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\documentclass[finalversion]{FPSAC2023}
\articlenumber{20}
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\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}

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\usepackage{amsfonts}
\usepackage{booktabs}
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\usepackage{tikz}
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\usepackage{varwidth}
\usepackage{subcaption}
\usepackage{multirow}
\usepackage{rotating}
\usepackage[boxsize=1em]{ytableau}

\usepackage{bbm}



\definecolor{darkred}{rgb}{0.7,0,0}
% Dark red emphasis
\newcommand{\defncolor}{\color{darkred}}
\newcommand{\defn}[1]{{\defncolor\emph{#1}}} % emphasis of a definition


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\newcommand{\Spin}{\mathrm{Spin}}

\DeclareMathOperator{\CARRY}{CARRY}

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\ytableausetup{smalltableaux, aligntableaux=center, textmode}

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\def\vtype{{\overrightarrow{\mathsf{type}}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\DeclareMathOperator{\dom}{dom} % get the dominant weight

\DeclareMathOperator{\Inc}{Inc}
\DeclareMathOperator{\KPro}{K-Pro}
\DeclareMathOperator{\maj}{maj}
\DeclareMathOperator{\Packed}{Pack}
\newcommand{\IncP}{\Inc_{\Packed}}
\DeclareMathOperator{\Pro}{Pro}
\DeclareMathOperator{\rev}{rev}
\DeclareMathOperator{\rot}{rot}
\DeclareMathOperator{\Row}{Row}

\DeclareMathOperator{\SYT}{SYT}
\DeclareMathOperator{\ASM}{\mathcal{ASM}}
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\newcommand{\blank}{\phantom{2}}

\newcommand{\com}[1]{\textcolor{blue}{$[\star$ #1 $\star]$}}


\newcommand{\osc}{\mathsf{O}}
\newcommand{\fan}{\mathsf{F}}
\newcommand{\vac}{\mathsf{V}}

\newcommand{\filling}{\Phi}
\newcommand{\tochord}{\mathsf{cd}}
\newcommand{\Mfan}{\mathsf{M}_{F}}
\newcommand{\Mfantoosc}{\mathsf{M}_{F \rightarrow O}}
\newcommand{\Mvactoosc}{\mathsf{M}_{V \rightarrow O}}
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\newcommand{\spincrystal}{\mathcal{B}_\mathsf{spin}}
\newcommand{\cboxcrystal}{\mathcal{C}_{\square}}
\newcommand{\bboxcrystal}{\mathcal{B}_{\square}}

\newcommand{\blowupNE}{\mathsf{blowup}^{\mathsf{NE}}}
\newcommand{\blowupSE}{\mathsf{blowup}^{\mathsf{SE}}}

%% end PPSS packages and commands %%%



%% define your title in the usual way
\title{Promotion for fans of Dyck paths}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[Pappe, Pfannerer, Schilling, \and Simone]{
Joseph Pappe\thanks{\href{mailto:jhpappe@ucdavis.edu}{jhpappe@ucdavis.edu}.}\addressmark{1}, 
Stephan Pfannerer\thanks{\href{mailto:stephan.pfannerer@tuwien.ac.at}{stephan.pfannerer@tuwien.ac.at}. Recipient of a DOC Fellowship of the Austrian Academy of Sciences.}\addressmark{2}, \\
Anne Schilling\thanks{\href{mailto:anne@math.ucdavis.edu}{anne@math.ucdavis.edu}. Partially supported by NSF grants DMS--1760329 and DMS--2053350.}\addressmark{1}, \and
Mary Claire Simone\thanks{\href{mailto:mcsimone@ucdavis.edu}{mcsimone@ucdavis.edu}.}\addressmark{1}}
%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of California, Davis, CA, USA \\ \addressmark{2}Institut f\"ur Diskrete Mathematik und Geometrie, TU Wien, Austria}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\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{We construct an injection from the set of $r$-fans of Dyck paths of length $n$ into the set of chord diagrams on $[n]$
that intertwines promotion and rotation. This is done in two different ways, namely as fillings of promotion matrices and in terms 
of Fomin growth diagrams. Our analysis uses the fact that $r$-fans of Dyck paths can be viewed as highest weight 
elements of weight zero in crystals of type $B_r$, which in turn can be analyzed using virtual crystals. On the level of 
Fomin growth diagrams, the virtualization process corresponds to the Roby--Krattenthaler blow up construction. 
Our construction generalizes to vacillating tableaux as well.
We give a cyclic sieving phenomenon on $r$-fans of Dyck paths using the promotion action.}

