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\title[Orbit Lengths for Promotion]{Identifying Orbit Lengths for Promotion}

\author[Catania, Kendrick, Russell, and Tymoczko]{\texorpdfstring{Elise Catania\thanks{\href{mailto:catan042@umn.edu}{catan042@umn.edu}}\addressmark{1}, Jack Kendrick\thanks{\href{mailto:jackgk@uw.edu}{jackgk@uw.edu}}\addressmark{2}, Heather M Russell\thanks{\href{mailto:hrussell@richmond.edu}{hrussell@richmond.edu}}\addressmark{3}, \and \\ Julianna S Tymoczko\thanks{\href{mailto:jtymoczko@smith.edu}{jtymoczko@smith.edu}}\addressmark{4}}{Elise Catania, Jack Kendrick, Heather M Russell, and Julianna S Tymoczko}}


\address{\addressmark{1}School of Mathematics, University of Minnesota Twin Cities, USA \\ \addressmark{2} Department of Mathematics, University of Washington, USA \\ \addressmark{3} Department of Mathematics and Statistics, University of Richmond, USA \\ \addressmark{4} Department of Mathematics and Statistics, Smith College, USA}


\received{\today}


\abstract{In this work we study Sch\"utzenberger's promotion operator on standard Young tableaux via a corresponding graphical construction known as $m-$diagrams. In particular, we prove that certain internal structures of SYT are preserved under promotion and correspond to distinct components of $m-$diagrams. By treating these structures as atomic parts of the $m-$diagram, we provide a simple algorithm for computing the promotion orbit length of rectangular SYT.}


\keywords{Young tableaux, jeu de taquin, promotion, web}


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\begin{document}
\include{example}

\maketitle


\section{Introduction}

Young tableaux are fundamental combinatorial objects. With various choices of fillings and rules for manipulations, Young tableaux encode representations and algebraic operations on representations \cite{Alexandersson, Haiman, pappe, Pfannerer}, Schubert varieties and the cohomology ring structure induced by Schubert classes \cite{thomas1, thomas2}, and other constructions in algebra and geometry \cite{pechenik2}.  
 
We study Sch\"utzenberger's \emph{promotion} operator on \emph{standard Young tableaux}, which is important in part because of its role in the cyclic-sieving phenomenon \cite{reiner, rhoades}.  A seminal --- and very surprising --- result of Haiman showed that the promotion orbit of an $m \times n$ rectangular standard Young tableaux has order at most $mn$ \cite{Haiman}.  This has been explored and extended in different contexts, using related operators \cite{pechenik, striker3, striker} or equivariant maps to other combinatorial objects admitting a group action \cite{armstrong, petersen2009promotion, russell, Julianna}.  
 
Yet, very little has been done to analyze or characterize the sizes of specific promotion orbits, despite analogous classification and stability questions in representation theory.  A result by Purbhoo and Rhee does identify all minimal orbits for rectangular tableaux \cite{purbhoo}.  In this paper, we do the following:
 \begin{enumerate}
\item define a notion of uniformly equivalent tableaux and investigate promotion via a graphical construction known as $m-$diagrams, which are essentially a union of noncrossing matchings;
 
\item prove that, while promotion of rectangular tableaux is not equivariant with rotation of $m-$diagrams, the \textit{rotational symmetry} of the $m-$diagram coincides with the length of promotion orbits. In particular, this allows us to analyze orbit lengths when the $m-$diagrams are disconnected (see \Cref{main_thm}).
\end{enumerate}

\section{Preliminaries}
\subsection{Tableaux}
We begin by fixing notations and conventions for tableaux.

\begin{definition}[Standard Young Tableau]
Suppose $\lambda = (\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k)$ is a partition of $n$. A \emph{Young diagram} of shape $\lambda$ is a grid of left- and top-aligned boxes with $\lambda_i$ boxes in the $i^{th}$ row for each $1 \leq i \leq k$. If all parts of the partition $\lambda$ are the same then we call $\lambda$ a rectangular partition and its Young diagram a \emph{rectangular Young diagram}.

