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\documentclass[finalversion]{FPSAC2025}
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\usepackage{amsfonts}
\usepackage{amsmath}
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\usepackage{pdfpages}

%\usepackage[margin= 0.9 in]{geometry}
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\usepackage[boxsize=1em]{ytableau}

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\newcommand\TAB[2][.3]{\TIKZ[scale=#1]{\Tableau{#2}}}


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\usepackage{tikz}
\usetikzlibrary{calc, shapes,arrows,positioning,cd}
\tikzset{>=stealth',
  head/.style = {fill = white, text=black},
  plaque/.style = {draw, rectangle, minimum size = 10mm},
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\tikzstyle{opLabel}=[fill=white, inner sep=2pt, rounded corners, pos=.3]
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\tikzstyle{TabNode}=[inner sep=2pt, fill=white, rounded corners]
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        \foreach \x [count = \c from 1] in {#1} {
		\foreach \y [count = \d from 1] in \x{
			\draw[ssyt\y, fill = ssyt\y, rounded corners=0] (\d,\c) rectangle (\d-1,\c-1);}}}
\newcommand\cTAB[3][.35]{\TIKZ[scale=#1, yscale=\PartDirection]{\SSYTcolors{#3}\Tableau{#2}}}	


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\usepackage{ytableau}
\usepackage{mathtools}
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\usepackage{comment}
\definecolor{darkred}{rgb}{0.7,0,0}
\definecolor{sapphire}{HTML}{1B4DC2}
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\newcommand{\defncolor}{\color{sapphire!70!black}}
\newcommand{\defn}[1]{{\defncolor\emph{#1}}} % emphasis of a definition
\newcommand{\CS}{{\mathsf{CS}}}
\newcommand{\wt}{{\mathsf{wt}}}
\newcommand{\Des}{{\mathsf{Des}}}
\newcommand{\DE}{{\mathsf{DE}}}
\newcommand{\row}{{\mathsf{row}}}
\newcommand{\skw}{{\mathsf{skew}}}
\newcommand{\std}{{\mathsf{std}}}
\newcommand{\cycle}{\mathsf{cycle}}
\newcommand{\rev}{\mathsf{rev}}
\newcommand{\LL}{\mathcal{L}}
\newcommand{\len}{\mathsf{len}}
\newcommand{\SYT}{\mathsf{SYT}}
\newcommand{\SSYT}{\mathsf{SSYT}}
\newcommand{\cM}{\mathcal{M}}
\def\Box{\mathsf{box}}
\def\ZZ{\mathbb{Z}}

% %%% TikZ -----
% \usepackage{tikz}
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%   pil/.style={->,thick},
%   junct/.style = {draw,circle,inner sep=0.5pt,outer sep=0pt, fill=black}
%   }
% \tikzstyle{opLabel}=[fill=white, inner sep=2pt, rounded corners, pos=.3]
% \tikzstyle{v}=[draw, fill =black, circle, inner sep=0pt, minimum size=2pt] %graph vertex





\usepackage[boxsize=1em]{ytableau}

%\usepackage[colorinlistoftodos]{todonotes}
%\newcommand{\Anne}[1]{\todo[size=\tiny,inline,color=red!30]{#1 \\ \hfill --- Anne}}
%\newcommand{\Sylvie}[1]{\todo[size=\tiny,inline,color=green!30]{#1 \\ \hfill --- Sylvie}}
%\newcommand{\Sarah}[1]{\todo[size=\tiny,inline,color=orange!30]{#1 \\ \hfill --- Sarah}}
%\newcommand{\Zajj}[1]{\todo[size=\tiny,inline,color=cyan!30]{#1 \\ \hfill --- Zajj}}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}

\theoremstyle{definition}
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\newtheorem{question}[theorem]{Question}
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\newtheorem{axiom}[theorem]{Axiom}

\numberwithin{equation}{section}


\usepackage{lipsum}

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\title{Crystal skeletons and their axioms}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author
\author[Brauner, Corteel, Daugherty, Schilling]{\texorpdfstring{
Sarah Brauner\thanks{\href{mailto:sarahbrauner@gmail.com}{sarahbrauner@gmail.com.} S.B. is supported by the NSF MSPRF DMS-2303060.}\addressmark{1}, 
Sylvie Corteel\thanks{\href{mailto:corteel@berkeley.edu}{corteel@berkeley.edu}. 
S.C. was partially supported by NSF grant DMS-2054482 and ANR grant  ANR-19-CE48-0011.}\addressmark{2},
Zajj Daugherty\thanks{\href{mailto:zdaugherty@reed.edu}{zdaugherty@reed.edu}}\addressmark{3}, \and
Anne Schilling\thanks{\href{mailto:aschilling@uvdavis.edu}{aschilling@ucdavis.edu}. 
A.S. was partially supported by  NSF grant DMS--2053350 and Simons Foundation grant MPS-TSM-00007191.}\addressmark{4}}{
Sarah Brauner, 
Sylvie Corteel,
Zajj Daugherty, and
Anne Schilling}}

%% then use \addressmark to match authors to institutions here
\address{
\addressmark{1}Division of Applied Mathematics, Brown University, Providence, RI, USA\\
\addressmark{2}Department of Mathematics, University of California, Berkeley, CA, USA and CNRS, IMJ-PRG, Sorbonne Universit\'e, France.\\
\addressmark{3}Department of Mathematics and Statistics, Reed College, Portland, OR, USA\\
\addressmark{4}Department of Mathematics, University of California, Davis, CA, USA}

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\received{\today}

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{
Crystal skeletons were introduced by Maas-Gari\'epy in 2023 by contracting quasi-crystal components in a crystal graph. On the representation 
theoretic level, crystal skeletons model the expansion of Schur functions into Gessel's quasisymmetric functions. Motivated by questions of Schur 
positivity, we give a new axiomatic approach to crystal skeletons in analogy to the local Stembridge axioms for crystals. In addition, we provide a 
combinatorial description of crystal skeletons, and prove many new properties, including a conjecture by Maas-Gari\'epy that crystal skeletons 
generalize dual equivalence graphs.
}

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

%% put your keywords here, or comment this out if you don't have them yet
\keywords{crystal graphs, Lusztig involution, dual equivalence graphs}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
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\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{fpsac.bib}
\appto\bibfont{\setlength{\emergencystretch}{1em}}
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\begin{document}


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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

Crystal graphs provide combinatorial tools to study the representation theory of Lie algebras. 
For instance, crystals are well-behaved with respect to taking tensor products and hence can be used to give combinatorial interpretations for 
Littlewood--Richardson coefficients. In type $A$, the character of an irreducible crystal $B(\lambda)$ of highest weight $\lambda$ is the Schur function $s_\lambda$. See~\cite{BumpSchilling.2017} for a detailed introduction.

