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\usepackage{lipsum}

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\usepackage{ytableau}
\usepackage[enableskew]{youngtab}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
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\newcommand\scalemath[2]{\scalebox{#1}{\mbox{\ensuremath{\displaystyle #2}}}}
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%% define your title in the usual way
\title{Growth Diagrams from Polygons in the Affine Grassmannian}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Tair Akhmejanov\thanks{\href{mailto:ta328@cornell.edu}{ta328@cornell.edu}}\addressmark{1}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, Cornell University, Malott Hall, Ithaca, NY 14850}

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\received{November 13, 2017}

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\revised{March 31, 2018}

%% put your English abstract here, or comment this out if you don't have one yet
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%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{We introduce growth diagrams arising from the geometry of the affine Grassmannian for $GL_m$. These affine growth diagrams are in bijection with the $c_{\vec\lambda}$ many components of the polygon space Poly($\vec\lambda$) for $\vec\lambda$ a sequence of minuscule weights and $c_{\vec\lambda}$ the Littlewood--Richardson coefficient. Unlike Fomin growth diagrams, they are infinite periodic on a staircase shape, and each vertex is labeled by a dominant weight of $GL_m$. Letting $m$ go to infinity, a dominant weight can be viewed as a pair of partitions, and we recover the RSK correspondence and Fomin growth diagrams within affine growth diagrams. The main combinatorial tool used in the proofs is the $n$-hive of Knutson--Tao--Woodward. The local growth rule satisfied by the diagrams previously appeared in van Leeuwen's work on Littelmann paths, so our results can be viewed as a geometric interpretation of this combinatorial rule.}

%% 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{growth diagram, Robinson--Schensted--Knuth correspondence,
Littlewood--Richardson coefficient, Knutson--Tao--Woodward hive, promotion, evacuation}

%% 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
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\usepackage[backend=bibtex]{biblatex}
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\begin{document}

\maketitle
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\section{Introduction}
The classical Robinson--Schensted correspondence is a bijection between permutations and pairs of same-shape standard Young tableaux. It was first written down in \cite{Robi} by Robinson in an attempted proof of the Littlewood--Richardson rule. Schensted later rediscovered the bijection in \cite{Sch} with the aim of understanding the longest increasing subsequences of a permutation, a point of view that was further generalized by Greene in \cite{Gre}. Then Fomin \cite{Fom} realized the bijection in terms of growth diagrams where standard Young tableaux are interpreted as saturated chains of partitions (see Figure \ref{Fomin}). These diagrams are square arrays of vertices, each labeled by a partition, where three of the vertex labels of a unit square determine the label of the fourth, southeast vertex via Fomin's local rules.

The combinatorics of the RS correspondence was realized geometrically by Steinberg \cite{Ste2}, Hesselink \cite{Hes}, and Spaltenstein \cite{Spa}, using the variety of flags preserved by a fixed nilpotent $\eta$, i.e.\ a Springer fiber. The components of this variety are indexed by standard Young tableaux of shape equal to the Jordan type of $\eta$. Recall that the relative position of two flags is a permutation given by the Bruhat decomposition. For two generic flags $f$ and $f'$ in components corresponding to tableaux $T$ and $S$ respectively, their relative position is the permutation given by the RS correspondence. These results were clarified and extended by van Leeuwen in \cite{vLee3}. In particular, each partition of Fomin's growth diagram is the Jordan type of $\eta$ restricted to $f_i \cap f_j'$. Hence, van Leeuwen completed this geometric interpretation of Fomin's combinatorial rules.

In a separate work \cite{vLee2}, van Leeuwen gave a combinatorial rule that realized the symmetry of the tensor product of irreducible representations in terms of Littelmann paths. Namely, for two irreducible representations $V_\lambda,V_\mu$ of a semisimple Lie algebra $\mathfrak g$, the multiplicity $c_{\lambda,\mu}^\nu$ in $V_\lambda\otimes V_\mu=\bigoplus_\nu c_{\lambda,\mu}^\nu V_\nu$ is equal to the number of $\lambda$-dominant paths of weight $\nu-\lambda$ that are obtained by repeated application of the Littelmann path operators on a fixed path of weight $\mu$ \cite{Litt}. Likewise, $c_{\mu,\lambda}^\nu$ is counted by an analogous set of paths, but with the roles of $\lambda$ and $\mu$ reversed. Since $c_{\lambda,\mu}^\nu$ is equal to $c_{\mu,\lambda}^\nu$, it is natural to ask for a bijection between these two sets of paths, and such a bijection is given in \cite{vLee2} that gradually transforms a path via a local rule. 

We show that this same combinatorial rule is related to the geometry of the affine Grassmannian. Using this rule, we construct new growth diagrams on a staircase shape that are in bijection with the components of the polygon space, the space of tuples in the affine Grassmannian that satisfy certain distance conditions. Within these affine growth diagrams we recover Fomin growth diagrams, as well as some classical bijections, including the RSK correspondence.

\begin{figure}[h]
\centering
\scalemath{.7}{
\begin{tikzpicture}
\draw[step=1cm] (0,0) grid (1,1);
\node at (0.15,.85) {$\gamma$};
\node at (1.15,.8) {$\beta$};
\node at (.15,-.2) {$\alpha$};
\node at (1.15,-.2) {$\delta$};
\end{tikzpicture}}
\hspace{20mm}
\scalemath{.75}{
\Ylinethick1.5pt
\begin{array}{|cc|cc|cc|cc|c}
\cline{1-8}
 \emptyset&& \emptyset&   & \emptyset& & \emptyset& &\emptyset\\
  &X& & & & & & & \\
\cline{1-8}
  \emptyset&&\scalebox{.5}{\yng(1)}&&\scalebox{.5}{\yng(1)}&&\scalebox{.5}{\yng(1)}&&\scalebox{.5}{\yng(1)}\\
    && & & & & & X& \\
\cline{1-8}
 \emptyset&&\scalebox{.5}{\yng(1)}&&\scalebox{.5}{\yng(1)}&&\scalebox{.5}{\yng(1)}&&\scalebox{.5}{\yng(2)}\\
   && & X& & & & & \\
\cline{1-8}
 \emptyset&&\scalebox{.5}{\yng(1)}&&\scalebox{.5}{\yng(2)}&&\scalebox{.5}{\yng(2)}&&\scalebox{.5}{\yng(2,1)}\\
  && & & & X& & & \\
\cline{1-8}
 \multicolumn{1}{c}{\emptyset}& \multicolumn{2}{r}{\scalebox{.5}{\yng(1)}}& \multicolumn{2}{r}{\scalebox{.5}{\yng(2)}}& \multicolumn{2}{r}{\scalebox{.5}{\yng(3)}}& \multicolumn{2}{r}{\scalebox{.5}{\yng(3,1)}}
\end{array}}
\hspace{10mm}
$1423\leftrightarrow (\scalemath{.5}{\begin{ytableau}1&2&3\\4\end{ytableau}},\scalemath{.5}{\begin{ytableau}1&2&4\\3\end{ytableau}})$
\caption{A Fomin growth diagram. Partitions $\alpha,\beta,\gamma$ of a unit square determine $\delta$. \label{Fomin}}
\end{figure}

\section{Geometry}
\subsection{Geometric Satake and the Polygon Space}
Let $G=GL_m(\mathbb C)$. Recall that the \textbf{dominant weights} of $G$ are weakly decreasing sequences of $m$ integers. The dual of a weight $\lambda$ is given by negating and reversing the sequence, and is denoted $\lambda^*$. Let $V_{\lambda}$ denote the irreducible representation corresponding to dominant $\lambda$. Although $G$ is not semisimple, we will say that the $\textbf{minuscule weights}$ are the fundamental weights $\omega_i=(1,\ldots,1,0,\ldots,0)$ and the dual fundamental weights $\omega_i^*=(0,\ldots,0,-1,\ldots,-1)$ for $1\leq i \leq m$ (where there are $i$ many $1$'s and $-1$'s respectively). For a sequence of dominant weights \mbox{$\vec\lambda=\left(\lambda^1,\lambda^2,\ldots,\lambda^n\right)$}, let \mbox{$c_{\vec\lambda}=\dim(V_{\lambda^1}\otimes\cdots\otimes V_{\lambda^n})^G$} be \textbf{the Littlewood--Richardson coefficient}, the dimension of the invariant space. 