A \emph{standard Young tableau (SYT)} is a filling of the Young diagram for $\lambda$ with the values $1, \ldots, n$ without repetition such that rows strictly increase left-to-right and columns strictly increase top-to-bottom. %
\end{definition}


Given an SYT $T,$ we use $|T|$ to denote the number of boxes in $T$ and $T_{ij}$ to denote the entry in the $i^{th}$ row and $j^{th}$ column of $T.$ Next, we review Sch\"utzenberger's jeu de taquin promotion on SYT. Promotion is a function from SYT of fixed shape $\lambda$ to SYT of the same shape \cite{Stanley}.

\begin{definition}[Promotion]\label{definition: promotion}
Given an SYT $T$, the \emph{promotion} of $T$ is the SYT $P(T)$ created as follows:
\begin{enumerate}
\item Erase 1 in the top left corner of $T$ and leave an empty box.
\item Given the configuration $\ytableausetup {boxsize=0.5 cm} \begin{ytableau} {} & b \\ a \\ \end{ytableau}$ and $b < a$ then slide $b$ left; else if $a < b$ slide $a$ up.
\item Repeat the above process until there are no nonempty boxes below or to the right of the empty box.
\item Decrement all entries by $1$ and insert the largest entry of $T$ into the empty box.
\end{enumerate}
\end{definition}

\begin{example} \Cref{fig:T and P(T)} gives an examples of an SYT $T$ and its promotion $P(T).$ This tableau will serve as our running example.
    \begin{figure}[ht]
        \centering
        \begin{tikzpicture}
            \node at (0,0) {\T};
            \node at (-2.5,0) {$T =$};
            \node at (5,0) {\PT};
            \node at (8,0) {$ = P(T)$};

            \draw[style=thick, ->] (2,0) -- (3,0);
        \end{tikzpicture}

        \caption{The tableau $T$ and its promotion $P(T)$}
        \label{fig:T and P(T)}
    \end{figure}
\end{example}

Note that while we define the promotion function on SYT, the same operation can be performed on any filling of a Young diagram such that the entries are non-repeating with strictly increasing rows and columns.

We are particularly interested in the orbits of tableaux under promotion. For a given SYT $T$, we denote the promotion orbit of $T$ by $\cO(T).$ The key observation in our approach is that certain internal structures of SYT are preserved during promotion. To this end, we introduce the following definition.

\begin{definition}[Equivalence]
Suppose $S$ and $T$ are skew tableaux with $|S|=|T|=n$ and contents $\{s_1, s_2, \ldots, s_n\}$ and $\{t_1, t_2, \ldots, t_n\}$ respectively. We say that $S$ is \textit{equivalent} to $T,$ or $S\equiv T,$ if $S$ and $T$ are the same after each tableau is left justified with the entries relabeled smallest to largest so that the content becomes $\{1, 2, \ldots, n\}.$ Moreover, we say $S$ is \textit{uniformly equivalent} to $T$ if the content of $S$ is $\{k+1, k+2, \ldots, k+n\}$ and the content of $T$ is $\{\ell +1, \ell +2, \ldots , \ell+n\}$ for some $k, \ell$ and $S\equiv T$. 
\end{definition}

Uniform equivalence is a powerful condition that allows us to treat uniformly proper subtableaux as atomic parts of the corresponding $m$-diagrams and webs (see the next definition for more). However, there are contexts in which more generality is useful.

\begin{definition}[Uniformly Proper Subtableau]
Suppose $T$ is a tableau with $n$ rows and $S\subsetneq T$ is a subtableau. We call $S$ {\em uniformly proper} if:
\begin{enumerate}
    \item $S$ has $n$ rows;
    \item $S$ is uniformly equivalent to a standard Young tableau.
\end{enumerate}
We call $S$ a uniformly proper {\em rectangular} subtableau if in addition all of its rows have the same length. If $T$ contains no uniformly proper subtableau, then we say that $T$ is \textit{minimal.}
\end{definition}

When an SYT $T$ is not minimal, and thus has some uniformly proper subtableau $S,$ we can write the tableau $T$ as the horizontal concatentation $T = T_1 ST_2,$ where $T_1$ and $T_2$ are also subtableaux of $T,$ see \Cref{fig:T=T_1ST_2}. More generally, given two skew tableaux $T$ and $S,$ we write $TS$ to refer to the tableau resulting from the horizontal concatenation of $T$ and $S$ when such an operation makes sense.