It is an important problem in representation theory and algebraic combinatorics to deduce the Schur function expansion of a symmetric function whose 
expansion in terms of Gessel's fundamental quasisymmetric function~\cite{Gessel.1984}  $F_\alpha$ is known. For example, combinatorial expressions 
for the quasisymmetric expansion of LLT polynomials, modified Macdonald polynomials~\cite{HHL.2005}, characters of higher Lie modules (or Thrall's 
problem)~\cite{GR.1993} or the plethysm of two Schur functions~\cite{LW.2012} exist, yet their Schur expansions are in general still elusive.
It is thus desirable to develop methods to deduce the Schur expansions from these quasisymmetric expansions. Whereas Schur functions are characters 
of irreducible crystals in type $A$, Gessel's fundamental quasisymmetric functions are characters of \defn{quasi-crystals}~\cite{MG.2023, CMRR.2023}, 
which are certain subcomponents of a crystal (see \S\ref{section.crystals}).

In~\cite{MG.2023}, Maas-Gari\'epy introduced the \defn{crystal skeleton} $\mathsf{CS}(\lambda)$ by contracting each quasi-crystal in $B(\lambda)$ 
to a vertex. Since there is a unique standard tableau $T\in \mathsf{SYT}(\lambda)$ in each quasi-crystal, it is natural to label the vertices 
of the crystal skeleton by standard tableaux. The crystal skeleton construction is the crystal analogue of Gessel's formula~\cite{Gessel.1984}
\[
    s_\lambda = \sum_{T \in \mathsf{SYT}(\lambda)} F_{\mathsf{Des}(T)},    
\]
where $\mathsf{Des}(T)$ is the descent composition of the standard tableau $T$ (see \S\ref{section.vertices}). As such, crystal skeletons have 
the potential to serve as a powerful tool in deriving Schur expansions from quasisymmetric expansions.

Our goal is to initiate this project by characterizing the crystal skeleton both combinatorially (see \Cref{section.crystalskeleton}) 
and axiomatically (see \Cref{sec:axioms}), 
in analogy to the local Stembridge axioms for crystals~\cite{Stembridge.2003}. Stembridge axioms have played a crucial role in crystal theory and 
have facilitated proofs of Schur positivity using crystals. For example, in~\cite{MS.2016} the Schur expansion of Stanley symmetric functions was analyzed 
by defining a crystal structure on the combinatorial objects underlying Stanley symmetric functions (decreasing factorizations of a permutation); there, the crystal 
structure was proved using Stembridge's axioms. We anticipate that our axioms will have similar applications for Schur positivity in cases where the 
quasisymmetric expansion is known.

Beyond applications to positivity questions, our analysis shows that the combinatorics of the crystal skeleton is interesting in its own right. We prove 
self-similarity properties, symmetries, and branching rules. We show that the edges of the crystal skeleton are labeled by certain intervals, and that including only minimal length intervals recovers the \defn{dual equivalence graph} developed 
by Assaf~\cite{Assaf.2007,Assaf.2015} and Roberts~\cite{Roberts.2014, Roberts.2014a}. This implies that crystal skeletons generalize dual equivalence graphs, verifying a conjecture of Maas-Gari\'epy. Our crystal skeleton axioms should give new axioms for the dual equivalence graphs. In addition, even though
crystal skeletons are obtained by contracting quasicrystals, we prove that they contain crystal graphs as subgraphs.
Hence we expect our new axioms to give a new perspective on crystal axioms as well.
Proofs are available in~\cite{BCDS24}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Crystals and quasi-crystals}
\label{section.crystals}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $\mathsf{SSYT}(\lambda)_n$ be the set of semistandard Young tableaux of shape $\lambda$ over the alphabet $[n]:=\{1,2,\ldots,n\}$ and 
$\SYT(\lambda)$ be the set of standard Young tableaux of shape $\lambda$. 

We use French notation for partitions and tableaux, where the sizes of the rows weakly decrease from bottom to top. Work of Robinson--Schensted--Knuth 
defines a bijection $\mathsf{RSK}$ between words in the alphabet $[n]$ and pairs $(P,Q)$ of a semistandard tableau $P$ over $[n]$ and a standard tableau 
$Q$ of the same shape. If $\mathsf{RSK}(w) = (P,Q)$, then $P$ is known as the \defn{insertion tableau} and $Q$ as the \defn{recording tableau}.

A semistandard tableau $b \in \SSYT(\lambda)_n$ gives rise to several combinatorial objects: 
\begin{itemize}
    \item  The (row) \defn{reading word} $\mathsf{row}(b)$ is the word obtained from $b$ by reading rows left to right, top to bottom. 
    \item The \defn{weight} $\mathsf{wt}(b)$ is the tuple $(\alpha_1,\alpha_2,\ldots,\alpha_n)$ with $\alpha_j$ the number of letters $j$ in $b$.
    \item The \defn{standardization} $\mathsf{std }(b)$
is obtained from $b$ and $\wt(b)$ by replacing the letters $i$ in $b$ from left to right by 
\[ \alpha_1+\alpha_2+\cdots+\alpha_{i-1}+1,\ldots,
\alpha_1+\alpha_2+\cdots+\alpha_i \textrm{ for all }1\leqslant i \leqslant \ell.\]  
The standardization of a word $w$ can be defined similarly. By construction, $\std(b) \in \SYT(\lambda)$ and $\std(w)$ is a permutation.
\end{itemize}

\begin{example}  
\[
         \text{Suppose} \quad
	b=
	\raisebox{0.3cm}{\begin{ytableau}
	4\\
	2&4\\
	1&3&3
	\end{ytableau}}.  \qquad \text{Then} \qquad \std(b) =  \raisebox{0.3cm}{\begin{ytableau}
	5\\
	2&6\\
	1&3&4
	\end{ytableau}}\ ,
\]
$\row(b) = 424133$, and $\wt(b) = (1,1,2,2)$. Note that $\row(\std(b)) = 526134 = \std(\row(b))$.
\end{example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Crystals and quasi-crystals on tableaux}