Let $\mathcal O=\mathbb{C}[[t]]$ be the ring of power series and $\mathcal K=\mathbb{C}((t))$ the field of Laurent series. Let $Gr=G(\mathcal{K})/G(\mathcal{O})$ be the \textbf{affine Grassmannian}, an ind-variety over $\mathbb{C}$. For each dominant weight $\lambda$ let $Gr(\lambda)=G(\mathcal O)t^\lambda$ be the orbit of $G(\mathcal O)$ in $Gr$ where $t^\lambda$ is the torus fixed point for $\lambda$. These are all of the $G(\mathcal O)$-orbits and are distinct. Then $Gr=\sqcup Gr(\lambda)$ and $\overline{Gr(\lambda)}=\cup_{\mu\leq \lambda} Gr(\mu)$ where $\mu\leq \lambda$ under the usual dominance order of weights. The orbits of $G(\mathcal K)$ on $Gr\times Gr$ are in bijection with orbits of $G(\mathcal O)$ on $Gr$. For $p,q\in Gr$ define the \textbf{distance} to be $d(p,q)=\lambda$ if $(p,q)$ are in the same $G(\mathcal K)$ orbit as $(t^0,t^\lambda)$. Note that $d(q,p)=\lambda^*$. One reason for the importance of the affine Grassmannian is the following theorem of Lusztig \cite{Lus}, Ginzburg \cite{Gin}, Beilinson--Drinfeld \cite{BD}, and Mirkovi\'c--Vilonen \cite{MV}. For general reductive groups one needs to take the Langlands dual $G^L$, but in our case $GL_m^L=GL_m$. 

\begin{thm}[Geometric Satake]{\label{Satake}}
The tensor category of perverse sheaves on the affine Grassmannian Gr with respect to the above stratification is equivalent to the tensor category of representations of $G$.
\end{thm}
For a sequence of dominant weights $\vec\lambda$ consider the twisted product of orbit closures,
\begin{align*}
\overline{Gr(\lambda^1)}\tilde\times \cdots \tilde\times \overline{Gr(\lambda^n)}:=\left\{\left(g_1,\ldots,g_n\right)\mid d(t^0,g_1)\leq\lambda^1, d(g_i,g_{i+1})\leq\lambda^{i+1}, 1\leq i\leq n-1 \right\},
\end{align*}
a sort of ``path space'' in $Gr$. Theorem \ref{Satake} gives a way to construct the invariant space $(V_{\lambda^1}\otimes\cdots\otimes V_{\lambda^n})^G$ geometrically from the following convolution morphism.
\begin{align*}
m_{\vec\lambda}:\overline{Gr(\lambda^1)}\tilde\times \overline{Gr(\lambda^2)}\tilde\times \cdots \tilde\times \overline{Gr(\lambda^n)}\rightarrow &Gr \\
(g_1,\ldots,g_n)\mapsto &g_n
\end{align*}
Define the polygon space to be the projective variety Poly$(\vec\lambda)=m_{\vec{\lambda}}^{-1}(t^0)$. The geometric Satake correspondence yields the following corollary.
\begin{cor}
The invariant space is isomorphic to the top homology of Poly$(\vec\lambda)$. $$\left(V_{\lambda^1}\otimes\cdots\otimes V_{\lambda^n}\right)^G\cong H_{top}\left(\text{Poly}(\vec\lambda)\right)$$
\end{cor}
Haines showed in \cite[Proposition 1.8]{Hai} that for $GL_m$, Poly$(\vec\lambda)$ is equidimensional, so as a corollary, the number of components of Poly$(\vec\lambda)$ is equal to $c_{\vec\lambda}$. We study the combinatorics of indexing the components when the $\lambda^i$ are minuscule. Note that in this case $c_{\vec\lambda}$ is a rectangular Kostka number. For $\lambda$ dominant minuscule, there are no dominant weights less than $\lambda$, so $\overline{Gr(\lambda)}=Gr(\lambda)$ and the inequalities above become equalities. Hence, Poly$(\vec\lambda)$ is the variety of $n$-gons, where a point in Poly$(\vec\lambda)$ is a tuple $(g_1,\ldots,g_{n-1},g_n=t^0)$ satisfying the $\vec\lambda$ distance conditions as in Figure \ref{polygon}.

\begin{figure}
\centering
\begin{minipage}{.49\textwidth}
\centering
	\scalemath{1}{
	\begin{tikzpicture}[decoration={markings,mark=at position 0.5 with {\arrow{>}}}] 
    \node(one) at (-1,0) {$t^0$};
    \node(two) at (0,1) {$g_1$};
    \node(three) at (2,1) {$g_2$};
    \node(four) at (3,0) {$g_3$};
    \node(five) at (2,-1) {$\cdots$};
    \node(six) at (0,-1) {$g_{n-1}$};
    \draw[->] (one) -- (two) node[midway,above]{\tiny $\lambda^1$};
    \draw[->] (two) -- (three) node[midway,above]{\tiny $\lambda^2$};
    \draw[->] (three) -- (four) node[midway,above]{\tiny $\lambda^3$};
    \draw[->] (four) -- (five) node[near start,below]{\tiny $\lambda^4$};
    \draw[->] (five) -- (six) node[midway,above]{\tiny $\lambda^{n-1}$};
    \draw[->] (six) -- (one) node[near end,below]{\tiny $\lambda^n$};
	\end{tikzpicture}}
\caption{A point of Poly($\vec\lambda$) \label{polygon}}
\end{minipage}
\hfill
\begin{minipage}{.49\textwidth}
\centering
\begin{tikzpicture}[decoration={markings,mark=at position 0.5 with {\arrow{>}}}] 
    \node(one) at (-1,0) {$t^0$};
    \node(two) at (0,1) {$g_1$};
    \node(three) at (2,1) {$g_2$};
    \node(four) at (3,0) {$g_3$};
    \node(five) at (2,-1) {$\cdots$};
    \node(six) at (0,-1) {$g_{n-1}$};
    \draw[->] (one) -- (two) node[above, near start, xshift=-3mm]{\tiny $\lambda^1=\mu^1$};
    \draw[->] (two) -- (three) node[midway,above]{\tiny $\lambda^2$};
    \draw[->] (three) -- (four) node[midway,above]{\tiny $\lambda^3$};
    \draw[->] (four) -- (five) node[near start,below]{\tiny $\lambda^4$};
    \draw[->] (five) -- (six) node[midway,below]{\tiny $\lambda^{n-1}$};
    \draw[->] (six) -- (one) node[near end,below,xshift=-3.5mm]{\tiny $\mu^{n^*}=\lambda^n$};
    \draw[->] (one) -- (three) node[midway,above]{\tiny $\mu^2$};
    \draw[->] (one) -- (four) node[midway,above]{\tiny $\mu^3$};
    \draw[->] (one) -- (five) node[midway,above]{\tiny $\mu^4$};
\end{tikzpicture}
\caption{The fan triangulation. \label{fan}}
\end{minipage}
\end{figure}

\subsection{Indexing Components with Affine Growth Diagrams}
Assume from now on that the $\lambda^i$ in $\vec\lambda$ are dominant minuscule. Fontaine, Kamnitzer, and Kuperberg showed in \cite{FKK} that the irreducible components of Poly$(\vec\lambda)$ can be indexed by minuscule paths for $\vec\lambda$, namely sequences of dominant weights $\vec\mu=(\mu^0=\emptyset,\mu^1,\ldots,\mu^n=\emptyset)$ such that $\mu^k-\mu^{k-1}=w\cdot \lambda^k$ for some element $w$ in the Weyl group. 
\begin{prop}[\cite{FKK}]
Each component $X$ of Poly$(\vec\lambda)$ has an open dense set $X^o$ such that for all $0\leq i\leq n$ the distance $d(t^0,g_i)=\mu^i$ is constant on $X^o$. The distances $\vec\mu=(\mu^0,\mu^1,\ldots,\mu^{n-1},\mu^n)$ form a minuscule path for $\vec\lambda$, and this is a bijection between components and minuscule paths.
\end{prop}