\begin{figure}[ht]
    \centering
    \ytableausetup{boxsize=0.6cm}
\begin{tikzpicture}
\node at (-4, 0) {\T};
\node at (-1.5, 0) {=};
\node at (0,0){\begin{ytableau}
1 & 2 \\
3  \\
4  \\
5  \\
\end{ytableau}};
\node at (2, 0) {\begin{ytableau}
    \none & 6 & 7 & 14 \\
     8 & 9 & 15 \\
    10 & 11 & 16 \\
    12 & 13 & 17
\end{ytableau}};
\node at (4, 0) {\begin{ytableau}
    \none & 19 \\
    18 & 21 \\
    20 & 23 \\
    22 & 24
\end{ytableau}};

\node at (-4, -1.75) {$T$};
\node at (0, -1.75) {$T_1$};
\node at (2, -1.75) {$S$};
\node at (4, -1.75) {$T_2$};
\end{tikzpicture}
    \caption{Decomposition of tableau $T$ with uniformly proper subtableau $S$}
    \label{fig:T=T_1ST_2}
\end{figure}

We will see in \Cref{section: main} that the problem of determining the orbit length $|\cO(T)|$ of any non-minimal tableau $T$ can be reduced to determining the orbit lengths of certain smaller minimal tableaux. 

\subsection{\texorpdfstring{$m-$}{m-}Diagrams}

Webs are planar graph morphisms in diagrammatic categories for certain representations of the quantum groups $U_q(\mathfrak{sl}_n)$ \cite{cautis, kuperberg1996spiders}. There is a rich literature on how promotion of tableaux corresponds to rotation of webs in the cases $n=2$ and $n=3$ {\cite{petersen2009promotion, russell, Julianna}; but this is constrained for larger $n$ by our limited understanding of webs in that case (though see \cite{gaetz2023rotation} for a rotation-invariant basis when $n=4$).

We instead focus on an intermediary between webs and tableaux: $m$-diagrams, which are collections of arcs that satisfy certain noncrossing conditions.

\begin{definition}[$m-$Diagram] \label{definition: m-diagrams}
A \emph{matching} $\mathcal{M}$ on the set $\{1, 2, \ldots, n\}$ is a collection of disjoint pairs
\[\mathcal{M} = \{(i_1, j_1), (i_2, j_2), \ldots, (i_k,j_k)\} \subseteq \{1, 2, \ldots, n\} \times \{1, 2, \ldots, n\}.\]
We often refer to the pair $(i,j)$ as an \emph{arc} and assume in our notation that $i \leq j$.  We say that an integer $i'$ is \emph{on the arc} $(i,j)$ if $i' \in \{i,j\}$ and \emph{below the arc} $(i,j)$ if $i<i'<j$.

A matching $\mathcal{M}$ is:
\begin{itemize}
    \item \emph{perfect} if every number $1, 2, \ldots, n$ is used on an arc;
    \item \emph{crossing} if it contains two arcs $(i,j), (i',j') \in \mathcal{M}$ with $i<i'<j<j'$;
    \item \emph{noncrossing} if it is not crossing;
    \item \emph{with repetition} if it contains at least one arc of the form $(i,i)$;
    \item \emph{standard} if every integer $i'$ below an arc $(i,j)$ is itself on an arc.
\end{itemize}
An $m$-diagram on the set $\{1, 2, \ldots, n\}$ is the union $\bigcup_{i=1}^r \mathcal{M}_i$ where each $\mathcal{M}_i$ is a noncrossing matching.  
\end{definition}

Note that an $m$-diagram itself may not be a noncrossing matching since arcs from one matching are allowed to cross arcs from another matching.  An $m$-diagram may have arcs of the form $(i,i)$ but is itself not a multiset, so each arc appears at most once in an $m$-diagram regardless of how many different matchings contain it.