We briefly review the crystal of type $A_{n-1}$ on tableaux. More details can be found in~\cite[Chapters 3, 8]{BumpSchilling.2017}. 
The \defn{crystal} $B(\lambda)_n$ is the set $\mathsf{SSYT}(\lambda)_n$ together with the maps
\begin{equation}
\begin{split}
	\mathsf{wt} &\colon B(\lambda)_n \to \mathbb{Z}^n_{\geqslant 0},\\
	e_i, f_i &\colon B(\lambda)_n \to  B(\lambda)_n \cup \{ \emptyset\} \qquad \text{for $i\in I = \{1,2,\ldots,n-1\}$.}
\end{split}
\end{equation}

The \defn{crystal raising} and \defn{crystal lowering} operators $e_i$ and $f_i$ are defined as follows.
The operators $e_i$ and $f_i$ act on the subword of $w=\mathsf{row}(b)$ containing only the letters $i$ and $i+1$, denoted by $w^{(i)}$.
Successively bracket (i.e. group) letters $i+1$ to the left of $i$. The subword of unbracketed letters is of the form $i^r (i+1)^s$. On this subword
\begin{equation*}
	e_i(i^r (i+1)^s) = \begin{cases} i^{r+1} (i+1)^{s-1} & \text{if $s>0$,}\\
	\emptyset & \text{else,}
	\end{cases}
	\quad
	f_i(i^r (i+1)^s) = \begin{cases} i^{r-1} (i+1)^{s+1} & \text{if $r>0$,}\\
	\emptyset & \text{else.}
	\end{cases}
\end{equation*}
These crystal operators on words preserve the recording tableau under RSK insertion. Hence they are well-defined on tableaux as well. 

\begin{example}
In $\SSYT(10,9,3,1)_4$, if
\[
b = \TIKZ[scale=.4]{
\filldraw[ssyt3] (6,0) rectangle (9,1) 
		(3,1) rectangle (5,2)
		(0,2) rectangle (2,3);
\filldraw[ssyt4] (9,0) rectangle (10,1) 
		(5,1) rectangle (9,2)
		(2,2) rectangle (3,3)
		(0,3) rectangle (1,4);
\Tableau{{1,1,1,1,2,2,3,3,3,4},{2,2,2,3,3,4,4,4,4}, {3,3,4},{4}}}
\qquad \text{then} \qquad
f_3b=
\TIKZ[scale=.4]{
\filldraw[ssyt3] (6,0) rectangle (9,1) 
		(3,1) rectangle (4,2)
		(0,2) rectangle (2,3);
\filldraw[ssyt4] (9,0) rectangle (10,1) 
		(4,1) rectangle (9,2)
		(2,2) rectangle (3,3)
		(0,3) rectangle (1,4);
\Tableau{{1,1,1,1,2,2,3,3,3,4},{2,2,2,3,4,4,4,4,4}, {3,3,4},{4}}}.
\]
\end{example}

One can define a \defn{quasi-crystal} inside of the crystal $B(\lambda)_n$ as follows. 

\begin{definition}
For a given crystal $B(\lambda)_n$, we define the \defn{quasi-crystal} $Q_T$ associated to a standard tableau $T \in \mathsf{SYT}(\lambda)$
by all elements $b\in B(\lambda)_n$ such that $\std(b) = T$.
\end{definition}

\begin{example}
Suppose $T \in \SYT(3,2,1)$ has $\row(T)= 645123$. Then $w = 322111$ is in the same quasi-component as $T$ in $B(3,2,1)_6$, since $\std(w) = \row(T)$.
\end{example}
It was shown in~\cite[Theorem\ 1]{MG.2023} that the components $Q_T$ are connected in $B(\lambda)_n$\footnote{Note that in~\cite{MG.2023} the quasi-crystal
components are defined in a slightly different way by fixing the descent composition. This is not quite accurate since the descent composition does not
uniquely specify a quasi-crystal component.}. Each quasi-crystal $Q_T$ contains exactly one standard tableau, namely $T$. It follows from~\cite[Section 2.5.2]{CMRR.2023} that for $b\in Q_T \subseteq B(\lambda)_n$ with $f_i(b) \in B(\lambda)_n$, we have $f_i(b) \in Q_T$ 
if and only if no $i+1$ and $i$ are bracketed in $\mathsf{row}(b)$.

\begin{example}\label{example.crystalandquasicrystals}
\Cref{figure.B21} shows the crystal $B(2,1)_3$ and its two quasi-crystal components.
\end{example}