Note that these distances are the ones along the edges of the fan triangulation of Figure \ref{fan}. Our first observation extends this result to any extroverted triangulation. An \textbf{extroverted triangulation} is a triangulation of the $n$-gon such that each triangle contains an external edge (see Figure \ref{hexagons}). Fix an extroverted triangulation $\tau$ and order its interior edges $(e_2,\ldots,e_{n-2})$ so that $e_i$ and $e_{i+1}$ share a vertex. Choose $e_1$ (respectively $e_{n-1}$) to be one of the two external edges sharing a vertex with $e_2$ (respectively $e_{n-2}$). Define $\sigma:[n]\rightarrow [n]$ so that $\lambda^{\sigma(1)}$ labels $e_1$, $\lambda^{\sigma(n-1)}$ labels $e_{n-1}$, and $\lambda^{\sigma(i)}$ labels the edge that completes the triangle with $e_{i-1}$ and $e_{i}$. Orient the edges so that either the tail of $e_i$ meets the tail of $e_{i-1}$, or their heads meets (see Figure \ref{triangulations to paths}). Let $E_\tau=(e_1,\ldots,e_{n-1})$.
\begin{figure}[b]
\centering
\scalemath{.5}{
\begin{tikzpicture}
\newdimen\R
   \R=1cm
   \draw (0:\R) \foreach \x in {60,120,...,360} {  -- (\x:\R) };
   \draw {(180:\R) -- (60:\R)};
   \draw {(180:\R) -- (0:\R)};
   \draw {(180:\R) -- (-60:\R)};
\end{tikzpicture}
\begin{tikzpicture}
\newdimen\R
   \R=1cm
   \draw (0:\R) \foreach \x in {60,120,...,360} {  -- (\x:\R) };
   \draw {(120:\R) -- (0:\R)};
   \draw {(180:\R) -- (0:\R)};
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\end{tikzpicture}
\begin{tikzpicture}
\newdimen\R
   \R=1cm
   \draw (0:\R) \foreach \x in {60,120,...,360} {  -- (\x:\R) };
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\begin{tikzpicture}
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\end{tikzpicture}
\begin{tikzpicture}
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\end{tikzpicture}
\begin{tikzpicture}
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   \R=1cm
   \draw (0:\R) \foreach \x in {60,120,...,360} {  -- (\x:\R) };
   \draw {(60:\R) -- (-60:\R)};
   \draw {(60:\R) -- (-120:\R)};
   \draw {(60:\R) -- (-180:\R)};
\end{tikzpicture}
\begin{tikzpicture}
\newdimen\R
   \R=1cm
   \draw (0:\R) \foreach \x in {60,120,...,360} {  -- (\x:\R) };
   \draw {(0:\R) -- (-120:\R)};
   \draw {(60:\R) -- (-120:\R)};
   \draw {(60:\R) -- (-180:\R)};
\end{tikzpicture}
\begin{tikzpicture}
\newdimen\R
   \R=1cm
   \draw (0:\R) \foreach \x in {60,120,...,360} {  -- (\x:\R) };
   \draw {(0:\R) -- (-120:\R)};
   \draw {(0:\R) -- (-180:\R)};
   \draw {(60:\R) -- (-180:\R)};
\end{tikzpicture}
\begin{tikzpicture}
\newdimen\R
   \R=1cm
   \draw (0:\R) \foreach \x in {60,120,...,360} {  -- (\x:\R) };
   \draw {(0:\R) -- (-120:\R)};
   \draw {(0:\R) -- (-180:\R)};
   \draw {(0:\R) -- (-240:\R)};
\end{tikzpicture}
\begin{tikzpicture}
\newdimen\R
   \R=1cm
   \draw (0:\R) \foreach \x in {60,120,...,360} {  -- (\x:\R) };
   \draw {(-60:\R) -- (60:\R)};
   \draw {(-60:\R) -- (120:\R)};
   \draw {(-60:\R) -- (180:\R)};
\end{tikzpicture}
\begin{tikzpicture}
\newdimen\R
   \R=1cm
   \draw (0:\R) \foreach \x in {60,120,...,360} {  -- (\x:\R) };
   \draw {(-120:\R) -- (120:\R)};
   \draw {(-120:\R) -- (60:\R)};
   \draw {(-120:\R) -- (0:\R)};
\end{tikzpicture}}
\hspace{5mm}
\scalemath{.5}{
\begin{tikzpicture}
\newdimen\R
   \R=1cm
   \draw (0:\R) \foreach \x in {60,120,...,360} {  -- (\x:\R) };
   \draw {(-120:\R) -- (120:\R)};
   \draw {(120:\R) -- (0:\R)};
   \draw {(-120:\R) -- (0:\R)};
   \fill[gray] (-120:\R)--(120:\R)--(0:\R)--cycle;
\end{tikzpicture}
\begin{tikzpicture}
\newdimen\R
   \R=1cm
   \draw (0:\R) \foreach \x in {60,120,...,360} {  -- (\x:\R) };
   \draw {(180:\R) -- (60:\R)};
   \draw {(180:\R) -- (-60:\R)};
   \draw {(-60:\R) -- (60:\R)};
   \fill[gray] (180:\R)--(60:\R)--(-60:\R)--cycle;
\end{tikzpicture}}
\caption{The 12 extroverted, and 2 non-extroverted, triangulations of the hexagon \label{hexagons}}
\end{figure}

\begin{prop}{\label{generalized path}}
Each component $X$ of Poly$(\vec\lambda)$ has an open dense set $X^o_\tau$ such that for every $e_i$ in $E_\tau$ the distance $\mu^i$ along $e_i$ is constant on $X^o_\tau$. This induces a bijection between the components of Poly$(\vec\lambda)$ and sequences of dominant weights, $\vec\mu=(\mu^0=\emptyset,\mu^1,\ldots,\mu^{n-1},\mu^n=\emptyset)$, such that $\mu^k-\mu^{k-1}=w\cdot \lambda^{\sigma(k)}$ for some $w$ in the Weyl group.
\end{prop}

For a fixed component $X$, taking the intersection of the open dense sets $X^o_\tau$ over all of the extroverted triangulations $\tau$ gives an open dense set on which all the distances $d(g_i,g_j)$ are constant. Call these the generic distances for the component $X$. Consider a quadrilateral with two external edge in the $n$-gon such that the corresponding generic distances in Poly$(\vec\lambda)$ are $\alpha,\beta,\gamma$ as in Figure \ref{local flip} where $n=6$. We give a local rule that given these three generic distances, determines the fourth generic distance $\delta$, thereby performing a ``quadrilateral flip''.

\begin{thm}[Local Growth Rule]{\label{local rule}}
For a generic point $(g_1,\ldots,g_n)$ in a component of Poly$(\vec\lambda)$, if the indicated distances of Figure \ref{local flip} are given by $\alpha,\beta,\gamma$, then $\delta=\text{sort}(\alpha+\beta-\gamma)$.
\end{thm}

Here weights are added component wise, so $\alpha +\beta=(\alpha_1+\beta_1,\ldots,\alpha_n+\beta_n)$. For three dominant weights $\alpha,\beta,\gamma$, the sequence $\alpha+\beta-\gamma$ need not be a weakly decreasing sequence, hence the sort operation orders the sequence of integers to make it a dominant weight.

\begin{figure}
\centering
\scalemath{.7}{
\begin{tikzpicture}
\draw[step=1cm] (0,0) grid (1,1);
\node at (0.15,.85) {$\gamma$};
\node at (1.15,.8) {$\beta$};
\node at (.15,-.2) {$\alpha$};
\node at (1.15,-.2) {$\delta$};
\end{tikzpicture}}
\hspace{5mm}
\scalemath{.7}{
\begin{tikzpicture}[decoration={markings,mark=at position 0.5 with {\arrow{>}}}] 
\newdimen\R
   \R=1cm
   \draw (0:\R) \foreach \x in {60,120,...,360} {  -- (\x:\R) };
   \draw {(120:\R) -- (60:\R)} node[midway, above, shift={(0,.2)}] {\phantom{\scalemath{.575}{\alpha}}};
   \draw {(180:\R) -- (240:\R)};
   \draw[postaction={decorate}] {(120:\R) -- (0:\R)} node[midway, above, shift={(.4,.35)}] {\scalemath{1}{\alpha}};
   \draw[postaction={decorate}] {(180:\R) -- (0:\R)} node[midway, shift={(-.2,.2)}] {\scalemath{1}{\gamma}};
   \draw[postaction={decorate}] {(180:\R) -- (-60:\R)} node[midway, below, shift={(-.2,-.35)}] {\scalemath{1}{\beta}};
\end{tikzpicture}}
\hspace{5mm}
\scalemath{.7}{
\begin{tikzpicture}[decoration={markings,mark=at position 0.5 with {\arrow{>}}}] 
\newdimen\R
   \R=1cm
   \draw (0:\R) \foreach \x in {60,120,...,360} {  -- (\x:\R) };
   \draw {(120:\R) -- (60:\R)} node[midway, above, shift={(0,.2)}] {\phantom{\scalemath{.575}{\alpha}}};
   \draw {(180:\R) -- (240:\R)};
   \draw[postaction={decorate}] {(120:\R) -- (0:\R)} node[midway, above, shift={(.55,.25)}] {\scalemath{1}{\alpha}};
   \draw[postaction={decorate}] {(120:\R) -- (-60:\R)} node[midway,shift={(.25,0)}] {\scalemath{1}{\delta}};
   \draw[postaction={decorate}] {(180:\R) -- (-60:\R)} node[midway, below, shift={(-.2,-.35)}] {\scalemath{1}{\beta}};
\end{tikzpicture}}
\caption{Dominant weights $\alpha,\beta,\gamma$ determine $\delta$ via one application of the local rule. \label{local flip}}
\end{figure}

As with Fomin growth diagrams, this rule allows us to construct growth diagrams where $\alpha,\beta,\gamma$ label three vertices of a unit square and determine the fourth, southeast vertex label $\delta$ according to the local rule in Theorem \ref{local rule}. We can organize the information of the generic distances of a fixed component $X$ of Poly$(\vec\lambda)$ into a growth diagram in a coherent manner. Consider an $n$-wide infinite staircase, such that each row contains $n-1$ unit squares. We will label the vertices of the diagram with the dominant weights that are the generic distances $d(g_i,g_j)$ of $X$. The diagram is organized so that a path through the staircase from the left to the right diagonal, together with the weight labels along the path, corresponds to an extroverted triangulation $\tau$, together with the generic distances of $X$ along the edges $E_\tau$. Begin by labelling the vertices along the left and right diagonals with the zero weight, $\emptyset$. Label the super-diagonal with $\vec\lambda$ periodically (see Figure \ref{begin labeling}).

\begin{figure}
\centering
\begin{minipage}{.49\textwidth}
\centering
\scalemath{.7}{
\begin{tikzpicture}
\draw[step=1cm] (-5,2) grid (0,3);
\draw[step=1cm] (-4,1) grid (1,2);
\draw[step=1cm] (-3,0) grid (2,1);
\draw[step=1cm] (-2,-1) grid (3,0);
\draw[step=1cm] (-1,-2) grid (4,-1);
\draw[step=1cm] (0,-3) grid (5,-2);
\draw (0,-3)--(6,-3);
\draw (-6,3)--(0,3);