We represent matchings and $m$-diagrams by drawing semicircular arcs in the upper half plane between the points $i$ and $j$ on the $x$-axis. The next lemma demonstrates a simple bijection between SYT and $m-$diagrams, the proof of which can be found in \cite[Lemma 1]{russell}, \cite[Proposition 2.4]{Julianna}.

\begin{lemma} \label{lemma: creating m-diagram from tableau}
Let $T$ be an SYT with $r$ rows and content $\{1, \ldots, n\}$.  For each $i \leq r-1$ the following recursive process constructs a noncrossing matching $\mathcal{M}_i$ on the integers filling rows $i$ and $i+1$ of $T$:
\begin{enumerate}
    \item If row $i$ of $T$ has $\lambda_i$ boxes then denote them by $t_i(1), t_i(2), \ldots, t_i(\lambda_i)$ and similarly for row $i+1$.
    \item Create an arc $(t_i(j), t_{i+1}(1))$ where 
    \[j = \max\{1 \leq s \leq \lambda_i: t_i(s) \leq t_{i+1}(1)\}.\]
    \item If $t_{i+1}(1), \ldots, t_{i+1}(\ell)$ are all on arcs then create an arc $(t_i(j), t_{i+1}(\ell+1))$ where 
    \begin{align*}
    j = \max\{1 \leq s \leq \lambda_i: t_i(s) \leq &t_{i+1}(\ell+1) \\ &\textup{ and } (t_i(s),t_{i+1}(j')) \textup{ is not an arc for any } j' \leq \ell\}.
    \end{align*}
\end{enumerate}
\end{lemma}

We rephrase the previous result in terms of $m$-diagrams as follows.  All of the proofs are immediate from the previous lemma together with the definitions.

\begin{corollary} \label{corollary: function varphi from tabs to m-diagrams}
The algorithm in \Cref{lemma: creating m-diagram from tableau} defines a function $\varphi(T)=M_T$ from $r$-row SYT to $m$-diagrams built from $r-1$ matchings.  In addition, all of the following hold:
\begin{itemize}
    \item The number of arcs in the matching $\mathcal{M}_i$ is equal to the number of boxes in row $i+1$ of $T$.  
    \item Each matching $\mathcal{M}_i$ is without repetition.
    \item The matching $\mathcal{M}_i$ is standard with respect to the entries in rows $i$ and $i+1$ of $T$.
    \item If $T$ is a standard Young tableau on a rectangular Young diagram then $\varphi(T)=M_T$ is the union of $r-1$ matchings $\mathcal{M}_i$ of the same cardinality.  Moreover every integer $1, 2, \ldots, |T|$ is one of three types: the end of an arc in $\mathcal{M}_i$ and the start of an arc in $\mathcal{M}_{i+1}$, or the start of an arc on $\mathcal{M}_1$, or the end of an arc in $\mathcal{M}_{r-1}$.
\end{itemize}  
\end{corollary}
An example of an $m-$diagram corresponding to an SYT is given in \Cref{fig:T and M_T}. We can define a rotation operation on $m$-diagrams algebraically as follows. Intuitively, this consists of connecting the endpoints of the boundary to form a circle, rotating the circle one step clockwise, and then disconnecting the circle back to a boundary line again.

\begin{figure}[H]
    \centering\scalebox{.97}{
    \begin{tikzpicture}
        \node at (0,0) {\T};
        \node at (0, -1.75) {$T$};
        \node at (9,0) {\MT};
        \node at (8.5,-1.75) {$\phi(T)$};
        \draw[style=thick, ->] (2.1,0)--(3,0);
    \end{tikzpicture}}
    \caption{Tableau $T$ and its corresponding $m-$diagram $\phi(T)$}
    \label{fig:T and M_T}
\end{figure} 

\begin{definition}[Rotation] \label{definition: rotation}
Denote by $\rho: \{1, 2, \ldots, n\} \rightarrow \{1, 2, \ldots, n\}$ the cyclic permutation given by $\rho(i)=i-1 \mod n$.  Then $\rho$ induces a map from the set of matchings on $\{1, 2, \ldots, n\}$ to itself by sending each arc $(a,b) \mapsto (\rho(a),\rho(b))$.  We call this map \emph{rotation} and denote the image of a matching $\mathcal{M}$ under rotation by $\rho(\mathcal{M})$.  
\end{definition}

The following two observations about rotation are almost immediate from the definitions.

\begin{lemma} \label{lemma: rotation preserves perfect and noncrossing matchings}
If $\mathcal{M}$ is a perfect matching then so is the rotation $\rho(\mathcal{M})$.  If $\mathcal{M}$ is a noncrossing matching then so is the rotation $\rho(\mathcal{M})$.
\end{lemma}