\begin{figure}
    \centering
\begin{tabular}{c@{\qquad\qquad}c}
 Crystal: & 
 Crystal skeleton:\\[5pt]
\TIKZ[scale=1.85]{
\draw[rounded corners, line width=.7pt,  dotted] (-.3, 0) to (-.3, -.3) to (.5,-.3) to (3.4,.75) to (3.4,1.3) to (.7,1.3) to 
 node[above, sloped]{$\alpha = (\leftcolor{2},\rightcolor{1})$} ++(-1,-1) to (-.3,0);
 \begin{scope}[shift={(4,0)}, scale = -1]
 \draw[rounded corners, line width=.7pt,  dotted] (-.3, 0) to (-.3, -.3) to (.5,-.3) to (3.4,.75) to (3.4,1.3) to (.7,1.3) to 
 	node[below, sloped]{$\alpha = (\leftcolor{1},\rightcolor{2})$} ++(-1,-1) to (-.3,0);
 \end{scope}
 \node[TabNode] (211)  at (0,0) {\cTAB{{1,1},{2}}{{1,1},{2}}};
 \node[TabNode] (212) at (1,-1) {\cTAB{{1,2},{2}}{{1,2},{2}}};
\node[TabNode] (311) at (1,1) {\cTAB{{1,1},{3}}{{1,1},{2}}};
\node[TabNode] (213) at (2,-1) {\cTAB{{1,3},{2}}{{1,2},{2}}};
\node[TabNode] (312) at (2,1) {\cTAB{{1,2},{3}}{{1,1},{2}}};
\node[TabNode] (322) at (3,1) {\cTAB{{2,2},{3}}{{1,1},{2}}};
\node[TabNode] (313) at (3,-1) {\cTAB{{1,3},{3}}{{1,2},{2}}};
 \node[TabNode] (323) at (4,0) {\cTAB{{2,3},{3}}{{1,2},{2}}};
\begin{scope}[every node/.style={LabEdge}]
\draw[->, shorten <=-2pt, shorten >=-2pt] (211) to node {\footnotesize$1$} (212);
\draw[->] (211) to node {\footnotesize$2$} (311);
\draw[->] (311) to node {\footnotesize$1$} (312);
\draw[->] (212) to node {\footnotesize$2$} (213);
\draw[->] (213) to node {\footnotesize$2$} (313);
\draw[->] (312) to node {\footnotesize$1$} (322);
\draw[->, shorten <=-2pt, shorten >=-2pt] (322) to node {\footnotesize$2$} (323);
\draw[->] (313) to node {\footnotesize$1$} (323);
\end{scope}
\node[below, shift={(0,-.25)}] at (312) {$*$};
\node[above, shift={(0,.25)}] at (213) {$*$};
 }
&
\TIKZ[scale=2]{
\node[TabNode] (312) at (0,.75) {$\cTAB{{1,2},{3}}{{1,1},{2}}$};
\node[TabNode] (213) at (0,-.75) {$\cTAB{{1,3},{2}}{{1,2},{2}}$};
\draw[->] (312) to node[LabEdge] {\footnotesize$I=[1,3]$} (213);
}
\end{tabular}
    \caption{Left: Crystal $B(2,1)_3$ with two quasi-crystal components indicated with dotted lines and standard tableaux indicated by $*$.
    Right: Corresponding crystal skeleton.}
    \label{figure.B21}
\end{figure}

The crystal $B(\lambda)_n$ of type $A_{n-1}$ enjoys a symmetry under the Sch\"utzenberger or Lusztig involution 
(see for example \cite[p.\ 79]{BumpSchilling.2017}). Let $B(\lambda)_n$ be the crystal of type $A_{n-1}$ with highest weight element $u_\lambda$ of 
highest weight $\lambda$ and lowest weight element $v_\lambda$; that is, the unique elements $u_\lambda$ and $v_\lambda$ such that 
$e_iu_\lambda=\emptyset$ and $f_iv_\lambda=\emptyset$ for all $1\leqslant i <n$. 

\begin{definition}
\label{definition.lusztig}
The \defn{Lusztig involution} $\eta \colon B(\lambda)_n \to B(\lambda)_n$
is defined as follows. Any $b\in B(\lambda)_n$ can be obtained from $u_\lambda$ by applying a sequence of lowering operators, that is, 
$b=f_{i_1}\cdots f_{i_k} u_\lambda$ for some sequence $1\leqslant i_j<n$. We define
\[ \eta(b) := e_{n-i_1} \cdots e_{n-i_k} v_\lambda. \]
In particular, $\eta(u_\lambda)=v_\lambda$ and $\eta(v_\lambda)=u_\lambda$.
\end{definition}

\begin{remark}
\label{remark.evacuation}
For the crystal of tableaux in type $A$, the Lusztig involution is given by \defn{evacuation}. 
For $T \in \mathsf{SSYT}(\lambda)$ and $w=w_1\ldots w_\ell = \mathsf{row}(T)$,
$\mathsf{evac}(T)$ is the RSK insertion tableau of the word $w^\# = (n+1-w_\ell) \ldots (n+1-w_1)$.
\end{remark}

\noindent We will see later that crystal skeletons are also invariant under the Lusztig involution, and in fact this invariance is one of the axioms 
which characterize them (see Axiom \ref{axiom.lusztig}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Crystal skeletons}\label{section.crystalskeleton}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In~\cite{MG.2023}, Maas-Gari\'epy introduced the \defn{crystal skeleton} by contracting the quasi-crystals in $B(\lambda)_n$ to a vertex, assuming 
that $n$ is sufficiently large. Since there is a unique standard tableau $T\in \mathsf{SYT}(\lambda)$ in each quasi-crystal in $B(\lambda)_n$,
it is natural to label the vertices of the crystal skeleton by standard tableaux.

\begin{definition}
\label{definition.CS}
Let $\lambda$ be a partition and consider the ambient crystal $B(\lambda)_n$ for $n \geqslant |\lambda|$.
The \defn{crystal skeleton} $\mathsf{CS}(\lambda)$ is an edge-labeled, directed graph whose vertices are in $\mathsf{SYT}(\lambda)$. 
For distinct $T,T' \in \mathsf{SYT}(\lambda)$, there is an edge from $T$ to $T'$ in $\mathsf{CS}(\lambda)$ if there exist $b\in Q_T$ and $b' \in Q_{T'}$ 
such that $f_i(b) = b'$ for some $1\leqslant i < n$.
\end{definition}

\noindent
This definition is rather abstract. One goal of our work is to give a concrete, combinatorial description of $\CS(\lambda)$ as a graph.
In addition, we will give a natural way to label the edges with intervals different from the edge-labels in~\cite{MG.2023}.
See \Cref{figure.B21} for an example.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Vertices of the crystal skeleton}
\label{section.vertices}
As discussed in \Cref{definition.CS}, the vertices of $\CS(\lambda)$ are naturally labeled by the set $\SYT(\lambda)$. We will also 
label the vertices of $\CS(\lambda)$ by certain compositions of $n$ as follows.