\Yboxdim{2mm}
\node at (-5.8,3.2) {$\emptyset$}; 
\node at (-4.8,3.2) {$\lambda^1$};
\node at (0.2,3.2) {$\emptyset$};

\node at (-4.8,2.2) {$\emptyset$}; 
\node at (-3.8,2.2) {$\lambda^2$};
\node at (1.2,2.2) {$\emptyset$};

\node at (-3.8,1.2) {$\emptyset$};  
\node at (-2.8,1.2) {$\lambda^3$};
\node at (2.2,1.2) {$\emptyset$};

\node at (-2.8,.2) {$\emptyset$};  
\node at (-1.8,.2) {$\lambda^4$};
\node at (3.2,.2) {$\emptyset$};

\node at (-1.8,-.8) {$\emptyset$};  
\node at (-.8,-.8) {$\lambda^5$};
\node at (4.2,-.8) {$\emptyset$};

\node at (-.8,-1.8) {$\emptyset$};  
\node at (0.2,-1.8) {$\lambda^6$};
\node at (5.2,-1.8) {$\emptyset$};

\node at (.2,-2.8) {$\emptyset$};
\node at (1.2,-2.8) {$\lambda^1$};
\node at (6.2,-2.8) {$\emptyset$};
%\node[circle,fill,inner sep=2pt] at (0,0) {}; %center dot
\end{tikzpicture}}
\caption{Empty affine growth diagram of type $\vec\lambda=(\lambda^1,\ldots,\lambda^6)$ \label{begin labeling}}
\end{minipage}
%\hspace{5mm}
\begin{minipage}{.49\textwidth}
\centering
\scalemath{.7}{
\begin{tikzpicture}
\draw[step=1cm] (-5,2) grid (0,3);
\draw[step=1cm] (-4,1) grid (1,2);
\draw[step=1cm] (-3,0) grid (2,1);
\draw[step=1cm] (-2,-1) grid (3,0);
\draw[step=1cm] (-1,-2) grid (4,-1);
\draw[step=1cm] (0,-3) grid (5,-2);
\draw (0,-3)--(6,-3);
\draw (-6,3)--(0,3);

\Yboxdim{2mm}
\node at (-5.8,3.2) {$\emptyset$}; 
\node at (-4.8,3.2) {$\yng(1)$};
\node at (-3.8,3.25) {$\yng(1,1)$};
\node at (-2.7,3.35) {$\young(:~,:~,~)$};
\node at (-1.7,3.35) {$\young(:~,:\empty,~)$};
\node at (0-.7,3.2) {$\yng(1)$};
\node at (0.2,3.2) {$\emptyset$};
%\node at (-3.5,2.5) {X};
\node at (-4.8,2.2) {$\emptyset$}; 
\node at (-3.8,2.2) {$\yng(1)$};
\node at (-2.75,2.35) {$\young(:~,:\empty,~)$};
\node at (-1.7,2.35) {$\young(:~,~,~)$};
\node at (-.725,2.35) {$\young(:~,:\empty,~)$};
\node at (0.175,2.35) {$\young(:\empty,:\empty,~)$};
\node at (1.2,2.2) {$\emptyset$};
%\node at (-2.4,1.5) {X};
\node at (-3.8,1.2) {$\emptyset$};  
\node at (-2.8,1.35) {$\young(:\empty,:\empty,~)$};
\node at (-1.725,1.35) {$\young(:\empty,~,~)$};
\node at (-.75,1.35) {$\young(:\empty,:\empty,~)$};
\node at (0.25,1.35) {$\young(:\empty,~,~)$};
\node at (1.2,1.35) {$\young(:\empty,:\empty,~)$};
\node at (2.2,1.2) {$\emptyset$};
%\node at (.65,.5) {X};
\node at (-2.8,.2) {$\emptyset$};  
\node at (-1.8,.35) {$\young(:\empty,:\empty,\empty)$};
\node at (-.75,.35) {$\young(:~,:\empty,~)$};
\node at (0.3,.35) {$\young(:~,~,~)$};
\node at (1.25,.35) {$\young(:~,:\empty,~)$};
\node at (2.2,.2) {$\yng(1)$};
\node at (3.2,.2) {$\emptyset$};
%\node at (1.5,-.5) {X};
\node at (-1.8,-.8) {$\emptyset$};  
\node at (-.8,-.8) {$\yng(1)$};
\node at (0.25,-.65) {$\young(:~,:\empty,~)$};
\node at (1.25,-.65) {$\young(:~,:~,~)$};
\node at (2.2,-.75) {$\yng(1,1)$};
\node at (3.2,-.8) {\yng(1)};
\node at (4.2,-.8) {$\emptyset$};
%\node at (-.5,-1.5) {X};
\node at (-.8,-1.8) {$\emptyset$};  
\node at (0.2,-1.65) {$\young(:\empty,:\empty,~)$};
\node at (1.275,-1.65) {$\young(:~,:\empty,~)$};
\node at (2.2,-1.8) {$\yng(1)$};
\node at (3.275,-1.65) {$\young(:~,:\empty,~)$};
\node at (4.175,-1.65) {$\young(:\empty,:\empty,~)$};
\node at (5.2,-1.8) {$\emptyset$};
%\node at (4.6,-2.5) {X};
\node at (.2,-2.8) {$\emptyset$};
\node at (1.2,-2.8) {\yng(1)};
\node at (2.2,-2.75) {\yng(1,1)};
\node at (3.3,-2.65) {$\young(:~,:~,~)$};
\node at (4.3,-2.65) {$\young(:~,:\empty,~)$};
\node at (5.2,-2.8) {\yng(1)};
\node at (6.2,-2.8) {$\emptyset$};
\end{tikzpicture}}
\caption{An affine growth diagram of type $\vec\lambda=(\omega_1,\omega_1,\omega_1^*,\omega_1^*,\omega_1,\omega_1^*)$ \label{first example}}
\end{minipage}
\end{figure}

\begin{Def}
An \textbf{affine growth diagram of type} $\boldsymbol{\vec\lambda}$ is a labelling of the remaining vertices by dominant weights such that the local rule is satisfied by every unit square.
\end{Def}

Oriented extroverted triangulations correspond to paths through the staircase diagram from the left diagonal to the right diagonal as follows. An oriented extroverted triangulation can be specified by first choosing an external edge and orienting it clockwise (R) or counterclockwise (L). Then attach a doubly extroverted triangle to this edge either to the right (R) or to the left (L), orienting the internal edge of this triangle accordingly. Continue shelling on triangles to the right (R) or to the left (L) until a second doubly extroverted triangle is reached, completing the triangulation. In terms of paths through the staircase diagram, an R corresponds to a horizontal step, while an L corresponds to a vertical step. For example, the fan triangulation for $n=6$ is encoded as $(1$R$)($RRRR$)$, and corresponds to the first line of the staircase diagram. The choice of the first external edge in the triangulation determines the row of the terminal vertex of the first step through the staircase (see Figure \ref{triangulations to paths}). 

\begin{figure}
\centering
\scalemath{.5}{
\begin{tikzpicture}
\draw[step=1cm] (-5,3) grid (0,2);
\draw[step=1cm] (-4,2) grid (1,1);
\draw[step=1cm] (-3,1) grid (2,0);
\draw[step=1cm] (-2,0) grid (3,-1);
\draw[step=1cm] (-1,-1) grid (4,-2);
\draw[step=1cm] (0,-2) grid (5,-3);

\Yboxdim{2mm}
\draw (0,-3)--(6,-3);
\draw (-6,3)--(0,3);

\node at (4,2){
\scalemath{1}{
\begin{tikzpicture}[decoration={markings,mark=at position 0.6 with {\arrow{>}}}] 
\newdimen\R
   \R=1cm
   \draw (0:\R) \foreach \x in {60,120,...,360} {  -- (\x:\R) };
   \draw[postaction={decorate},ultra thick] (180:\R) -- (120:\R);
   \draw[postaction={decorate},ultra thick] (180:\R) -- (60:\R);
   \draw[postaction={decorate},ultra thick] (180:\R) -- (0:\R);
   \draw[postaction={decorate},ultra thick] (180:\R) -- (-120:\R);
   \draw[postaction={decorate},ultra thick] (180:\R) -- (-60:\R);
\end{tikzpicture}}
};

\node at (-4,-2){
(1R)(RRRR)
};
\draw[black, ultra thick] (-6,3) -- (0,3);
\end{tikzpicture}}
\scalemath{.5}{
\begin{tikzpicture}
\draw[step=1cm] (-5,3) grid (0,2);
\draw[step=1cm] (-4,2) grid (1,1);
\draw[step=1cm] (-3,1) grid (2,0);
\draw[step=1cm] (-2,0) grid (3,-1);
\draw[step=1cm] (-1,-1) grid (4,-2);
\draw[step=1cm] (0,-2) grid (5,-3);

\Yboxdim{2mm}
\draw (0,-3)--(6,-3);
\draw (-6,3)--(0,3);