We would like for rotation to respect promotion in the sense that rotating the $m$-diagram for SYT $T$ produces the $m$-diagram for the promotion $P(T),$ but usually it does not. For example, it is not true that the $m-$diagram $\phi(P(T))$ is equal to $\rho(\phi(T))$ for the tableau $T$ in our running example. However, the next section describes a different way to connect rotations to promotions. The arguments in \Cref{section: main} utilize the idea of \textit{sub-diagrams} and \textit{components} of an $m-$diagram, which we introduce here. 

\begin{definition}[Sub-diagrams and Components] Consider an $m-$diagram $M =$
    {\parfillskip=0pt \par } \noindent
    $ \cup_{i=1}^r(a_i, b_i)$ where each $(a_i, b_i)$ is an arc. The proper subcollection $C=\{(a_{i_k}, b_{i_k})\}_{k=1}^s$ forms a \textit{sub-diagram} of $M$ if whenever $a$ is an endpoint of some arc in $C,$ all arcs with $a$ as an endpoint are in $C$ and no $(a_j, b_j)$ for $j\notin\{i_1, \ldots, i_s\}$ crosses any arc in $C.$ The sub-diagram is \textit{uniform} if its endpoints are all adjacent and a \textit{component} if it has no proper subcollection that forms a sub-diagram. Two sub-diagrams $C, C'$ are equivalent if $\rho^N(C) = C'$ for some $N.$ 
\end{definition}

An example of components and sub-diagrams of an $m-$diagram is given in \Cref{fig:components}.

\begin{figure}[ht]
    \centering
    \MTcomponents
    \caption{The components of the $m-$diagram $\phi(T)$ are $C_1, C_2, C_3,$ and $C_4.$ The components $C_1, C_2, C_3$ are uniform, whereas $C_4$ is nonuniform. Note that the components $C_1$ and $C_3$ are equivalent since $\rho^{12}(C_3) = C_1.$ Any union of components forms a sub-diagram of $\phi(T).$ For example, $C_1\cup C_2$ forms a uniform sub-diagram, whereas $C_3\cup C_4$ forms a non-uniform sub-diagram.}
    \label{fig:components}
\end{figure}

\section{Determining Orbit Lengths}\label{section: main}
Our main result is an identification between the orbit lengths of rectangular SYT and the rotatational symmetry of their corresponding $m-$diagrams. This identification allows us to explicitly compute the orbit length of a non-minimal tableau by only considering a subset of its, typically much smaller, subtableaux. To this end, we provide a simple algorithm for the computation of orbit lengths.

\begin{theorem}\label{main_thm}
    Fix a rectangular SYT $T$ and its corresponding $m-$diagram $M=\phi(T).$ Let $N$ be the smallest positive integer such that $\rho^N(M) = M.$ Then, the equality $|\cO(T)| = N$ holds.
\end{theorem}

The proof of \Cref{main_thm} involves a combination of three lemmas that relate the internal structure of $T$ and $P(T)$ to the geometric structures of $M$ and $\rho(M).$ We begin by identifying all uniformly proper subtableaux of $T$ with uniform sub-diagrams of $M.$ 

\begin{lemma}{\label{lem: components subtableaux bijection}}
    For a fixed rectangular SYT $T$ and its corresponding $m-$diagram $M=\varphi(T),$ we have the following correspondences:
    \begin{enumerate}
        \item The set of uniformly proper rectangular subtableaux of $T$ is in bijection with the uniform sub-diagrams of $M.$
        \item The set of minimal uniformly proper rectangular subtableaux of $T$ is in bijection with the uniform components of $M.$
    \end{enumerate}
\end{lemma}

Then, we show that the action of promotion on $T$ preserves the set of uniformly proper subtableaux that do not contain the minimal element of the tableau.

\begin{lemma}{\label{lem: promotion preserves subtableau}} If $T=T_1ST_2,$ then $P(T) = T_1'S'T_2'$ where $T_1'T_2'\equiv P(T_1T_2)$ and $S'$ is a uniformly proper rectangular subtableau of $P(T)$ that is uniformly equivalent to $S.$
\end{lemma}