%% 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 bases, promotion, Fomin growth and chord diagrams, Dyck paths}

%% 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
%% but you can use bibtex if you want
\usepackage[backend=bibtex]{biblatex}
\addbibresource{FPSAC2023_growthdiagrams_for_fans.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\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}

Interest in invariant subspaces goes back to Rumer, Teller and Weyl~\cite{Weyl.1932}, who studied the quantum mechanical description of
molecules. In particular, they devised diagrammatic bases for the invariant spaces. For SL$(n)$, a diagrammatic basis for the invariant space
was constructed by Cautis, Kamnitzer and Morrison~\cite{CKM.2014}, generalizing Kuperberg's webs~\cite{Kuperberg.1996} for SL$(2)$ and SL$(3)$.

The dimension of the invariant subspace of a tensor product $V^{\otimes N}$ of an irreducible representation $V$ of a Lie algebra $\mathfrak{g}$
is equal to the number of highest weight elements of weight zero in $\mathcal{B}^{\otimes N}$, where $\mathcal{B}$ is the crystal basis associated
to $V$~\cite{Westbury.2016, PfannererRubeyWestbury.2020}. The symmetric group acts on $V^{\otimes N}$ by permuting tensor positions.
By Schur--Weyl duality, this action commutes with the action of the Lie group. In particular, the symmetric group acts on the invariant space
of $V^{\otimes N}$. It was shown by Westbury~\cite{Westbury.2016} that the action of the long cycle corresponds to the action of promotion
on highest weight elements of weight zero in $\mathcal{B}^{\otimes N}$. In this setting promotion is defined using Henriques' and Kamnitzer's
commutor~\cite{HK.2006}, see~\cite{FK.2014, Westbury.2016, Westbury.2018}. 

In general, it is desirable to have a correspondence between highest weight elements of weight zero in $\mathcal{B}^{\otimes N}$ and diagram bases,
such as chord diagrams, which intertwine promotion and rotation. For Kuperberg's webs~\cite{Kuperberg.1996}, this was achieved by
Petersen, Pylyavskyy, Rhoades~\cite{PPR.2009} and Patrias~\cite{Patrias.2019} by showing that the growth algorithm of Khovanov and
Kuperberg~\cite{KK.1999} intertwines promotion with rotation. For the vector representation of the symplectic group and the adjoint representation
of the general linear group, such a correspondence between highest weight elements of weight zero and chord diagrams which intertwines
promotion and rotation was given in~\cite{PfannererRubeyWestbury.2020}.

In this paper, we construct an injection from the set of $r$-fans of Dyck paths of length $n$ into the set of chord 
diagrams on $[n]$ that intertwines promotion and rotation. There is a natural correspondence between $r$-fans of Dyck paths  
and highest weight elements in the tensor product of the spin crystal of type $B_r$.
We present this injection in two different ways: 
(1) as fillings of promotion matrices~\cite{Lenart.2008} (see \cref{section.promotion diagrams});
(2) in terms of Fomin growth diagrams~\cite{Fomin.1986,Krattenthaler.2006} (see \cref{section.fomin growth}).

While the first description shows that the map intertwines promotion and rotation, the second description shows injectivity.
Our proof strategy uses virtualization of crystals (see for example~\cite{BumpSchilling.2017}) and results of~\cite{PfannererRubeyWestbury.2020}
for oscillating tableaux of weight zero (or equivalently highest weight words of weight zero for the vector representation type $C_r$):
\begin{enumerate}
\item Find a virtual crystal morphism for the spin crystals of type $B_r$ into the $r$-th tensor power of the crystal of the vector representation of 
type $C_r$ (\cref{section.virtual}).
\item Use virtualization to map a fan of Dyck paths to an oscillating tableau (\cref{section.highest weight}).
\item Show that this virtualization commutes with promotion and the filling rules.
\item Show that blowing up the filling of the growth diagram corresponds to the filling of the oscillating tableau. In this sense, the blow up on growth diagrams
is the analogue of the virtualization on crystals.
\end{enumerate}
An overview of our strategy is shown in \cref{figure.overview r fans}. Our analysis also generalizes to vacillating tableaux, details of which appear
in the long version of this paper~\cite{PPSS.2022}.

