highest_weight_vector():RC.highest_weight_vector()};
sage: Xi = B.crystal_morphism(defn)
sage: t.pp()
  1  1  1  1  1  1  1  1  3  3
  2  2  2  2  2  2  5
  3  3  4  5  5
  4
sage: ascii_art(Xi(t))
-1[ ][ ]0  -2[ ][ ]-2  -2[ ][ ][ ][ ]1  -3[ ][ ][ ]-3
           -2[ ]-1 
\end{lstlisting}
\end{ex}



\begin{ex}
Let $\g$ be of type $A_3$ and
\[
(\nu, J) = 
\begin{tikzpicture}[scale=.35,baseline=-18]
 \rpp{2}{-1}{2}
\begin{scope}[xshift=6cm]
 \rpp{3,1}{-1,-1}{1,2}
\end{scope}
\begin{scope}[xshift=13cm]
 \rpp{3}{-1}{0}
\end{scope}
\end{tikzpicture}.
\]
Then 
\[
\Psi(\nu, J) = \begin{tikzpicture}[baseline]
\matrix [tab] 
 {
 	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw,fill=gray!30]{1}; & 
	\node[draw]{2}; & 
	\node[draw]{3}; \\
  	\node[draw,fill=gray!30]{2}; &  
	\node[draw,fill=gray!30]{2}; & 
	\node[draw,fill=gray!30]{2}; &
	\node[draw]{3}; &
	\node[draw]{4}; &
	\node[draw]{4}; \\
  	\node[draw,fill=gray!30]{3}; & 
	\node[draw]{4}; \\
 };
\end{tikzpicture}.
\]
\end{ex}







\section{Statistics}
\label{sec:statistics}

Consider a marginally large tableau $T \in \TT(\infty)$. A \emph{$k$-segment} of $T$ is a maximal sequence of $k$-boxes in $T$. Let $\seg'(T)$ be the total number of segments of $T$ that are not $i$-sequences in the $i$-th row. In other words, this is the total number of segments of $T$ minus the hieght of $T$.

\begin{dfn}[\cite{LS12,LS14}]
The \emph{segment statistic $\seg$} on marginally large tableaux is defined type-by-type as follows.
\begin{itemize}
\item[$A_n$:] Define $\seg(T) := \seg'(T)$.
\item[$B_n$:] Let $e_B(T)$ be the number of rows $i$ the contain both a $0$-segment and $\bi$ segment. Define $\seg(T) := \seg'(T) - e_B(T)$.
\item[$C_n$:] Define $\seg(T) := \seg'(T)$.
\item[$D_{n+1}$:] Let $e_D(T)$ be the number of rows $i$ that contain an $\bi$-segment but neither a $(n+1)$-segment nor $\overline{n+1}$-segment. Define $\seg(T) = \seg'(T) + e_D(T)$.
\item[$G_2$:] Let $e_G(T)$ be 1 if $T$ contains a $0$-segment and $\bon$-segment in the first row and 0 otherwise. Define $\seg(T) = \seg'(T) - e_G(T)$.
\end{itemize}
\end{dfn}

Define $\delta_{(r)}$ by $\delta$ on $\RC(\lambda)$ and then embedding into $\RC(\lambda - \Lambda_r)$ where $r = \max \{a \in I \mid \inner{h_a}{\lambda} \neq 0 \}$. Equivalently $\delta_{(r)} = \delta^r$ but returning only the $b$ from the first application of $\delta$.

\begin{dfn}
The \emph{repeat statistic $\rpt$} on rigged configurations is given recursively as follows. Consider $(\nu, J) \in \RC(\lambda_{\nu})$. Start with $r = n$ and $s = 0$. Let $x_r^c$ denote the smallest colabel of $\nu^{(r)}$ and project $(\nu, J)$ into $\RC(\lambda_{(r)})$, where $\lambda_{(r)} = \lambda - x_r^c \Lambda_r$. Let $c_r = \inner{h_r}{\lambda_{(r)}}$, and let $b^{(r)} = (b_1, \dotsc, b_{c_r})$ be the values returned by $\delta_{(r)}^{c_r}$ (in $\RC(\lambda')$). Increase $s$ by the number of distinct elements ocurring in $b$ (note any particular value in $b$ occurs sequentially). We also do the following modifications depending on the type:
\begin{itemize}
\item[$B_n$:] If $0, \overline{r} \in b$, subtract 1 from $s$.
\item[$D_{n+1}$:] If $\overline{r} \in b$ and $n+1, \overline{n+1} \notin b$, then add 1 to $s$.
\item[$G_2$:] If $0, \bon \in b$, subtract 1 from $s$
\end{itemize}
Now recurse with $r-1$ unless $r = 1$, and $\rpt$ is the final value of $s$.
\end{dfn}