\node at (4,2){
\scalemath{1}{
\begin{tikzpicture}[decoration={markings,mark=at position 0.6 with {\arrow{>}}}] 
\newdimen\R
   \R=1cm
   \draw (0:\R) \foreach \x in {60,120,...,360} {  -- (\x:\R) };
   \draw[postaction={decorate},ultra thick] (60:\R) -- (0:\R);
   \draw[postaction={decorate},ultra thick] (60:\R) -- (-60:\R);
   \draw[postaction={decorate},ultra thick] (60:\R) -- (-120:\R);
   \draw[postaction={decorate},ultra thick] (120:\R) -- (-120:\R);
   \draw[postaction={decorate},ultra thick] (180:\R) -- (-120:\R);
\end{tikzpicture}}
};

\node at (-4,-2){
(3R)(RRLL)
};
\draw[black, ultra thick] (-4,1) -- (-1,1) -- (-1,3) -- (0,3);
\end{tikzpicture}}
\caption{Extroverted triangulations correspond to paths through staircase diagram.\label{triangulations to paths}}
\end{figure}

Fix a sequence of minuscule weights $\vec\lambda=(\lambda^1,\ldots,\lambda^n)$ and fix a component $X$ of Poly$(\vec\lambda)$. Let $\tau$ be an oriented extroverted triangulation with corresponding path $p_\tau$ through the staircase. Label the vertices along the path $p_\tau$ with the corresponding generic distances of $X$. Since, by Proposition \ref{generalized path}, the component $X$ is determined by these generic distances along the interior edges of $\tau$, repeated application of the local rule fills in the entire diagram, thereby determining all of the generic distances. See Figure \ref{first example} where the affine growth diagram can be filled in by starting from the vertex labels along one of the paths in Figure \ref{triangulations to paths}, or by starting from any other path from diagonal to diagonal.

\begin{thm}
Affine growth diagrams of type $\vec\lambda$ are in bijection with the components of Poly$(\vec\lambda)$.
\end{thm}

In particular, if we start with a minuscule path, i.e.\ the generic distances of a component along the fan triangulation, then this is the top line of the affine growth diagram. The next line of the growth diagram represents the neighboring fan triangulation. In other words, repeated application of the local rule to determine this second line rotates the fan triangulation. The $i$th line represents the distances along the $i$th fan triangulation. Rotating $n$ times gets back to the initial fan triangulation, hence establishes the following proposition, which is not combinatorially obvious from the local growth rule alone.

\begin{prop}
Affine growth diagrams are $n$ periodic.
\end{prop}

\section{Classical Bijections}
Suppose that the $\lambda^i$ are either $\omega_1$ or $\omega_1^*$. In this case, there is a natural way to mark affine growth diagrams. Consider again starting with a minuscule path, the top line of an affine growth diagram, and determining the second line from repeated applications of the local rule (see Figure \ref{first line}). Suppose that $\lambda^1=\omega_1$, so that $\mu^1=\omega_1$ as well. To compute $\nu^1$, the local rule says to subtract one from the first row of $\mu^2$ and sort to get $\nu^1$. This can happen in two ways. If $\mu^2$ had a box in the first row, then $\nu^1$ has one fewer positive box than $\mu^2$. If $\mu^2$ did not have a box in the first row, then $\nu^1$ has one more negative box than $\mu^2$. In other words, going from $\mu^2$ to $\nu^{1}$ either removes a positive box, or adds a negative box. This relation between $\mu^{i}$ and $\nu^{i-1}$ continues to hold for all $i$. Since $\mu^1=(1,0,\ldots,0)$, $\nu^0=\emptyset$ and $\mu^6=\emptyset$, $\nu^5=(0,\ldots,0,-1)$, the differences between $\mu^i$ and $\nu^{i-1}$ begin by being a removal of a positive box and finish by being an addition of a negative box, with a switch occurring somewhere in the middle. This switch is guaranteed to happen exactly once if $m\geq n/2$. Mark the unit square across which this switch occurs (see Figure \ref{marking}). 

\begin{figure}
\begin{minipage}{.49\textwidth}
\centering
\scalemath{.5}{
\begin{tikzpicture}
\draw[step=2cm,gray] (0,0) grid (14,2);
\fill[white] (-.1,-.1) rectangle (1.98,1.98);
\fill[white] (12.02,.02) rectangle (14.1,2.1);

\node at (-.2,2.2) {\scalemath{1.25}{\mu^0=\emptyset}};
\node at (2,2.3) {\scalemath{1.25}{\mu^1= {\tiny\yng(1)}}};
\node at (4.2,2.2) {\scalemath{1.25}{\mu^2}};
\node at (6.2,2.2) {\scalemath{1.25}{\mu^3}};
\node at (8.2,2.2) {\scalemath{1.25}{\mu^4}};
\node at (10.2,2.2) {\scalemath{1.25}{\mu^5}};
\node at (12.7,2.2) {\scalemath{1.25}{\mu^6=\emptyset}};
\node at (2.2,.25) {\scalemath{1.25}{\nu^0=\emptyset}};
\node at (4.2,.25) {\scalemath{1.25}{\nu^1}};
\node at (6.2,.25) {\scalemath{1.25}{\nu^2}};
\node at (8.2,.25) {\scalemath{1.25}{\nu^3}};
\node at (10.2,.25) {\scalemath{1.25}{\nu^4}};
\node at (12.7,.25) {\scalemath{1.25}{\nu^5={\tiny\yng(1)}^*}};
\node at (14.75,.25) {\scalemath{1.25}{\nu^6=\emptyset}};

\end{tikzpicture}
}
\caption{First row of a diagram.{\label{first line}}}
\end{minipage}
\hspace{-10mm}
\begin{minipage}{.60\textwidth}
\centering
\scalemath{.5}{
\noindent\makebox[\textwidth]{
\begin{tikzpicture}
\draw[step=2cm,gray] (0,0) grid (14,2);
\fill[white] (-.1,-.1) rectangle (1.98,1.98);
\fill[white] (12.02,.02) rectangle (14.1,2.1);

\node at (.3,2.26) {$\emptyset$};
\node at (2.3,2.26) {{\tiny$\yng(1)$}};
\node at (4.3,2.4) {{\tiny$\yng(1,1)$}};
\node at (6.4,2.4) {{\tiny$\yng(2,1)$}};
\node at (8.3,2.26) {{\tiny$\yng(2)$}};
\node at (10.3,2.26) {{\tiny$\yng(1)$}};
\node at (12.3,2.26) {$\emptyset$};

\node at (2.3,.26) {$\emptyset$};
\node at (4.3,.26) {{\tiny$\yng(1)$}};
\node at (6.4,.26) {{\tiny$\yng(2)$}};
\node at (8.5,.54) {{\tiny$\young(:~~,:\empty,~)$}};
\node at (10.4,.54) {{\tiny$\young(:~,:\empty,~)$}};
\node at (12.3,.54) {{\tiny$\young(:\empty,:\empty,~)$}};
\node at (14.3,.26) {$\emptyset$};

\node at (7,1) {X};

\end{tikzpicture}
}
}
\caption{Example of marking the first row. \label{marking}}
\end{minipage}
\end{figure}

\begin{prop}
If for all $i$, $\lambda^i=\omega_1$ or $\omega_1^*$, and $m\geq n/2$, then any affine growth diagram of type $\vec\lambda$ has exactly one mark in every row and column, hence gives an affine permutation.
\end{prop} 

The group $\widetilde{S_n}$ of \textbf{affine permutations} consists of bijections $f:\mathbb Z\rightarrow \mathbb Z$ such that $f(x+n)=f(x)+n$. Such an $f$ is uniquely determined by its values on $[n]$, written as $[f(1),\cdots, f(n)]$, and called the \textbf{window} of $f$. From an affine permutation we obtain a classical permutation by reducing the values $[f(1),\cdots, f(n)]$ mod $n$. In this way, the marked squares of an affine growth diagram give a classical permutation. For example, the left diagram of Figure $\ref{oscillating examples}$ has permutation $341265$. This can be seen by moving the right, blue triangle underneath the left, red triangle to get a permutation matrix.

\begin{prop}
The permutation associated to an affine growth diagram is a fixed-point-free involution.
\end{prop}

A subset of minuscule paths are the oscillating tableaux, i.e.\ those consisting of weights that don't have negative parts. An \textbf{oscillating tableau} of length $n$ is a sequence of partitions $(\mu^0=\emptyset,\mu^1,\ldots,\mu^{n-1},\mu^n=\emptyset)$ that differ by a box at each step. Starting with a fixed-point-free involution written into an empty staircase diagram, we give a procedure to determine the weight label at each vertex such that the result is an affine growth diagram with first row an oscillating tableaux. This procedure induces a bijection, thereby rediscovering the following Stanley--Sundaram bijection \cite{Sun}.

\begin{prop}[\cite{Sun}]
For $n$ even there is a bijection between fixed-point-free involutions of $S_{n}$ and length $n$ oscillating tableaux.
\end{prop}