The key insight in the proof of \Cref{lem: promotion preserves subtableau} is that the sliding path of the empty box is entirely horizontal as it passes through a uniformly proper subtableau. This ensures that the internal structure of each uniformly proper subtableau is preserved during promotion. This is illustrated in \Cref{fig: sliding path}.

\begin{figure}[ht]
    \centering
    \begin{tikzpicture} [scale=.75]
    \node at (.5,3.5) {b};
    \node at (.5,1.75) {$T_1$};
    \node at (2.5,1.75) {$S$};
    \node at (5,1.75) {$T_2$};
\node at (3,-.5) {$T$};

\draw[thick,densely dotted] (0,3)--(1,3)--(1,4);
\draw[thick, densely dotted] (5,0)--(5,1)--(6,1);

\draw[thick](0,0)--(0,4)--(6,4)--(6,0)--(0,0);
\draw[thick] (1,0)--(1,3)--(2,3)--(2,4);
\draw[thick] (4,0)--(4,3)--(5,3)--(5,4);
\begin{scope}[thick,color=Cerulean, decoration={markings,mark=at position 0.5 with {\arrow{>}}}]
\draw[postaction={decorate}] (0.75,3.5)--(4.5,3.5);
\draw[postaction={decorate}] (4.5,3.5)--(4.5,0.5);
\draw[postaction={decorate}] (4.5,0.5)--(5.5,0.5);
\end{scope}
    \draw [-stealth](6.5,2) -- (8.5,2);
\node at (9.5,1.75) {$T_1$};
    \node at (11.5,1.75) {$S$};
    \node at (14,1.75) {$T_2$};
\node at (12,-.5) {$P(T)$};

\draw[thick](9,0)--(9,4)--(15,4)--(15,0)--(9,0);
\draw[thick] (10,0)--(10,4);
\draw[thick] (13,0)--(13,4);
    \end{tikzpicture}
    \caption{The sliding path of $b$ through $T$ and structure of $P(T)$}
    \label{fig: sliding path}
\end{figure}

The combination of \Cref{lem: components subtableaux bijection,lem: promotion preserves subtableau} shows that the sub-diagrams of $M$ are simply rotated to form the sub-diagrams of the $m-$diagram $\phi(P(T)).$

\begin{lemma}\label{lem: rotate components}
    Let $M=\varphi(T)$ and choose a sub-diagram $C.$ Suppose that $E_C = \{(a_{i_k}, b_{i_k})\}_{k=1}^s$ is the set of endpoints of $C$ and $T_C$ is the tableau corresponding to $C.$
    \begin{enumerate}
        \item The arcs with endpoints $\rho(E_C)$ in $\varphi(P(T))$ form a sub-diagram of $\varphi(P(T)).$
        \item If $E_C$ does not contain 1, the arcs with endpoints $\rho(E_C)$ in $\varphi(P(T))$ are $\rho(C).$
        \item If $E_C$ does contain 1, the arcs with endpoints $\rho(E_C)$ in $\varphi(P(T))$ form the $m-$diagram $\varphi(P(T_C))$ when endpoints are appropriately relabelled.
    \end{enumerate}
\end{lemma}

The proof of \Cref{main_thm} is then a straightforward combination of \Cref{lem: components subtableaux bijection,lem: promotion preserves subtableau,lem: rotate components}. Now, since the orbit length $|\cO(T)|$ is identified with the rotational symmetry of the $m-$diagram $\phi(T),$ we can calculate the order of promotion of $T$ by calculating the order of rotational symmetry of its corresponding $m-$diagram. We outline a simple process for this when 1 and $n$ are not in the same component of $\phi(T)$ in \Cref{algorithm}. 

\begin{theorem}\label{algorithm}
    Let $M$ be an $m-$diagram on $n$ vertices such that the vertices 1 and $n$ are not in the same component. The order of rotational symmetry of $M$ can be determined as follows:
    \begin{enumerate}
        \item Let $k$ be the number of non-equivalent components of $M.$ Color each vertex of the $m-$
        diagram a color from $[k]$ such that two vertices are the same color if and only if they belong to equivalent components.
        \item Define $N$ to be the smallest positive integer such that for any vertex $i$, $i- N\mod n$ has the same color as $i$ and for any component $C,$ the rotation $\rho^N(C)$ is an equivalent component.
        \item For each component $C$ of $M$ with a vertex in $[N],$ let $N_C$ be the number of vertices of $C$ that are less than or equal to $N$ and $T_C$ the SYT corresponding to $C.$
        \item Let $\ell$ be the smallest positive integer such that $|\mathcal{O}(T_C)| {\big|} \ell N_C$ for each $C.$
    \end{enumerate}
    Then, $M$ has rotational symmetry of order $\ell N.$
\end{theorem}