For $T \in \mathsf{SYT}(\lambda)$, the letter $i$ is a \defn{descent} if the letter $i+1$ is in a higher row of $T$.
Denote the descents of $T$ by $d_1<d_2<\cdots<d_k$. The \defn{descent composition} is defined as
\[
	\mathsf{Des}(T) = ( d_1, d_2 - d_1, \ldots, d_k - d_{k-1}, n-d_k), \quad \text{where $n = |\lambda|$.}
\]

\begin{example}\label{example.descentcomposition}
We color $T \in \SYT(3,2,1)$ by its descent composition below: 
\[ T = 
	\TIKZ[scale=.4]{
		\SSYTcolors{{1,2,4},{2,3},{4}}
		\Tableau{{1,3,6},{2,4},{5}}
		}
. \]
In particular, $\Des(T) = (1,2,1,2)$. 
\end{example}

In what follows, it will be useful to label the vertices of $\CS(\lambda)$ by both $T \in \SYT(\lambda)$ and its corresponding descent composition 
$\Des(T)$. In particular, our axiomatic description of crystal skeletons in \Cref{sec:axioms} labels vertices by the compositions $\Des(T)$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Edges of the crystal skeleton}
The edges of the crystal skeleton are more subtle. In~\cite{MG.2023}, the edges of $\mathsf{CS}(\lambda)$ are indexed by the minimal index $j$ 
such that $f_j(b)=b'$ for $b$ and $b'$ in the respective quasicrystal components. We give two alternative characterizations of the edges in $\CS(\lambda)$.

First, let $\pi = \row(T)$ for $T \in \mathsf{SYT}(\lambda)$ with $n=|\lambda|$. Let $I=[i,i+2m] \subseteq [n]$ be an interval of length $2m+1\geqslant 3$ 
and $\pi|_I$ be the subword of $\pi$ restricted to the letters in $I$. 

\begin{definition}
The letters $I$ in $\pi$ form a \defn{Dyck pattern interval} if the RSK insertion tableau of $\pi|_I$ has shape $(m+1,m)$ and the RSK insertion tableau 
of $\pi|_{[i,i+m]}$ has shape $(m+1)$. 
\end{definition}

\noindent 
We show that the edges of $\CS(\lambda)$ are labeled by Dyck pattern intervals.

\begin{theorem}[{\cite[Theorem\ 3.9]{BCDS24}}]
\label{theorem.short pattern}
There is a bijection between the edges in $\mathsf{CS}(\lambda)$ and Dyck pattern intervals that occur in $\row(T)$ for  $T \in \mathsf{SYT}(\lambda)$. 
\end{theorem}

\begin{example}\label{example.shortpattern}
We describe in detail the edge below found in $\CS(3,2,1)$ in \Cref{figure.CS321}:
\[ T=   
\TIKZ[scale=.4]{
		\SSYTcolors{{1,1,1},{2,2},{3}}
		\Tableau{{1,2,3},{4,5},{6}}
		}
    \quad \xrightarrow[]{~I = [1,5]~} \quad T' =  
\TIKZ[scale=.4]{
		\SSYTcolors{{1,1,2},{2,2},{3}}
		\Tableau{{1,2,5},{3,4},{6}}
		}
    . \]
 In this case $\pi = \row(T) = 645123$, $\pi|_{[1,5]} = 45123$, which has RSK insertion tableau of shape $(3,2)$, and $\pi|_{[1,3]}=123$ is increasing. 
 Thus $I$ is a Dyck pattern interval on $\pi$. 

 To see why there is an edge from $T$ to $T'$ in $\CS(3,2,1)$, note that 
  the word $w = 622111$ is in the same quasi-component of $B(3,2,1)_6$ as $\pi$, since $\std(w) = \pi$. Similarly, $w' = 622112$ is in the same 
  quasi-component as $\pi' := \row(T') = 634125$ in $B(3,2,1)_6$, since $\std(w') = \pi'$. Next, observe that in $B(3,2,1)_6$, we have
  \[ f_{1}(w) = f_1(62211\textcolor{viridian}{1})  = 62211\textcolor{viridian}{2} = w'. \]
  The idea behind \Cref{theorem.short pattern} is that the Dyck pattern interval detects this edge in $B(\lambda)_n$. Note that $\Des(T) = (3,2,1)$ 
  and $\Des(T') = (2,3,1)$. The way descent compositions change between edges in $\CS(\lambda)$ is described more generally in Axiom \ref{axiom.labels} 
  in \Cref{sec:axioms}.
  \end{example}

Second, note that in \Cref{example.shortpattern}, one can obtain $T'$ from $T$ by the left action of the 3-cycle $(543)$ on $T$. We show 
in \cite{BCDS24} that this is true in general: given an edge between $T$ and $T'$ in $\CS(\lambda)$ labeled by the interval $I = [i,i+2m]$, one has 
\[ (j+m, j+m-1, \ldots, j+1, j) \cdot T = T',\]
where $j \in [i,i+m]$. Refer to \cite[Section 3.2.2]{BCDS24} for details on how to specify $j$.

\begin{example}
 \Cref{figure.CS321} shows the crystal skeleton $\mathsf{CS}(3,2,1)$. Each edge is labeled by the Dyck pattern interval in black, and 
 corresponding cycle in blue. The vertices of $\CS(3,2,1)$ are indexed by the set $\SYT(3,2,1)$; each $T \in \SYT(3,2,1)$ is colored by its 
 corresponding descent composition as in \Cref{example.descentcomposition}. The gray components correspond to the pieces 
 $G_{\lambda^{-}}$ obtained from the branching rules discussed in~\eqref{equation.branching} in \Cref{sec:axioms}.
\end{example}
\begin{figure}[t]
\begin{center}
\scalebox{1}{\includegraphics{fig-321-fpsac-lambdam.pdf}}
\end{center}
\caption{The crystal skeleton $\mathsf{CS}(3,2,1)$ with edges labeled by the intervals and cycles. Thick arrows highlight the top subcrystal as in Axiom \ref{axiom.crystal}, and the outlined components are the connected components in $G_{[1, 5]}$ as in \eqref{equation.branching}.
\label{figure.CS321}}
\end{figure}

\begin{remark}\label{remark.selfsimilarity}
 Using \Cref{theorem.short pattern}, we prove that $\CS(\lambda)$ exhibits \defn{self-similarity} in the following sense.  For $T\in \mathsf{SYT}(\lambda)$ and an interval $[a,b]$, we define $T_{[a,b]}$ to be the skew tableau $T$ restricted to the interval $[a,b]$. For a skew tableau $T$, let
$\mathsf{jdt}(T)$ be the jeu de taquin straightening of $T$, and write $\mu=\mathsf{shape}(\mathsf{jdt}(T_{[a,b]}))$. We show that $\mathsf{CS}(\mu)$ is a subgraph of $\mathsf{CS}(\lambda)$ with the following edge relabeling: if $I$ is an edge label in $\CS(\lambda)$,
then the corresponding edge label in $\CS(\mu)$ is $\{i-a+1 \mid i \in I\}$. See \cite[Section~4.3]{BCDS24}.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Generalizing dual equivalence graphs}