Suppose $n$ is even and let $\vec\lambda=(\omega_1,\ldots,\omega_1,\omega_1^*,\ldots,\omega_1^*)$, a sequence of $n/2$ many $\omega_1$'s followed by $n/2$ many $\omega_1^*$'s. In this case, all minuscule paths are oscillating tableaux $(\mu^0=\emptyset,\mu^1,\ldots,\mu^{n-1},\mu^n=\emptyset)$ that consist of partitions that increase from $\emptyset$ to $\mu^{n/2}$ and then decrease back down to $\emptyset$. This is the same information as a pair of same-shape standard Young tableaux. In this case, the markings of the corresponding affine growth diagrams are contained in the squares indicated in Figure \ref{full RS}. They give a permutation in $S_{n/2}$, and this is the (transpose) RS correspondence. Furthermore, we recover Fomin growth diagrams.

\begin{thm}
The positive parts of the weights of the left, red, indicated square of Figure \ref{full RS} give a Fomin growth diagram to the northwest, and the negative parts give a Fomin growth diagram to the southeast. The analogous statement holds for the right, blue squares.
\end{thm}

Hence, the RS bijection is contained in the Stanley--Sundaram bijection. In particular, starting with an oscillating tableau $\vec\mu$ that encodes the pair of same-shape standard Young tableaux $(P,Q)$, suppose that $(P,Q)$ corresponds to the permutation $\sigma=
   \bigl(\begin{smallmatrix}
    1 & 2 & 3 & \cdots & k-1 & k \\        
    \sigma_1 & \sigma_2 & \sigma_3 & \cdots &  \sigma_{k-1}  & \sigma_k
  \end{smallmatrix}\bigr)$ under the RS correspondence. Then under Stanley--Sundaram, $\vec\mu$ corresponds to the fixed point-free-involution
$
   \pi=\bigl(\begin{smallmatrix}
    \sigma_1 & \sigma_2 & \cdots & \sigma_{k-1} & \sigma_k & k+1 & k+2 & \cdots & 2k-1 & 2k \\        
    2k & 2k-1 & \cdots & k+2 & k+1 &\sigma_k & \sigma_{k-1} & \cdots &  \sigma_{2}  & \sigma_1
  \end{smallmatrix}\bigr). \label{involution}
$

\begin{figure}
\begin{minipage}{.49\textwidth}
\scalemath{.7}{
\begin{tikzpicture}
\draw[step=1cm,gray] (-6,-3) grid (6,3);
\fill[white] (-6,-3.1) rectangle (0,-2);
\fill[white] (-6.1,-3.1) rectangle (-1.0001,-1.0001);
\fill[white] (-6,-3) rectangle (-2,0);
\fill[white] (-6,-3) rectangle (-3,1);
\fill[white] (-6,-3) rectangle (-4,2);
\fill[white] (-6.1,-3.1) rectangle (-5,3);
\fill[white] (6,3.1) rectangle (0,2);
\fill[white] (6.1,3.1) rectangle (1.0001,1.0001);
\fill[white] (6,3) rectangle (2,0);
\fill[white] (6,3) rectangle (3,-1);
\fill[white] (6,3) rectangle (4,-2);
\fill[white] (6.1,3.1) rectangle (5,-3);

\draw[blue,ultra thick] (0,0)--(0,-3)--(3,-3)--(3,0)--cycle;
\draw[red,ultra thick] (0,0)--(0,3)--(-3,3)--(-3,0)--cycle;
%\Ylinethick1.5pt
\Yboxdim{2mm}
\node at (6.2,-2.8) {$\scalemath{1}{\emptyset}$};
\node at (5.2,-2.8) {\yng(1)};
\node at (4.3,-2.8) {\yng(2)};
\node at (3.3,-2.8) {\yng(3)};
\node at (2.3,-2.8) {\yng(2)};
\node at (1.2,-2.8) {\yng(1)};
\node at (.2,-2.8) {$\scalemath{1}{\emptyset}$};
\node at (.6,-2.5) {\scalemath{1}{\text{X}}};
\node at (5.2,-1.8) {$\scalemath{1}{\emptyset}$};
\node at (4.2,-1.8) {\yng(1)};
\node at (3.3,-1.8) {\yng(2)};
\node at (2.3,-1.8) {\yng(1)};
\node at (1.3,-1.8) {$\scalemath{1}{\emptyset}$};
\node at (0.2,-1.5) {$\young(:\empty,:\empty,:\empty,\empty)$};
\node at (-.8,-1.8) {$\scalemath{1}{\emptyset}$};  
\node at (1.6,-1.5) {\scalemath{1}{\text{X}}};
\node at (4.2,-.8) {$\scalemath{1}{\emptyset}$};
\node at (3.2,-.8) {\yng(1)};
\node at (2.3,-.8) {$\scalemath{1}{\emptyset}$};
\node at (1.3,-.5) {$\young(:\empty,:\empty,:\empty,\empty)$};
\node at (0.3,-.5) {$\young(::\empty,::\empty,::\empty,~~)$};
\node at (-.8,-.5) {$\young(:\empty,:\empty,:\empty,\empty)$};
\node at (-1.8,-.8) {$\scalemath{1}{\emptyset}$};  
\node at (2.6,-.5) {\scalemath{1}{\text{X}}};
\node at (3.2,.2) {$\scalemath{1}{\emptyset}$};
\node at (2.2,.5) {$\young(:\empty,:\empty,:\empty,~)$};
\node at (1.3,.5) {$\young(::\empty,::\empty,::\empty,~~)$};
\node at (0.325,.5) {$\young(:::\empty,:::\empty,:::\empty,~~~)$};
\node at (-.7,.5) {$\young(::\empty,::\empty,::\empty,~~)$};
\node at (-1.8,.5) {$\young(:\empty,:\empty,:\empty,\empty)$};
\node at (-2.8,.2) {$\scalemath{1}{\emptyset}$};  
\node at (-2.5,.5) {\scalemath{1}{\text{X}}};
%\node[circle,fill,inner sep=2pt] at (0,0) {};
\node at (2.2,1.2) {$\scalemath{1}{\emptyset}$};
\node at (1.2,1.5) {$\young(:\empty,:\empty,:\empty,~)$};
\node at (0.3,1.5) {$\young(::\empty,::\empty,::\empty,~~)$};
\node at (-.7,1.5) {$\young(:\empty,:\empty,:\empty,~)$};
\node at (-1.7,1.2) {$\scalemath{1}{\emptyset}$};
\node at (-2.8,1.2) {$\yng(1)$};
\node at (-3.8,1.2) {$\scalemath{1}{\emptyset}$};  
\node at (-1.5,1.5) {\scalemath{1}{\text{X}}};
\node at (1.2,2.2) {$\scalemath{1}{\emptyset}$};
\node at (0.2,2.5) {$\young(:\empty,:\empty,:\empty,~)$};
\node at (-.7,2.2) {$\scalemath{1}{\emptyset}$};
\node at (-1.7,2.2) {$\yng(1)$};
\node at (-2.7,2.2) {$\yng(2)$};
\node at (-3.8,2.2) {$\yng(1)$};
\node at (-4.8,2.2) {$\scalemath{1}{\emptyset}$}; 
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\node at (-3.8,3.25) {$\yng(1,1)$};
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\node at (-4,-1) {$\sigma=321$};
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\node at (-4,-1) {$\sigma=132$};
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\end{minipage}
\begin{minipage}{.49\textwidth}
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\node at (-.8,-1.8) {$\scalemath{1}{\emptyset}$};  
\node at (0.2,-1.65) {$\young(:\empty,:\empty,~)$};
\node at (1.275,-1.65) {$\young(:~,:\empty,~)$};
\node at (2.325,-1.65) {$\young(:~~,:\empty,~)$};
\node at (3.275,-1.825) {\yng(2)};
\node at (4.175,-1.825) {\yng(1)};
\node at (5.2,-1.8) {$\scalemath{1}{\emptyset}$};
\node at (2.5,-2.5) {\scalemath{1}{\text{X}}};
\node at (.2,-2.8) {$\scalemath{1}{\emptyset}$};
\node at (1.2,-2.8) {\yng(1)};
\node at (2.3,-2.8) {\yng(2)};
\node at (3.3,-2.775) {\yng(2,1)};
\node at (4.3,-2.775) {\yng(1,1)};
\node at (5.2,-2.8) {\yng(1)};
\node at (6.2,-2.8) {$\scalemath{1}{\emptyset}$};
\draw (0,-3)--(6,-3);
\draw (-6,3)--(0,3);

\node at (-4,-1) {$\sigma=312$};
\node at (4,2) {$\pi=546213$};
\end{tikzpicture}
}
\end{minipage}
\caption{The six affine growth diagrams of the RS correspondence. \label{full RS}}
\end{figure}

\begin{figure}
\begin{minipage}{.5\textwidth}
%\centering
\scalemath{.7}{
\begin{tikzpicture}
\draw[step=1cm] (-5,2) grid (0,3);
\draw[step=1cm] (-4,1) grid (1,2);
\draw[step=1cm] (-3,0) grid (2,1);
\draw[step=1cm] (-2,-1) grid (3,0);
\draw[step=1cm] (-1,-2) grid (4,-1);
\draw[step=1cm] (0,-3) grid (5,-2);
\draw (0,-3)--(6,-3);
\draw (-6,3)--(0,3);