\begin{figure}
\centering
\scalebox{0.6}{
\begin{tikzpicture}[thick]

	%outer labels
	\draw(-8, 0) node[text width = 2cm]{\textbf{Type B spin}};
	\draw(-8, -4) node[text width = 2cm]{\textbf{Type C vector}};

	%vertices of commuting diagram
	\node[rectangle, draw = black, text width = 1.9cm] (topleft) at (-6,0) {Promotion Fillings};
	\node[rectangle, draw = black, text width = 1.75cm] (topmid) at (0,0) {$r$-fans of Dyck paths};
	\node[rectangle, draw = black, text width = 2.7cm](topright) at (6,0) {Growth Diagram Fillings};
	\node[rectangle, draw = black, text width = 1.9cm](botleft) at (-6,-4) {Promotion Fillings};
	\node[rectangle, draw = black, text width = 2.5cm](botmid) at (0,-4) {Standard oscillating tableaux of weight $0$};
	\node[rectangle, draw = black, text width = 2.7cm](botright) at (6,-4) {Growth Diagram Fillings};

	%edges
	\draw[<-] (topleft)--(topmid) node[midway, above] {Promotion} node[midway, below] {Filling Rule};
	\draw[->] (topmid)--(topright) node[midway, above] {\cref{construction.Burgegrowthrule}};
	\draw[<->] (topleft)--(botleft);
	\draw[right hook->] (topmid)--(botmid);
	\draw[<->] (topright)--(botright);
	\draw[<-] (botleft)--(botmid) node[midway, above] {Promotion} node[midway, below] {Filling Rule};
	\draw[->] (botmid)--(botright) node[midway, above] {Growth Diagram} node[midway, below]{Filling Rule};
	\draw[<->] (topleft) to [out= 45,in=135, looseness = .5] (topright);
	\draw[<->] (botleft) to [out= -45,in=-135, looseness = .5] (botright);

	%edge labels
	\draw (-6,-2) node[fill = white] {\cref{proposition.promotion fillings}};
	\draw (0,-2) node[fill=white] {virtualization};
	\draw (6, -2) node[fill=white] {Roby--Krattenthaler ``Blow up"/``Blocksum"};
	\draw (-1, 2) node[text width = 1] {\cref{thm:fans-main}};
	\draw (-.25, -6) node[text width = 1] {\cite{PfannererRubeyWestbury.2020}};
\end{tikzpicture}
}
\caption{Overview of strategy and results for $r$-fans of Dyck paths
\label{figure.overview r fans}}
\end{figure}

Having the injective map to chord diagrams gives a first step towards a diagrammatic basis for the invariant subspaces. In addition,
Fontaine and Kamnitzer~\cite{FK.2014} as well as Westbury~\cite{Westbury.2016} tied the promotion action on highest weight elements
of weight zero to the cyclic sieving phenomenon introduced by Reiner, Stanton and White~\cite{RSW.2004}. In \cref{section.cyclic sieving},
we make this cyclic sieving phenomenon more concrete by providing the polynomial in terms of the energy function. We also conjecture another
polynomial, which is the $q$-deformation of the number of $r$-fans of Dyck paths, to give a cyclic sieving phenomenon.

The paper is organized as follows. In \cref{section.crystal}, we give a brief review of crystal bases and virtual crystals and provide
the virtual crystals for the spin representation of type $B_r$ into type $C_r$. We also define promotion on crystals via the crystal
commutor. In \cref{section.chord}, we give the various filling rules to construct the map to chord diagrams. \cref{section.main results}
is reserved for the statements of our main results. The long version~\cite{PPSS.2022} of this extended abstract contains all proofs and more details on
vacillating tableaux.