Dual equivalence graphs were first introduced by Haiman~\cite{Haiman.1992}. The vertices of the \defn{dual equivalence graph} $\mathsf{DE}(\lambda)$
are also indexed by $\SYT(\lambda)$. 
The edges in $\mathsf{DE}(\lambda)$ are undirected and given by the elementary dual equivalence relations $D_i$ ($1< i <
|\lambda|$) defined on permutations as follows: 
\begin{equation}
\label{equation.dual equivalence}
\begin{split}
	\ldots i \ldots i+1 \ldots i-1 \ldots &\stackrel{i}{\longleftrightarrow} \ldots i-1 \ldots i+1 \ldots i \ldots  \\
	\ldots i \ldots i-1 \ldots i+1 \ldots &\stackrel{i}{\longleftrightarrow} \ldots i+1 \ldots i-1 \ldots i \ldots  
\end{split}
\end{equation}
The operator $D_i$ is not defined for other configurations of the letters $i-1,i,i+1$ in the permutation.
Note that descents in the permutation do not change under $D_i$. Hence $D_i$ is defined on a standard Young tableau $T$ as well using the 
reading word $\mathsf{row}(T)$.

We prove that crystal skeletons generalize dual equivalence graphs.

\begin{theorem}[{\cite[Conjecture~5.3]{MG.2023}, \cite[Theorem~4.1]{BCDS24}}]
\label{thm:DEinsideCS}
The dual equivalence graph $\mathsf{DE}(\lambda) = (V,E)$ is a subgraph of the crystal skeleton $\mathsf{CS}(\lambda)$ (disregarding edge labels and 
directions), where
\[ V  = \{ T \in \SYT(\lambda) \}, \quad \quad E = \left\{ \textrm{Dyck pattern intervals $I$ in } \CS(\lambda)~\big|~ |I| = 3 \right\}.  \] 
\end{theorem}
\noindent In \Cref{figure.CS321}, $\DE(3,2,1)$ is obtained by including the edges labeled by intervals of the form $I = [i, i+2]$, or equivalently, 
the edges where the corresponding cycle is a transposition.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Axiomatic characterization of the crystal skeleton}
\label{sec:axioms}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We now give an axiomatic characterization of the crystal skeleton. We state the axioms in \Cref{section.axiomstatement}. 
In \Cref{section.axiomscharacterize}, we show that these axioms characterize crystal skeletons.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The axioms}\label{section.axiomstatement}

Fix $n \in \ZZ_{\geqslant 1}$. 
Let $G$ be a finite, directed, vertex- and edge-labeled graph, with underlying vertex set $V$ and edge set $E$, satisfying the following:
\begin{itemize}
    \item The vertices are labeled by compositions $\alpha=(\alpha_1,\ldots,\alpha_\ell)$ of $n$, 
so that the labeled vertex set is $V_L=\{(v,\alpha) \mid v\in V\}$ with $\alpha \models n$. Sometimes we identify $\alpha$ with the partition 
$(\alpha^{(1)},\ldots,\alpha^{(\ell)})$ of $[n]$, where $ \alpha^{(i)} = \{\alpha_1+\cdots + \alpha_{i-1}+1,\ldots,
\alpha_1+\cdots+\alpha_i\}. $
\item The edges are labeled by (odd-length) intervals  $I \subseteq [n]$, so that the labeled edge set is 
$E_L=\{(vw,I) \mid vw \in E\}$.
\end{itemize}

With $G$ as above, we define the Lusztig involution on $G$ as follows. 

\begin{definition}[Lusztig involution] \label{def.LIonCS}
The \defn{Lusztig involution} $\mathcal{L}_n$ on $G$ is defined by 
\begin{itemize}
    \item relabeling the vertices by replacing $(v,\alpha)$ by 
$(v,\rev(\alpha))$, and 
\item reversing all edge directions and changing the edge label 
\begin{align*} I=[a,b] \quad \quad \textrm{ to } \quad \quad I^{\LL}:=[n+1-b,n+1-a]. \end{align*}
\end{itemize}
\end{definition}

\begin{example} 
Suppose $n=6$. Then $\LL_6$ acts by
\[ \LL_6: \bigg( (v,(3,2,1)) \xrightarrow{[4,6]} (w,(3,1,2)) \bigg) \longmapsto \bigg((w,(2,1,3)) \xrightarrow{[1,3]} (v,(1,2,3)) \bigg). \]
\end{example}

We will also restrict our graphs via \defn{branched graphs} as follows.

\begin{definition}[Branched graph]
Define $G_{[1,n-1]} = (V_L',E_L')$, where 
\[ V_L'=\{ (v, \alpha \setminus \{ n \}) \mid (v, \alpha) \in V_L \} \quad \textrm{ and } \quad E_L' = \{(vw,I) \mid (vw,I)\in E_L, \  I \subseteq [n-1] \}.\]
\end{definition}

\begin{example}
In \Cref{figure.CS321}, the graph $G_{[1,5]}$ is isomorphic to the portion of $\CS(3,2,1)$ shaded in gray. To obtain $G_{[1,5]}$ from the 
gray subgraph, keep the edge labels the same and replace the vertex $\Des(T)$ for $T \in \SYT(3,2,1)$ with $\Des(T|_{[1,5]})$ as 
in Remark \ref{remark.selfsimilarity}.
\end{example}

We are now ready to state the axioms for crystal skeletons. Graph isomorphisms are considered to preserve vertex and edge labels.