If the vertices $1$ and $n$ are in the same component of $\phi(T),$ we can still apply the algorithm of \Cref{algorithm} by first pre-processing $\phi(T)$ in the following way:
\begin{enumerate}
    \item Let $C_1$ be the component of $\phi(T)$ containing both 1 and $n.$ Suppose the first $k$ vertices of $\phi(T)$ are contained in $C_1$ and let $T_1$ be the Young tableau to $C_1.$
    \item Determine the tableau $P^k(T_1)$ and its corresponding $m-$diagram $\phi(P^k(T_1)).$ Note that while $T_1$ is not an SYT, the promotion operation can still be performed since the entries of $T_1$ strictly increase both row and column-wise.
    \item Define $M$ to be the $m-$diagram $\rho^k(\phi(T))$ with the rightmost component replaced by $\phi(P^k(T_1)).$
\end{enumerate}

Note that by \Cref{lem: promotion preserves subtableau}, the $m-$diagram $M$ defined in the above process exactly corresponds to $\phi(P^k(T))$ and so the rotational symmetry of $M$ corresponds to the orbit length $|\cO(T)|.$ Moreover, our approach of identifying uniform components of $\phi(T)$ with the uniformly proper subtableaux of $T$ allows us to calculate $\phi(P^k(T)),$ and thus $P^k(T),$ without requiring the $k-$fold evaluation of promotion on $T$ itself.

Following the algorithm in \Cref{algorithm}, we see that the orbit length $|\cO(T)|$ is determined by examining orbit lengths of a set of minimal standard Young tableaux. While there is no known method for identifying the orbit lengths of minimal tableaux, typically these tableaux will be much smaller than $T$ and thus directly computing their orbit lengths will be much less computationally expensive than computing the entire promotion orbit of $T.$ Furthermore, while there may be many components of $\phi(T)$ that have at least one vertex in $[N],$ only those that contain vertices outside of $[N]$ need be considered in the calculation of $k:$ if all vertices of the component $C$ are contained in $[N],$ then $|\cO(T)|{\big |}N_C$ by \cite{Haiman}.

\begin{example} We can now calculate the promotion orbit length of our running example $T.$ Since 1 and 24 are in the same component in the $m-$diagram $\phi(T),$ we first pre-process the $m-$diagram following the above steps. We begin by identifying $C_1,$ the minimal component of $\phi(T)$ that contains both 1 and 24 as vertices.
\begin{center}
    \begin{tikzpicture}
        \node at (0,0) {\MTpreprocess};
        \node at (2.5, 0.75) {\color{Cerulean} $C_1$};
        \node at (0,-1) {$\phi(T)$};
    \end{tikzpicture}
\end{center}
We then form the tableau $T_1$ corresponding to $T_1.$ Since the first vertex not contained in $C_1$ is vertex 2, we need only promote $T_1$ once. 
\begin{center}
    \begin{tikzpicture}
        \node at (0,0) {\begin{ytableau}
            1 & 19 \\
            18 & 21 \\
            20 & 23 \\
            22 & 24
        \end{ytableau}};
        \node at (0,-1.75) {$T_1$};
        \draw[style=thick, ->] (1, 0) -- (2, 0);
        \node at (3,0) {\begin{ytableau}
            17 & 18 \\
            19 & 20 \\
            21 & 22 \\
            23 & 24
        \end{ytableau}};
        \node at (3, -1.75) {$P(T_1)$};
    \end{tikzpicture}
\end{center}
After computing the promotion $P(T_1),$ we form the new $m-$diagram $M$ by computing the rotation $\rho(\phi(T))$ and replacing the rightmost component with $\phi(P(T_1)).$ We then use the steps described in \Cref{algorithm} to determine the rotational symmetry of $M.$
\begin{center}
    \begin{tikzpicture}
        \node at (0,0) {\MTpostprocess};
        \node at (-4.5, 0.75) {$C_1$};
        \node at (-2, 0.75) {\color{Cerulean} $C_2$};
        \node at (1, 0.75) {$C_3$};
        \node at (3.5, 0.75) {\color{Cerulean} $C_4$};
        \node at (0,-.75) {$M$};
    \end{tikzpicture}
\end{center}
Using the same notation as in \Cref{algorithm}, we see that $N = 12$ since the components $C_1$ and $C_3$ are equivalent and the components $C_2$ and $C_4$ are equivalent. Since the components $C_1$ and $C_2$ are entirely contained in the first 12 vertices of $M,$ we conclude that $M$ has rotational symmetry of order 12. It follows that $|\cO(T)| = 12.$