We thank S. Hopkins, V. Reiner, M. Rubey and B. Westbury for discussions.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Crystal bases}
\label{section.crystal}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Crystal bases form a combinatorial skeleton of representations of quantum groups associated to Lie algebras. They were first introduced
by Kashiwara~\cite{Kashiwara.1990} and Lusztig~\cite{Lusztig.1990}.
Axiomatically, for a given root system $\Phi$ with index set $I$ and weight lattice $\Lambda$, a \defn{crystal} is a nonempty set $\mathcal{B}$
together with maps
$e_i, f_i \colon \mathcal{B} \to \mathcal{B} \sqcup \{\emptyset\}$, 
$\varepsilon_i, \varphi_i \colon \mathcal{B} \to \mathbb{Z}$,
$\mathsf{wt} \colon \mathcal{B} \to \Lambda$
for $i\in I$, satisfying certain conditions (see for example~\cite[Definition 2.13]{BumpSchilling.2017}). The operators $e_i$ and $f_i$ are
called \defn{raising} and \defn{lowering operators}. The map $\mathsf{wt}$ is the \defn{weight map}.
The map $\varepsilon_i$ (resp. $\varphi_i$) measures how often $e_i$ (resp. $f_i$) can be applied
to the given crystal element. For all crystals considered in this paper,
$\varepsilon_i(b) = \max\{k \geqslant 0 \mid e_i^k (b) \neq \emptyset \}$ for $b\in \mathcal{B}$ and similarly for $\varphi_i(b)$.
An element $b \in \mathcal{B}$ is called \defn{highest weight} if $e_i(b) = \emptyset$ for all $i \in I$.

Here we define certain crystals for the root systems $B_r$ and $C_r$ explicitly.
Let $\mathbf{e}_i \in \mathbb{Z}^r$ be the $i$-th unit vector with $1$ in position $i$ and 0 everywhere else.

\begin{definition}
The \defn{spin crystal} of type $B_r$, denoted by $\mathcal{B}_{\mathsf{spin}}$, consists of all $r$-tuples $\epsilon = (\epsilon_1,\epsilon_2,\ldots,\epsilon_r)$,
where $\epsilon_i \in \{\pm\}$. The weight of $\epsilon$ is
$\mathsf{wt}(\epsilon) = \frac{1}{2} \sum_{i=1}^r \epsilon_i \mathbf{e}_i$.
The crystal operator $f_r$ annihilates $\epsilon$ unless $\epsilon_r=+$. If $\epsilon_r=+$, $f_r$ acts on $\epsilon$ by changing $\epsilon_r$ from $+$ to $-$
and leaving all other entries unchanged.
The crystal operator $f_i$ for $1\leqslant i <r$ annihilates $\epsilon$ unless $\epsilon_i=+$ and $\epsilon_{i+1}=-$. In the latter case, $f_i$ acts on $\epsilon$
by changing $\epsilon_i$ to $-$ and $\epsilon_{i+1}$ to $+$.
The crystal operator $e_i$ is defined similarly.
\end{definition}

The crystal $\mathcal{B}_{\mathsf{spin}}$ of type $B_3$ is depicted in \cref{figure.virtual and spin}.

\begin{figure}
\centering
\scalebox{0.9}{
\includegraphics[width=11cm]{virtual_crystals1.pdf}}
\caption{Left: One component of the crystal $\widehat{\mathcal{V}} = \mathcal{C}_\square^{\otimes 3}$ of type $C_3$.
Middle: The virtual crystal $\mathcal{V}$ inside $\widehat{\mathcal{V}}$ of type $B_3$.
Right: The spin crystal $\mathcal{B}_{\mathsf{spin}}$ of type $B_3$.
\label{figure.virtual and spin}}
\end{figure}

\begin{definition}
\label{definition.vector crystal}
The \defn{crystal for the vector representation} $\mathcal{C}_\square$ of type $C_r$
consists of the elements $\{1,2,\ldots,r,\overline{r},\ldots, \overline{2},\overline{1}\}$. The crystal operator
$f_i$ for $1\leqslant i<r$ maps $i$ to $i+1$, maps $\overline{i+1}$ to $\overline{i}$ and annihilates all other elements. The crystal operator $f_r$ maps
$r$ to $\overline{r}$ and annihilates all other elements. The crystal operators $e_i$ are defined similarly.
Furthermore, $\mathsf{wt}(i) = \mathbf{e}_i$ and $\mathsf{wt}(\overline{i}) = -\mathbf{e}_i$.
\end{definition}

A remarkable property of crystals is that they respect \defn{tensor products}. Given two crystals $\mathcal{B}$ and $\mathcal{C}$ associated to
the same root system $\Phi$, the tensor product $\mathcal{B} \otimes \mathcal{C}$ as a set is the Cartesian product $\mathcal{B} \times \mathcal{C}$.
The weight of $b \otimes c \in \mathcal{B} \otimes \mathcal{C}$ is the sum of the weights $\mathsf{wt}(b\otimes c) = \mathsf{wt}(b) + \mathsf{wt}(c)$.
For more information, see~\cite[Section 2.3]{BumpSchilling.2017}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Virtual crystals}
\label{section.virtual}
Stembridge~\cite{Stembridge.2003} characterized crystals associated to simply-laced root systems
in terms of local rules on the crystal graph. Crystals for non-simply-laced root systems can be constructed using virtual 
crystals~\cite[Chapter 5]{BumpSchilling.2017}.