\begin{axiom}[Axioms for crystal skeletons]
\label{axioms}
Let $n$ be a positive integer and $G$ be a finite, connected, vertex- and edge-labeled graph with labeled vertex set 
$V_L$ and labeled edge set $E_L$ as above. 
We call $G$ a \defn{$\mathsf{CS}$-graph} if the following axioms hold:

\begin{enumerate}[label=\bf {A}\arabic{enumi}.,ref=\bf {A}\arabic{enumi}]\setcounter{enumi}{-1}
\item
\label{axiom.intervals} (\emph{Intervals})
Suppose $(v,\alpha) \xrightarrow{~I~} (w, \beta)$ is an edge in $G$. Then the interval $I \subseteq [n]$ satisfies
\begin{enumerate}[label=(\alph*)]
\item \label{axiom.interval types-decomp} $I=I^- \cup \{k\} \cup I^+$ where  $|I^-|=|I^+|>0$; and  
\item \label{axiom.interval types-out} $I^- \cup \{k\} \subseteq \alpha^{(j)}$ and $I^+ \subseteq \alpha^{(j+1)}$ for some $1\leqslant j < \ell$, 
where $\ell$ is the length of $\alpha$.
\end{enumerate}
\[
\TIKZ[scale=.75, thick]{
\draw[thin, black!20] (-1.5, 0) to (8.5,0);
\draw[Bracket-Bracket] (0,0) to (3,0);
\draw[Bracket-Bracket] (4,0) to (7,0);
\begin{scope}[every node/.style={draw, thin, inner sep=1.75pt, fill=white, rounded corners}]
\node at (1.5,0) {\small $I^-\vphantom{I_-}$};
\node at (3.5,0) {\small $k\vphantom{I_-^-}$};
\node at (5.5,0) {\small $I^+\vphantom{I_-}$};
\end{scope}
\draw[rounded corners=2, sapphire] (-1+.2, .25) to (-1+.2, .5) to 
	node[pos=.15, inner sep=.5pt, fill=white]{$\ \!\cdots$} 
	node[above] {$\alpha^{(j)}$} (4-.15-.05, .5) to (4-.15-.05, .25);
\draw[rounded corners=2, sapphire] (8+.2, .25) to (8+.2, .5) to 
	node[pos=.15, inner sep=.5pt, fill=white]{$\ \!\cdots$} 
	node[above] {$\alpha^{(j+1)}$} (4-.15+.05, .5) to (4-.15+.05, .25);
}
\]

\item 
\label{axiom.outgoing}(\emph{Outgoing edges}) 
For each $(v, \alpha) \in V_L$ and each interval $I$ satisfying~\ref{axiom.intervals}, exactly one of the following holds: Either
\begin{enumerate}[label=(\roman*)]
\item there is exactly one outgoing edge $(v, \alpha) \xrightarrow{~I~} \fbox{\small $\phantom{(v, \alpha)}$}$ \ labeled by $I$; or 
\item \label{axiom.outgoing.incoming} 
there is an incoming edge $(u,\gamma) \xrightarrow{~J~} (v, \alpha)$ with $J \subseteq I$ and $\gamma$ dominating $\alpha$.
\end{enumerate}

\item\label{axiom.labels} (\emph{Labels})
Let $(v,\alpha) \xrightarrow{~I~} (w,\beta)$ be an edge as in \ref{axiom.intervals}. Given $\alpha$ and $I$, then $\beta$ must be of the form 
\[\beta = (\alpha^{(1)}, \dots, \alpha^{(j-2)}, 
	\TIKZ[yscale=.45]{\draw[thin] (0,.5) to (0,0) to (2,0) to +(0,.5); \node[circle, draw, inner sep=1pt] at (1,.5) {$*$};}\ ,
	\alpha^{(j+2)}, \dots, \alpha^{(\ell)}),\]
	\def\circasterisk{\TIKZ{\node[circle, draw, inner sep=1pt] at (1,.5) {$*$};}}
where \circasterisk\  is one of
\begin{enumerate}[label=\bf \Roman{enumii}.,ref=\bf \Roman{enumii}]
\item \label{axiom.length pres}(\emph{length-preserving}) \ 
	$\circasterisk = (\alpha^{(j-1)}, 			\hspace{5pt}
				\alpha^{(j)} \setminus \{k\}, 	\hspace{5pt}
				\alpha^{(j+1)} \cup \{k\})$;
\item  \label{axiom.length inc} (\emph{length-increasing}) \ 
	$\circasterisk = (\alpha^{(j-1)}, 			\hspace{5pt}
				\alpha^{(j)} \setminus \{k\}, 	\hspace{5pt}
				I^+ \cup \{k\}, 				\hspace{5pt}
				\alpha^{(j+1)} \setminus I^+)$; 
	or
\item \label{axiom.length dec} (\emph{length-decreasing})  \ 
	$\circasterisk = (\alpha^{(j-1)} \cup I^-,  		\hspace{5pt}
				\alpha^{(j+1)} \cup \{k\})$, 
		given $\alpha^{(j)}=I^-\cup \{k\}$.
\end{enumerate}