\Yboxdim{2mm}
\node at (-5.8,3.2) {$\emptyset$}; 
\node at (-4.8,3.2) {$\yng(1)$};
\node at (-3.8,3.25) {$\yng(1,1)$};
\node at (-2.7,3.2) {$\yng(1)$};
\node at (-1.7,3.25) {$\emptyset$};
\node at (0-.7,3.2) {$\yng(1)$};
\node at (0.2,3.2) {$\emptyset$};
\node at (-3.5,2.5) {X};
\node at (-4.8,2.2) {$\emptyset$}; 
\node at (-3.8,2.2) {$\yng(1)$};
\node at (-2.75,2.35) {$\young(:~,:\empty,~)$};
\node at (-1.8,2.35) {$\young(:\empty,:\empty,~)$};
\node at (-.725,2.35) {$\young(:~,:\empty,~)$};
\node at (0.175,2.35) {$\young(:\empty,:\empty,~)$};
\node at (1.2,2.2) {$\emptyset$};
\node at (-2.4,1.5) {X};
\node at (-3.8,1.2) {$\emptyset$};  
\node at (-2.8,1.35) {$\young(:\empty,:\empty,~)$};
\node at (-1.725,1.35) {$\young(:\empty,~,~)$};
\node at (-.675,1.35) {$\young(:~,~,~)$};
\node at (0.25,1.35) {$\young(:\empty,~,~)$};
\node at (1.2,1.35) {$\young(:\empty,:\empty,~)$};
\node at (2.2,1.2) {$\emptyset$};
\node at (.65,.5) {X};
\node at (-2.8,.2) {$\emptyset$};  
\node at (-1.8,.35) {$\young(:\empty,:\empty,\empty)$};
\node at (-.75,.35) {$\young(:~,:\empty,~)$};
\node at (0.3,.35) {$\young(:\empty,:\empty,~)$};
\node at (1.25,.35) {$\young(:~,:\empty,~)$};
\node at (2.2,.2) {$\yng(1)$};
\node at (3.2,.2) {$\emptyset$};
\node at (1.5,-.5) {X};
\node at (-1.8,-.8) {$\emptyset$};  
\node at (-.8,-.8) {$\yng(1)$};
\node at (0.2,-.7) {$\emptyset$};
\node at (1.2,-.8) {$\yng(1)$};
\node at (2.2,-.7) {$\yng(1,1)$};
\node at (3.2,-.8) {\yng(1)};
\node at (4.2,-.8) {$\emptyset$};
\node at (-.5,-1.5) {X};
\node at (-.8,-1.8) {$\emptyset$};  
\node at (0.2,-1.65) {$\young(:\empty,:\empty,~)$};
\node at (1.275,-1.65) {$\young(:~,:\empty,~)$};
\node at (2.325,-1.65) {$\young(:~,:~,~)$};
\node at (3.275,-1.7) {$\young(:~,:\empty,~)$};
\node at (4.175,-1.7) {$\young(:\empty,:\empty,~)$};
\node at (5.2,-1.8) {$\emptyset$};
\node at (4.6,-2.5) {X};
\node at (.2,-2.8) {$\emptyset$};
\node at (1.2,-2.8) {\yng(1)};
\node at (2.2,-2.75) {\yng(1,1)};
\node at (3.3,-2.775) {\yng(1)};
\node at (4.3,-2.775) {$\emptyset$};
\node at (5.2,-2.8) {\yng(1)};
\node at (6.2,-2.8) {$\emptyset$};

\node at (-4.5,-1) {$\pi=341265$};

\draw[red, thick] (-6,3) -- (-.025,3) -- (-.025,-3) -- cycle;
\draw[blue, thick] (6,-3) -- (0,-3) -- (0,3) -- cycle;
\end{tikzpicture}}
\end{minipage}
\begin{minipage}{.5\textwidth}
%\centering
\scalemath{.7}{
\begin{tikzpicture}
\draw[step=1cm] (-5,2) grid (0,3);
\draw[step=1cm] (-4,1) grid (1,2);
\draw[step=1cm] (-3,0) grid (2,1);
\draw[step=1cm] (-2,-1) grid (3,0);
\draw[step=1cm] (-1,-2) grid (4,-1);
\draw[step=1cm] (0,-3) grid (5,-2);
\draw (0,-3)--(6,-3);
\draw (-6,3)--(0,3);

\Yboxdim{2mm}
\node at (-5.8,3.2) {$\emptyset$}; 
\node at (-4.8,3.3) {$\yng(1,1)$};
\node at (-3.7,3.35) {$\yng(2,2,1)$};
\node at (-2.7,3.25) {$\yng(2,1)$};
\node at (-1.7,3.25) {$\yng(2,2)$};
\node at (0-.8,3.3) {$\yng(1,1)$};
\node at (0.2,3.2) {$\emptyset$};
%\node at (-1.5,2.5) {X};
\node at (-4.8,2.2) {$\emptyset$}; 
\node at (-3.8,2.35) {$\yng(1,1,1)$};
\node at (-2.8,2.2) {$\yng(1)$};
\node at (-1.8,2.25) {$\yng(1,1)$};
\node at (-.725,2.25) {$\emptyset$};
\node at (0.175,2.45) {$\young(:\empty,:\empty,~,~)$};
\node at (1.2,2.2) {$\emptyset$};
%\node at (-2.4,1.5) {X};
\node at (-3.8,1.2) {$\emptyset$};  
\node at (-2.85,1.45) {$\young(:\empty,:\empty,~,~)$};
\node at (-1.725,1.45) {$\young(:~,:\empty,~,~)$};
\node at (-.8,1.5) {$\young(:\empty,:\empty,~,~,~)$};
\node at (0.25,1.5) {$\young(::\empty,::\empty,:~,~~,~~)$};
\node at (1.2,1.5) {$\young(:\empty,:\empty,~,~,~)$};
\node at (2.2,1.2) {$\emptyset$};
%\node at (-.35,.5) {X};
\node at (-2.8,.2) {$\emptyset$};  
\node at (-1.8,.25) {$\yng(1)$};
\node at (-.8,.35) {$\young(:\empty,:\empty,\empty)$};
\node at (0.3,.35) {$\young(::\empty,:~,~~)$};
\node at (1.2,.35) {$\young(:\empty,:\empty,\empty)$};
\node at (2.2,.3) {$\yng(1,1)$};
\node at (3.2,.2) {$\emptyset$};
%\node at (1.5,-.5) {X};
\node at (-1.8,-.8) {$\emptyset$};  
\node at (-.8,-.55) {$\young(:\empty,:\empty,~,~)$};
\node at (0.25,-.55) {$\young(::\empty,::\empty,~~,~~)$};
\node at (1.2,-.55) {$\young(:\empty,:\empty,~,~)$};
\node at (2.3,-.55) {$\young(:~,:~,:\empty,~)$};
\node at (3.2,-.65) {$\young(:\empty,:\empty,\empty)$};
\node at (4.2,-.8) {$\emptyset$};
%\node at (.5,-1.5) {X};
\node at (-.8,-1.8) {$\emptyset$};  
\node at (0.2,-1.55) {$\young(:\empty,:\empty,~,~)$};
\node at (1.275,-1.75) {$\emptyset$};
\node at (2.25,-1.65) {$\yng(1,1,1)$};
\node at (3.25,-1.825) {\yng(1)};
\node at (4.175,-1.75) {\yng(1,1)};
\node at (5.2,-1.8) {$\emptyset$};
%\node at (2.5,-2.5) {X};
\node at (.2,-2.8) {$\emptyset$};
\node at (1.2,-2.75) {\yng(1,1)};
\node at (2.3,-2.65) {\yng(2,2,1)};
\node at (3.3,-2.775) {\yng(2,1)};
\node at (4.3,-2.775) {\yng(2,2)};
\node at (5.2,-2.75) {\yng(1,1)};
\node at (6.2,-2.8) {$\emptyset$};
%\node[circle,fill,inner sep=2pt] at (0,0) {}; %center dot
\draw (0,-3)--(6,-3);
\draw (-6,3)--(0,3);

\node at (-.4,2.6) {$2$};
\node at (-3.4,1.6) {$2$};
\node at (-1.4,1.4) {$1$};
\node at (-1.4,-.4) {$1$};

\node at (.6,-2.5) {$2$};
\node at (3.6,-1.5) {$1$};
\node at (1.6,-1.5) {$1$};
\node at (1.6,.5) {$2$};

\draw[red, thick] (-6,3) -- (-.025,3) -- (-.025,-3) -- cycle;
\draw[blue, thick] (6,-3) -- (0,-3) -- (0,3) -- cycle;

\end{tikzpicture}}
\end{minipage}
\caption{Standard and semistandard oscillating tableau examples. \label{oscillating examples}}
\end{figure}


\subsection{Knuth Version}
In the previous subsection the $\lambda^i$ were assumed to be $\omega_1$ or $\omega_1^*$. Dropping this assumption, affine growth diagrams can now be marked with natural numbers, the number of boxes that perform a ``switch'' across a unit square (see the right diagram of Figure \ref{oscillating examples}). The result is an $n\times n$ symmetric matrix such that for all $i$ either $r_i=0$ or $c_i=0$, where $r_i$, respectively $c_i$, denotes the sum of the entries of the $i$th row, respectively column, up to and including the main diagonal entry. 