In this paper, we utilize virtual crystals to construct Fomin growth diagrams and the promotion operators for type $B_r$ using results for type $C_r$.
Hence let us briefly review the set-up for virtual crystals. Let $X \hookrightarrow Y$ be an embedding of Lie algebras such that the fundamental
weights $\omega_i$ and simple roots $\alpha_i$ map as
$\omega_i^X \mapsto \gamma_i \sum_{j \in \sigma(i)} \omega_j^Y$ and
$\alpha_i^X \mapsto \gamma_i \sum_{j \in \sigma(i)} \alpha_j^Y$.
Here $\gamma_i$ is a multiplication factor, $\sigma \colon I^X \to I^Y/\operatorname{aut}$ is a bijection and $\operatorname{aut}$ is an
automorphism on the Dynkin diagram for $Y$.

\begin{definition}
If there is an embedding of Lie algebras $X \hookrightarrow Y$, then $\mathcal{V} \subseteq \widehat{\mathcal{V}}$ is a \defn{virtual crystal}
for the root system $\Phi^X$ if
\begin{enumerate}
\item[{\bf V1.}] The ambient crystal $\widehat{\mathcal{V}}$ is a crystal associated to a representation for the root system $\Phi^Y$ with
crystal operators $\widehat{e}_i$, $\widehat{f}_i$, $\widehat{\varepsilon}_i$, $\widehat{\varphi}_i$ for $i \in I^Y$ and weight function $\widehat{\mathsf{wt}}$.
\item[{\bf V2.}] If $b\in \mathcal{V}$ and $i\in I^X$, then $\widehat{\varepsilon}_j(b)$ has the same value for all $j\in \sigma(i)$ and that value is a multiple of
$\gamma_i$. The same is true for $\widehat{\varphi}_j(b)$.
\item[{\bf V3.}] The subset $\mathcal{V} \sqcup \{\emptyset\} \subseteq \widehat{\mathcal{V}} \sqcup \{\emptyset\}$ is closed under the virtual crystal
operators
$e_i:= \prod_{j\in \sigma(i)} \widehat{e}_j^{\gamma_i}$ and
$f_i:= \prod_{j\in \sigma(i)} \widehat{f}_j^{\gamma_i}$
and $\varepsilon_i(b) = \max\{k\geqslant 0 \mid e_i^k(b) \neq \emptyset\}$, 
$\varphi_i(b) = \max\{k \geqslant 0 \mid f_i^k(b) \neq \emptyset\}$ for $b\in \mathcal{V}$.
\end{enumerate}
\end{definition}

The tensor product of two virtual crystals for the same embedding $X \hookrightarrow Y$ is again a virtual crystal
(see for example~\cite[Theorem 5.8]{BumpSchilling.2017}).

We will now apply the theory of virtual crystals to the embedding $B_r \hookrightarrow C_r$. In this setting $I^{C_r} = I^{B_r} = \{1,2,\ldots,r\}$,
$\sigma(i)=\{i\}$, $\gamma_i=2$ for $1\leqslant i<r$ and $\gamma_r=1$.
We consider as the ambient crystal $\widehat{\mathcal{V}} = \mathcal{C}_\square^{\otimes r}$.
Define an ordering $<$ on the set $[r] \cup [\bar{r}]$ as follows
$1 < 2 < \dots < r < \bar{r} < \dots < \bar{1}$.
Denote by $|\cdot |$ the map from $[r] \cup [\bar{r}]$ to $[r]$ that sends letters to their corresponding unbarred values.