From our definition of $\rpt$, for a fixed $r$ we have $\inner{h_a}{\lambda} = 0$ for all $a > r$. So the values $p_i^{(a)}$ are equal on $\RC(\infty)$ and $\RC(\lambda)$ for all $a > r$ and $i \in \ZZ$. So the map $\delta_{(r)}$ just starts each time at $\nu^{(r)}$ and is well defined since all strings in $\nu^{(a)}$ for $a < r$ are non-singular after applying $\delta$. Furthermore by Lemma~\ref{lemma:embed_RC}, the statistic $\rpt$ is well defined. Thus we have the following.

\begin{prop}
Let $(\nu, J) \in \RC(\infty)$. Then
\[
\rpt(\nu, J) = \seg(\Psi(\nu, J)).
\]
\end{prop}

%In the definition of $\rpt$, on applying $\delta_{(r)}$ we begin in $\nu^{(r)}$ by selecting the smallest length string with the largest colabel. Suppose the smallest selected string in $\nu^{(a)}$ (where $a \geq r$) of is length $\ell$, then all colabels of length $i < \ell$ remain fixed by the definition of $\delta$ and the vacancy numbers. Therefore on each successive application of $\delta_{(r)}$, we either act on the same string as before or a larger string in $\nu^{(r)}$.

%Let $\ell'$ denote the string selected before $\ell$, then the colabel decreases for all strings of length $\ell' \leq i < \ell$. We note that a colabel of 0 indicates a string is singular and can be selected by $\delta$ and all colabels are positive by the validity condition. Thus all subsequently selected strings by $\delta$ (i.e., except in $\nu^{(r)}$) have a tendency to be stay small.

From the definition of $\rpt$, it is clear that it is a direct translation of $\seg$ through the bijection $\Psi$. It would be good if there was a non-recursive translation of $\seg$ on rigged configurations.

\begin{ex}
Consider the rigged configuration $(\nu, J)$ obtained from Example~\ref{ex:full_bijection}:
\[
(\nu, J) =
\begin{tikzpicture}[scale=.35,baseline=-18]
 \rpp{2}{0}{2}
\begin{scope}[xshift=6cm]
 \rpp{2,1}{-2,-1}{0,0}
\end{scope}
\begin{scope}[xshift=12cm]
 \rpp{4}{1}{2}
\end{scope}
\begin{scope}[xshift=19cm]
 \rpp{3}{-3}{-2}
\end{scope}
\end{tikzpicture}.
\]
Thus we begin with $(\nu, J) \in \RC(\lambda_{\nu})$ with $\lambda_{\nu} = 4\Lambda_4 + 5\Lambda_3 + 4\Lambda_2 + 3\Lambda_1$, so we have
\[
\begin{tikzpicture}[scale=.35,baseline=-18]
 \rpp{2}{0}{2}
\begin{scope}[xshift=6cm]
 \rpp{2,1}{-2,-1}{1,1}
\end{scope}
\begin{scope}[xshift=12cm]
 \rpp{4}{1}{2}
\end{scope}
\begin{scope}[xshift=19cm]
 \rpp{3}{-3}{0}
\end{scope}
\end{tikzpicture}.
\]
We thus project onto $\RC(4\Lambda_3 + 3\Lambda_2 + 2\Lambda_1)$, and since $\inner{h_4}{\lambda_{(4)}} = 0$, there is nothing more to do. Next, after projecting onto $\RC(3\Lambda_3 + 3\Lambda_2 + 2\Lambda_1)$ and then applying $\delta_{(3)}^3$, we obtain
\[
\begin{tikzpicture}[scale=.35,baseline=-18]
 \rpp{2}{0}{2}
\begin{scope}[xshift=6cm]
 \rpp{2,1}{-2,-1}{0,1}
\end{scope}
\begin{scope}[xshift=13cm]
 \rpp{1}{1}{1}
\end{scope}
\begin{scope}[xshift=19cm]
 \rpp{1}{-1}{-1}
\end{scope}
\end{tikzpicture}
\]
with $b_{(3)} = (4,5,5)$. Projecting onto $\RC(\lambda_2 + 2\Lambda_1)$, we then obtain after applying $\delta_{(2)}$, we obtain
\[
\begin{tikzpicture}[scale=.35,baseline=-18]
 \rpp{2}{0}{0}
\begin{scope}[xshift=6cm]
 \rpp{2}{-2}{-2}
\end{scope}
\begin{scope}[xshift=13cm]
 \draw (0,-1.5) node {$\emptyset$};
\end{scope}
\begin{scope}[xshift=19cm]
 \draw (0,-1.5) node {$\emptyset$};
\end{scope}
\end{tikzpicture}
\]
with $b_{(2)} = (5)$. Finally, we apply $\delta_{(1)}^2$ and obtain $b_{(1)} = (3, 3)$. Therefore, we have $\rpt(\nu, J) = 2 + 1 + 1 = 4$. It is also easy to check that $\seg(T) = 4$.
\end{ex}