\item\label{axiom.fans} (\emph{Fans})
Suppose $(v,\alpha) \xrightarrow{~I~} (w,\beta)$ is an edge in $G$ with $\beta$ satisfying axiom~\ref{axiom.labels}~\ref{axiom.length inc}.
Then this edge is part of a \defn{fan}
\begin{center}
\TIKZ[>=latex,line join=bevel, yscale=.8]{
%%
\node (N0) at (0,0) [draw,draw=none] {$(v,\alpha)$};
\node (N1) at (-7,3) [draw,draw=none] {$(w_1,\beta_1)$};
\node (N2) at (-4,3) [draw,draw=none] {$(w_2,\beta_2)$};
\node (N4) at (0,3) [draw,draw=none] {$(w_{m-1},\beta_{m-1})$};
\node (N5) at (4,3) [draw,draw=none] {$(w_m,\beta_m)$};
  \draw[->, bend left=10] (N0)  to node[opLabel, pos=.5](I1){$I_1$} (N1);
  \draw[->, bend left=10] (N0)  to node[opLabel, pos=.5](I2){$I_2$} (N2);
  \draw[->] (N0)  to node[opLabel, pos=.5](Ia-1){$I_{m-1}$} (N4);
  \draw[->] (N0)  to node[opLabel, pos=.5]{$I_m$} (N5);
  \draw[->] (N1)  to node[opLabel, pos=.5]{$J_1$} (N2);
  \draw[->] (N2)  to node[opLabel, pos=.5]{$\cdots$} (N4);
  \draw[->] (N4)  to node[opLabel, pos=.5]{$J_{m-1}$} (N5);
  \draw[->] (N5) to[bend left=30] node[opLabel, pos=.5]{$J_m$} (N0);
\path (I2) to node{$\cdots$} (Ia-1);
}
\end{center}
where $(w,\beta)$ is one of the $(w_j,\beta_j)$ and 
\begin{align*}
    I_1&=[i,i+2m], \quad I_2=[i+1,i+2m-1],  \ldots,  I_m=[i+m-1,i+m+1] \\
    J_1&=[i+2m-1,i+2m+1], \quad J_2=[i+2m-2,i+2m],\ldots,  J_m=[i+m,i+m+2]
\end{align*}
for some $i$ and $m$. The edges labeled $I_1,\ldots, I_m$ satisfy
axiom~\ref{axiom.labels}~\ref{axiom.length inc}, the edges labeled $J_1,\ldots,J_{m-1}$ satisfy axiom~\ref{axiom.labels}~\ref{axiom.length pres}, and
the edge labeled $J_m$ satisfies axiom~\ref{axiom.labels}~\ref{axiom.length dec}.

\item\label{axiom.lusztig} (\emph{Lusztig involution}) 
$G$ and $G_{[1,n-1]}$ are invariant under Lusztig involution:
\begin{enumerate}[label=(\alph*)]
\item \label{axiom.lusztig n}
$\mathcal{L}_n(G) \cong G$; and
\item \label{axiom.lusztig n-1}
$\mathcal{L}_{n-1}(G_{[1,n-1]}) \cong G_{[1,n-1]}$.
\end{enumerate}

\item\label{axiom.crystal} (\emph{Top subcrystal}) Let $s = \min \{ \len(\alpha) \mid (v,\alpha) \in V_L\}$ be the minimal length of all vertex labels.
Let $G^s$ be the induced subgraph of $G$ with vertex set 
\[ \{(v,\alpha) \in V_L \mid \len(\alpha) =s\}. \] Then
$G^s$ is isomorphic to the crystal graph $B(\lambda)_s$ for some partition $\lambda$ with $\ell(\lambda)=s$.
\end{enumerate}
\end{axiom}

\begin{remark}
Note that the crystal $B(\lambda)_s$ appearing in axiom~\ref{axiom.crystal} has a unique vertex $(u,\lambda)$ with label $\lambda$.
Hence $G$ itself has a unique such vertex and so we will write $G_\lambda$ for the $\CS$-graph $G$ containing $B(\lambda)_s$.
\end{remark}

\begin{remark}
We refer to Axioms \ref{axioms} as $GL_n$-axioms.
There are several alternative but equivalent axioms defined in \cite{BCDS24}. 
First, one can replace Axioms \ref{axiom.lusztig} and \ref{axiom.crystal} by assuming a branching condition holds (see \eqref{equation.branching}), 
as well as a certain connectivity condition. We refer to these as $S_n$-axioms. Second, we give local axioms most closely related to
Stembridge axioms for crystals~\cite{Stembridge.2003} using commutation relations. See \cite[Section~5]{BCDS24}. 
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The characterization}\label{section.axiomscharacterize}

Our main results state that the axioms in \Cref{section.axiomstatement} are necessary and sufficient conditions for a graph to be a crystal 
skeleton. We first prove that the crystal skeleton from \Cref{definition.CS} is indeed a $\CS$-graph.

\begin{theorem}[{\cite[Theorem~5.7]{BCDS24}}]
\label{theorem.CS is CS-graph}
    For any partition $\lambda$, the crystal skeleton $\CS(\lambda)$ is a $\CS$-graph with vertex labeling given by descent 
    compositions and edge labeling given by Dyck pattern intervals. 
\end{theorem}
The proof of \Cref{theorem.CS is CS-graph} uses the combinatorial properties of $\CS(\lambda)$ to
show that it satisfies each of the conditions in Axiom \ref{axioms}. 

\begin{example}\label{example.topsubcrystal}
In \Cref{figure.CS321}, the edges drawn in bold show a graph isomorphic to the crystal $B(3,2,1)_{3}$. More generally, the graph 
$\CS(\lambda)$ will contain the crystal $B(\lambda)_{\ell(\lambda)}$.
\end{example}

Next, we show that \emph{any} $\CS$-graph is a crystal skeleton $\CS(\lambda)$ for some $\lambda$.

\begin{theorem}[{\cite[Section 5.5]{BCDS24}}]\label{theorem.axiomscharacterize}
Axioms \ref{axioms} uniquely characterize $G_\lambda$. In particular, $G_\lambda \cong \CS(\lambda)$.
\end{theorem}

$\CS$-graphs exhibit branching rules mirroring the symmetric group: 
    \begin{equation}
    \label{equation.branching}
    	G_{[1,n-1]} \cong \bigcup_{\lambda^- \subseteq \lambda, |\lambda/\lambda^-|=1} G_{\lambda^-},
    \end{equation}    
    where each of the $G_{\lambda^-}$ are $\CS$-graphs. An example of this decomposition is shown in \Cref{figure.CS321}; each 
    connected component $G_{\lambda^-}$ is shaded in gray.
\begin{proof}[Proof idea of \Cref{theorem.axiomscharacterize}]
Our proof is by induction, using \eqref{equation.branching}. In particular, we show that given the collection of $\{ G_{\lambda^-} \}$ 
from \eqref{equation.branching}, the graph $G_\lambda$ can be uniquely recovered. This argument is quite subtle, and heavily utilizes 
Axioms \ref{axiom.fans}, \ref{axiom.lusztig} and \ref{axiom.crystal}.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{
This work began at the conference ``Community in Algebraic and Enumerative Combinatorics'' in 
January 2024. We thank the organizers of this workshop and the Banff International Research Station for hosting it.
}

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\end{document}