\begin{definition}
\label{definition.V def}
Let $\mathcal{V} \subseteq \widehat{\mathcal{V}}$ be given by
$\mathcal{V} \coloneqq \{ v_r \otimes v_{r-1} \otimes \dots \otimes v_1 \in \widehat{\mathcal{V}}  \mid v_i > v_j \text{ and } |v_i| \not = |v_j| \text{ for all } i > j\}$.
Let $f_{i} = \widehat{f}^2_{i}$, $e_{i} = \widehat{e}^2_{i}$ for $1 \leqslant i <r$ and $f_{r} = \widehat{f}_{r}$, $e_{r} = \widehat{e}_{r}$.
\end{definition}

\begin{definition}
\label{definition.virtual_spintovector}
Let $\Psi \colon \mathcal{B}_{\mathsf{spin}} \to \mathcal{V}$ be the map
$\Psi(\epsilon_{1} \epsilon_{2} \cdots \epsilon_{r}) = v_{r} \otimes v_{r-1} \otimes \dots \otimes v_{1}$,
where $v_r > v_{r-1} > \cdots > v_1$ such that if $\epsilon_i= +$ then $v$ contains an $i$ and if $\epsilon_i = -$ then $v$ contains an $\bar{i}$ for all
$1\leqslant i\leqslant r$.
\end{definition}

\begin{proposition}
The map $\Psi$ is a bijective map that intertwines the crystal operators on $\mathcal{B}_{\mathsf{spin}}$ and $\mathcal{V}$.
Furthermore, $\mathcal{V}$ is a virtual crystal for the embedding of Lie algebras $B_{r} \hookrightarrow C_{r}$.
\end{proposition}

An example of the virtual crystal construction for $\mathcal{B}_{\mathsf{spin}}$ is given in \cref{figure.virtual and spin}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Highest weights of weight zero}
\label{section.highest weight}

A weight $\lambda \in \Lambda$ is called \defn{minuscule} if $\langle \lambda, \alpha^\vee \rangle \in \{0,\pm1\}$ for all coroots $\alpha^\vee$.
A crystal $\mathcal{B}$ is called \defn{minuscule} if $\mathsf{wt}(b)$ is minuscule for all $b \in \mathcal{B}$. Note that $\mathcal{B}_{\mathsf{spin}}$
is a minuscule crystal (see for example~\cite[Chapter 5.4]{BumpSchilling.2017}).
A weight $\lambda$ is called \defn{dominant} if $\langle \lambda,\alpha_i^\vee \rangle \geqslant 0$ for all $i\in I$. 
Except for spin weights, dominant weights can be identified with partitions. A \defn{partition} $\lambda$ is a sequence
$\lambda = (\lambda_1,\lambda_2, \ldots,\lambda_\ell)$ such that $\lambda_1 \geqslant \lambda_2 \geqslant \cdots \geqslant \lambda_\ell \geqslant 0$.
We identify partitions that differ by trailing zeroes.

Let $\mathcal{B}_1,\mathcal{B}_2,\ldots,\mathcal{B}_n$ be minuscule crystals. For any highest weight element
$u = u_n\otimes \cdots \otimes u_1 \in \mathcal{B}_n \otimes \cdots \otimes \mathcal{B}_1$,
we may bijectively associate a sequence of dominant weights $\emptyset = \mu^0, \mu^1, \ldots, \mu^n$, where $\mu^q := \sum_{i=1}^q \mathsf{wt}(u_i)$.
If $\mu^n$ is zero, $u$ is a \defn{highest weight element of weight zero}.
Note that the number of highest weight elements of weight zero in a tensor product of crystals is equal to the dimension of the invariant 
subspace~\cite{Westbury.2016, PfannererRubeyWestbury.2020}.

\begin{definition}[Sundaram~\cite{Sundaram.1990}]
An \defn{$r$-symplectic oscillating tableau} $\osc$ of length $n$ and shape $\mu$ is a sequence of partitions
$\osc = (\emptyset = \mu^0, \mu^1, \ldots, \mu^{n} = \mu)$
such that the Ferrers diagrams of two consecutive partitions differ by exactly one cell, and each partition $\mu^i$ has at most $r$ nonzero parts.
\end{definition}

Next we relate highest weight elements of weight zero in $\mathcal{B}_{\mathsf{spin}}^{\otimes n}$ of type $B_r$ and $r$-fans of Dyck paths.