In \cite{Rob}, Roby extended the Stanley--Sundaram bijection to semistandard oscillating tableaux. A \textbf{semistandard oscillating tableau} of length $n$ is a sequence of partitions $(\emptyset,\mu^1,\ldots,\mu^{n-1},\emptyset)$ such that for all $i$, $\mu^i$ and $\mu^{i-1}$ differ by a vertical strip. As before, write such a symmetric natural-number matrix into an empty affine growth diagram. There is a procedure to determine the weight label at each vertex such that the result is an affine growth diagram with the first row a semistandard oscillating tableau. Again, this induces a bijection, thereby rediscovering the Stanley--Sundaram/Roby bijection. 

\begin{prop}
There is a bijection between natural-number symmetric matrices such that for all $i$ either $r_i=0$ or $c_i=0$ and semistandard oscillating tableaux. Furthermore, $c_i$ is the number of boxes added in the $i$th step (going from $\mu^{i-1}$ to $\mu^i$), and $r_i$ is the number of boxes removed.
\end{prop}

In the case when $\vec\lambda=(\omega_{i_1},\ldots,\omega_{i_k},\omega_{i_{k+1}}^*,\ldots,\omega_{i_{k+l}}^*)$, a minuscule path is a semistandard oscillating tableau that grows for $k$ steps to $\mu^k$ and then decreases for $l$ steps back to the empty partition. This is the same information as a pair of (row-strict) same-shape semistandard Young tableaux with shape $\mu^k$. Knuth \cite{Knu} extended the Robinson--Schensted correspondence to give a bijection between natural-number matrices and pairs of same-shape semistandard Young tableaux. Roby \cite{Rob} showed how to see the full RSK correspondence in terms of Fomin growth diagrams. As before, the (transpose) RSK bijection lives inside of the Stanley--Sundaram/Roby bijection.

\begin{thm}
If $\vec\lambda$ is as described in the previous paragraph, then the positive parts of the weights of the $k\times l$ rectangle inside of an affine growth diagram is a Fomin--Roby growth diagram to the northwest, and the negative parts give a Fomin--Roby growth diagram to the southeast.
\end{thm}

\section{Promotion and Evacuation}

We can rephrase everything in terms of dominant $SL_m$ weights (see Figure \ref{prom and evac}). In this case a minuscule path begins at the zero weight and grows to a rectangle. Fontaine and Kamnitzer showed in \cite{FK} that, in this case, rotation of minuscule paths gives promotion of rectangular tableaux. From different geometric considerations, such diagrams were also considered by Speyer \cite{Spe} in the special case that $\lambda^i=\omega_1$ for all $i$.

Let $\partial$ denote the promotion operation on tableaux computed via deflation, and $\partial^*$ the inverse promotion operation computed via inflation, so that $\partial\circ \partial^*=$ id $=\partial^*\circ\partial$. Suppose that tableau $T$ labels line $i$ of the affine growth diagram. Then line $i+1$ is labeled by $\partial(T)$ and line $i-1$ by $\partial^*(T)$. The $n$-periodicity of the affine growth diagram recovers the $n$-periodicity of promotion on rectangular tableaux, $\partial^n(T)=T=(\partial^*)^n(T)$. Likewise, traversing successive vertical lines to the left is given by promotion and to the right by inverse promotion. Furthermore, the vertical line that begins where horizontal line $i$ ends is labeled by the Sch\"utzenberger evacuation tableau $ev(T)$. The $n+i$ horizontal line is labeled by the dual evacuation procedure applied to $ev(T)$, which recovers the identity $ev^*\circ ev(T)=\partial^n(T)$, and the fact that $ev^*=ev$ in the rectangular case. We can also recover the identity $ev\circ \partial = \partial^*\circ ev$. These results are summarized in Figure \ref{prom and evac}.

\begin{figure}[h]
\centering
\scalemath{.5}{
\begin{tikzpicture}
\draw[step=1cm] (-6,-3) grid (6,3);
\fill[white] (-6.1,-3.1) rectangle (-.01,-2.01);
\fill[white] (-6.1,-3.1) rectangle (-1.01,-1.01);
\fill[white] (-6.1,-3.1) rectangle (-2.01,-.01);
\fill[white] (-6.1,-3.1) rectangle (-3.01,.99);
\fill[white] (-6.1,-3.1) rectangle (-4.01,1.99);
\fill[white] (-6.1,-3.1) rectangle (-5.01,2.99);
 \fill[white] (6.1,3.1) rectangle (.01,2.01);
\fill[white] (6.1,3.1) rectangle (1.01,1.01);
\fill[white] (6.1,3.1) rectangle (2.01,.01);
\fill[white] (6.1,3.1) rectangle (3.01,-.99);
\fill[white] (6.1,3.1) rectangle (4.01,-1.99);
\fill[white] (6.1,3.1) rectangle (5.01,-2.99);

\Yboxdim{2mm}
\node at (.5,4.5) {$ev(T)$};
\node at (.25,4) {$\downarrow$};
\node at (2.75,3.5) {$\partial^*\circ ev(T)=ev\circ\partial(T)$};
\node at (1.25,3) {$\downarrow$};
\node at (-6.5,3) {$T\rightarrow$};
\node at (-5.75,2) {$\partial(T)\rightarrow$};
\node at (-2,-2) {$\partial^*(T)\rightarrow$};
\node at (-2.5,-3) {$ev^*\circ ev(T)=\partial^6(T)=T\rightarrow$};
\node at (-5.8,3.2) {$\emptyset$}; 
\node at (-4.8,3.2) {$\yng(1)$};
\node at (-3.7,3.2) {$\yng(2)$};
\node at (-2.7,3.25) {$\yng(2,1)$};
\node at (-1.7,3.35) {$\yng(2,2,1)$};
\node at (0-.6,3.35) {$\yng(3,2,2)$};
\node at (0.3,3.35) {$\yng(3,3,3)$};
\node at (-4.8,2.2) {$\emptyset$}; 
\node at (-3.8,2.2) {$\yng(1)$};
\node at (-2.8,2.25) {$\yng(1,1)$};
\node at (-1.7,2.35) {$\yng(2,1,1)$};
\node at (-.6,2.35) {$\yng(3,2,1)$};
\node at (0.35,2.35) {$\yng(3,3,2)$};
\node at (1.35,2.35) {$\yng(3,3,3)$};
\node at (-3.8,1.2) {$\emptyset$};  
\node at (-2.8,1.2) {$\yng(1)$};
\node at (-1.725,1.275) {$\yng(2,1)$};
\node at (-.675,1.275) {$\yng(3,2)$};
\node at (0.35,1.35) {$\yng(3,3,1)$};
\node at (1.35,1.35) {$\yng(3,3,2)$};
\node at (2.35,1.35) {$\yng(3,3,3)$};
\node at (-2.8,.2) {$\emptyset$};  
\node at (-1.8,.275) {$\yng(1,1)$};
\node at (-.75,.275) {$\yng(2,2)$};
\node at (0.35,.35) {$\yng(3,2,1)$};
\node at (1.35,.35) {$\yng(3,2,2)$};
\node at (2.35,.35) {$\yng(3,3,2)$};
\node at (3.35,.35) {$\yng(3,3,3)$};
\node at (-1.8,-.8) {$\emptyset$};  
\node at (-.8,-.75) {$\yng(1,1)$};
\node at (0.3,-.65) {$\yng(2,1,1)$};
\node at (1.3,-.65) {$\yng(2,2,1)$};
\node at (2.35,-.65) {$\yng(3,2,1)$};
\node at (3.35,-.65) {\yng(3,2,2)};
\node at (4.35,-.65) {$\yng(3,3,3)$};
\node at (-.8,-1.8) {$\emptyset$};  
\node at (0.2,-1.75) {$\yng(1,1)$};
\node at (1.275,-1.75) {$\yng(2,1)$};
\node at (2.325,-1.75) {$\yng(3,1)$};
\node at (3.35,-1.65) {\yng(3,1,1)};
\node at (4.35,-1.65) {\yng(3,2,2)};
\node at (5.35,-1.65) {$\yng(3,3,3)$};
\node at (.2,-2.8) {$\emptyset$};
\node at (1.2,-2.8) {\yng(1)};
\node at (2.3,-2.8) {\yng(2)};
\node at (3.3,-2.775) {\yng(2,1)};
\node at (4.3,-2.65) {\yng(2,2,1)};
\node at (5.35,-2.65) {\yng(3,2,2)};
\node at (6.35,-2.65) {$\yng(3,3,3)$};
%\node[circle,fill,inner sep=2pt] at (0,0) {}; %center dot
\draw (0,-3)--(6,-3);
\draw (-6,3)--(0,3);
\end{tikzpicture}}
\caption{An $SL_m$ growth diagram with promotion and evacuation indicated. \label{prom and evac}}
\end{figure}

\acknowledgements{I would like to thank Allen Knutson for suggesting the question and for many helpful discussions, Bruce Westbury for pointing out the occurrence of the local rule in \cite{vLee2}, and Alex Yong and David Speyer for helpful discussions.}

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\end{document}