\end{example}

\section{Extending Results to Semistandard Young Tableaux}
Thus far, we have only considered rectangular SYT. However, our results generalize to rectangular (column) semistandard Young tableaux in a natural way.

\begin{definition}
    A \emph{(column) semistandard Young tableau (SSYT)} is a filling of the Young diagram for the partition $\lambda$ with the numbers $1, 2, \ldots, n$ so that each number appears at least once, rows strictly increase left-to-right, and columns weakly increase top-to-bottom. 

    The \emph{content} of an SSYT $T$ is the collection of integers filling its boxes with repetition.  If each number $i$ appears $e_i$ times in $T$ then we denote the content $\{1^{e_1}, 2^{e_2}, \ldots, n^{e_n}\}$.  Note that the content may be a multiset. 
\end{definition}

\begin{remark}
Usually \emph{semistandard} refers to tableaux in which columns strictly increase and rows are weakly increasing.  For this reason, we usually stress \emph{column} in our terminology.  Note that each column semistandard tableau is the transpose of a (row) semistandard tableau. Since the promotion orbit length is preserved under the transposition of tableaux, it is sufficient to characterize the orbits of column SSYT.
\end{remark}

The definition of promotion given in \Cref{definition: promotion} is easily generalizable to a column SSYT $T$ by repeating steps (1) to (3) until there are no copies of 1 remaining in $T$ and then inserting the maximum entry of $T$ into every empty box. Moreover, $m-$diagrams can be constructed from SSYT using the same process as in \Cref{lemma: creating m-diagram from tableau}. Our \Cref{main_thm} can also be used to determine the promotion orbit lengths of an SSYT $T.$

\begin{corollary}\label{cor: apply to SSYT}
    Let $T$ be an SSYT and $M=\phi(T)$ be its corresponding $m-$diagram. If the vertices $1$ and $n$ are not in the same component, the orbit length $|\cO(T)|$ can be determined using the algorithm in \Cref{algorithm}.
\end{corollary}

There are several key observations that allow us to extend our result to SSYT. Firstly, there is a natural map $\psi$ from the set of SSYT to SYT of the same shape. In essence, the map $\psi$ expands the content of $T$ so that any entry appears in the tableau only once while ensuring that the relative ordering of entries is preserved. If the number 1 appears $k$ times in the SSYT $T,$ then the $k-$fold promotion of the SYT $\psi(T)$ corresponds to 1 promotion of $T$ and so the equality $\psi(P(T)) = P^k(\psi(T))$ holds. Moreover, there is a well defined ``contraction'' map $\kappa$ on the $m-$diagram $\phi(\psi(T))$ so that the equality $\kappa(\phi(\psi(T))) = \phi(T)$ holds. The contraction map also respects rotation in the following sense: if the number 1 appears $k$ times in the SSYT $T,$ then $k$ rotations of the $m-$diagram $\phi(\psi(T))$ followed by contraction corresponds to 1 rotation of the $m-$diagram $\phi(T),$ i.e. the equality $\kappa(\rho^k(\phi(\psi(T)))) = \rho(\phi(T))$ holds.

Combining these observations with the arguments arising from the study of SYT completes the proof of \Cref{cor: apply to SSYT}. As with SYT, if the vertices 1 and $n$ are in the same component of $\phi(T),$ the $m-$diagram can be pre-processed as outlined above and then \Cref{cor: apply to SSYT} can be applied accordingly.

\acknowledgements{We would like to thank Meredith Pan and Risa Vandegrift for helpful discussions during the early stages of this work.}

\printbibliography


\end{document}