\begin{definition}
\label{definition.r-fan}
An \defn{$r$-fan of Dyck paths} $\fan$ of length $n$ is a sequence
$\fan = (\emptyset = \mu^0, \mu^1, \ldots, \mu^{n} = \emptyset)$
of partitions $\mu^i$ with at most $r$ parts such that the Ferrers diagram of two consecutive partitions differs by exactly one cell in each part.
In other words, $\mu^i$ differs from $\mu^{i+1}$ by $(\pm 1, \pm 1, \ldots, \pm 1)$ for $0\leqslant i<n$.
\end{definition}

Since $\mathcal{B}_{\mathsf{spin}}$ of type $B_r$ is minuscule, by the above discussion $\epsilon = \epsilon_n \otimes \cdots \otimes \epsilon_1 \in
\mathcal{B}_{\mathsf{spin}}^{\otimes n}$ is highest weight if and only if $\sum_{i=1}^q \mathsf{wt}(\epsilon_i)$ is dominant for all $1\leqslant q \leqslant n$.
Hence highest weight elements of weight zero can be identified with an $r$-fan of Dyck paths of length $n$: the $j$-th entry of $\epsilon_i$ is $+$ if and only
if the $j$-th Dyck path has an up-step at position $i$. In particular, for a highest weight element $\epsilon$ of weight zero, the sequence of dominant weights
$\mu^q := \sum_{i=1}^q 2\mathsf{wt}(\epsilon_i)$ for $0\leqslant q \leqslant n$ defines an $r$-fan of Dyck paths consistent with \cref{definition.r-fan}.

\begin{example}
\label{example.3fan}
For $r=3$ and $n=4$, $\fan = ((000),(111),(220), (111), (000))$ is a 3-fan of Dyck paths corresponding to
$\epsilon = (-, -, -) \otimes (-, -, +) \otimes (+, +, -) \otimes (+, +, +) \in \mathcal{B}_{\mathsf{spin}}^{\otimes 4}$.
\end{example}

Following \cref{definition.virtual_spintovector}, we obtain an embedding from the set of $r$-fans of Dyck paths into the set of oscillating tableaux.

\begin{definition}
For an $r$-fan of Dyck paths $\fan = (\emptyset = \lambda^0, \ldots, \lambda^{n} = \emptyset)$ define the oscillating
tableau $\fantoosc(\fan) = (\mu^0, \dots, \mu^{rn})$ as follows. Let $v^t = \Psi(\lambda^t-\lambda^{t-1})$
for $1\leqslant t \leqslant n$ with $\Psi$ as in \cref{definition.virtual_spintovector}. Then
$\mu^{tr + s} = \lambda^t + \sum_{i=1}^s \mathsf{wt}(v_i^{t+1})$ for $0\leqslant t < n$, $0\leqslant s < r$.
\end{definition}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Promotion via crystal commutor}

For finite crystals $B_\lambda$ of classical type of highest weight $\lambda$, Henriques and Kamnitzer~\cite{HK.2006} introduced the crystal
commutor as follows. Let $\eta_{B_\lambda} \colon B_\lambda \to B_\lambda$ be the Lusztig involution, which maps the highest weight vector to the
lowest weight vector and interchanges the crystal operators $f_i$ with $e_{i'}$, where $w_0(\alpha_i) = -\alpha_{i'}$ under the longest element $w_0$.
This can be extended to tensor products of such crystals by mapping each connected component to itself using the above.
Then the \defn{crystal commutor} is defined as
\[
	\sigma_{B_\lambda, B_\mu} \colon B_\lambda \otimes B_\mu \to B_\mu \otimes B_\lambda \quad \text{with} \quad
	b \otimes c \mapsto \eta_{B_\mu \otimes B_\lambda} (\eta_{B_\mu}(c) \otimes \eta_{B_\lambda}(b)).
\]
\begin{definition}[\cite{FK.2014,Westbury.2016,Westbury.2018}]
Let $C$ be a crystal and $u \in C^{\otimes n}$ a highest weight element. Then \defn{promotion} $\mathsf{pr}$ on $u$ is defined as
$\sigma_{C^{\otimes n-1},C}(u)$.
\end{definition}

\begin{example}
Promotion is
$\sigma_{\mathcal{B}_{\mathsf{spin}}^{\otimes 3},\mathcal{B}_{\mathsf{spin}}}(\epsilon)
	= (-,-,-) \otimes (-,+,+) \otimes (+,-,-) \otimes (+,+,+)$
for $\epsilon$ in \cref{example.3fan}.
\end{example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Promotion via local rules}
\label{section.promotion local rules}