There is another statistic on rigged configurations which has a natural crystal interpretation. The \emph{difference statistic $\diff_a$} is defined by
\[
\diff_a(\nu, J) := \min_i \{ p_i^{(a)} - \max J_i^{(a)} \}
\]
for some $(\nu, J) \in \RC(\lambda)$. The difference statistic is measuring how far $(\nu, J)$ from being marginally valid in $\nu^{(a)}$, and so it can be interpreted as the largest $c_a$ such that $(\nu, J) \in \RC(\lambda - c_a \Lambda_a)$ (under the natural projection). We can also combine this into a single statistic $\diff(\nu, J) = \sum_{a \in I} \diff_a(\nu, J)$.

On highest weight crystals, the \emph{remove statistic $\rem_a$} is the largest $c_a$ such that $b \in B(\lambda)$ is non-zero under the natural projection to $B(\lambda - c_a \Lambda_a)$. Moreover, we can also combine this into a single statistic $\rem(\nu, J) = \sum_{a \in I} \rem_a(\nu, J)$. Given these interpretations, the following is an immediate consequence.

\begin{prop}
\label{prop:diff_rem_lambda}
Let $\Psi \colon \RC(\lambda) \longrightarrow B(\lambda)$ be an isomorphism. Then
\[
\diff_a(\nu, J) = \rem_a(\Psi(\nu, J)).
\]
\end{prop}

Furthermore, the difference statistic can be extended to $\RC(\infty)$, where it is measuring how far the rigged configuration is from being valid in $\nu^{(a)}$. Thus we see that the weight $\sum_{a \in I} \diff_a(\nu, J) \Lambda_a$ denotes the (unique) minimal weight that we need to project $(\nu, J) \in \RC(\infty)$ onto in order to guarantee the result is non-zero. We can also extend $\rem_a$ to $B(\infty)$ by considering the smallest weight $\lambda$ such that $b \in B(\infty)$ is non-zero under the projection onto $B(\lambda)$, and then $\rem_a(b) = \inner{\lambda}{h_a}$. We also have following the analog of Proposition~\ref{prop:diff_rem_lambda}.

\begin{prop}
Let $\Psi \colon \RC(\infty) \longrightarrow B(\infty)$ be the canonical isomorphism. Then
\[
\diff_a(\nu, J) = \rem_a(\Psi(\nu, J)).
\]
\end{prop}

We can also interpret $\rem_a$ on (marginally large) tableaux, with no columns of height $n$ in types $B_n$ or $D_{n+1}$, as being the number of basic columns of height $a$ (possibly not full height) that can be removed from a tableaux $T \in \TT(\lambda)$ and sliding the entries of those rows left such that the result is a classical tableaux. Columns of height $n$ in type $B_n$ are counted twice, and in type $D_n$ they contribute to both $\rem_n$ and $\rem_{n+1}$. Additionally, we can extend this interpretation to $T \in \TT(\infty)$.

%Given the context of tableaux, we can define a modified version of $\rem$ by $\widetilde{\rem} := \sum_{a \in I} a \rem_a$ which counts the number of boxes (instead of the number of columns) that can removed.

\bibliography{RC_MLT}{}
\bibliographystyle{amsalpha}
%\bibliographystyle{amsplain}
\end{document}
RC_to_MLT.bbl
\newcommand{\etalchar}[1]{$^{#1}$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\begin{thebibliography}{KKM{\etalchar{+}}92}

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\end{thebibliography}