Adapting local rules of van Leeuwen~\cite{vanLeeuwen.1998}, Lenart~\cite{Lenart.2008} gave a combinatorial realization of the crystal commutor
$\sigma_{A,B}$ by constructing an equivalent bijection between the highest weight elements of $A \otimes B$ and $B \otimes A$ respectively. 
We define Lenart's construction in the case of promotion.
The\defn{ local rules} of Lenart~\cite{Lenart.2008} can be stated as follows:
four weight vectors $\lambda, \mu, \kappa, \nu \in \Lambda$ depicted in a square diagram
$\scalebox{0.6}{
	\begin{tikzpicture}[scale=0.5]
	\draw (0,0) rectangle (2,2);
	\node at (0,2)[anchor=south west]{$\lambda$};
	\node at (2,2)[anchor=south west]{$\nu$};
	\node at (0,0)[anchor=south west]{$\kappa$};
	\node at (2,0)[anchor=south west]{$\mu$};
	\end{tikzpicture}}
$
satisfy the local rule, if $\mu = \dom_{W}(\kappa+\nu-\lambda)$, where $W$ is the Weyl group of the root system $\Phi$ for $A$ and $B$ and $\dom_W(\rho)$ is the dominant weight in the Weyl orbit of $\rho$.


Let $C$ be a minuscule crystal and $(\emptyset = \mu^0,\mu^1,\dots,\mu^{n}=\mu)$ the sequence of partitions corresponding to the highest weight element $u \in C^{\otimes n}$. Let $(\emptyset \eqqcolon \hat \mu^0,\hat \mu^1,\dots, \hat \mu^{n} \coloneqq \mu)$ be the sequence of partitions obtained by constructing a skewed grid as in ~\eqref{eq:local_commutor} and recursively computing $\hat \mu^{1}, \dots, \hat \mu^{n-1}$ using Lenart's local rules. 
\begin{equation}\label{eq:local_commutor}
\scalebox{0.7}{
\tikzset{ short/.style={ shorten >=7pt, shorten <=7pt } }
    \begin{tikzpicture}[baseline=(current bounding box.center), scale=1.4]
    \node at (0,0+0.13)[anchor=west] {$\mu^0$};
    \node at (1,0+0.13)[anchor=west] {$\mu^1$};
    \node at (2,0+0.13)[anchor=west] {$\mu^2$};
    \node at (4,0+0.13)[anchor=west] {$\mu^{n-1}$};
    \node at (5,0+0.13)[anchor=west] {$\mu^{n}$};

    \node at (1,-1+0.13)[anchor=west] {$\hat\mu^0$};
    \node at (2,-1+0.13)[anchor=west] {$\hat\mu^1$};
    \node at (4,-1+0.13)[anchor=west] {$\hat\mu^{n-2}$};
    \node at (5,-1+0.13)[anchor=west] {$\hat\mu^{n-1}$};
    \node at (6,-1+0.13)[anchor=west] {$\hat\mu^{n}$};

    \draw (0,0) -- (2,0);
    \draw[dotted,short] (2,0) -- (4,0);
    \draw (4,0) -- (5,0);
    \draw (1,-1) -- (2,-1);
    \draw[dotted,short] (2,-1) -- (4,-1);
    \draw (4,-1) -- (6,-1);
    \draw (1,0) -- (1,-1);
    \draw (2,0) -- (2,-1);
    \draw (4,0) -- (4,-1);
    \draw (5,0) -- (5,-1);
    \end{tikzpicture}}
\end{equation}

It follows from Lenart ~\cite[Theorem~4.4]{Lenart.2008} that $(\emptyset = \hat \mu^0,\hat \mu^1,\dots, \hat \mu^{n}=\mu)$ is the unique sequence of 
partitions corresponding to the highest weight element $\mathsf{pr}(u)$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Chord diagrams}
\label{section.chord}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Promotion-evacuation diagrams}
\label{section.promotion diagrams}

\begin{figure}
\centering
\begin{center}
\scalebox{0.8}{
\begin{tabular}{p{0.20\textwidth}|p{0.12\textwidth}|p{0.15\textwidth}|p{0.30\textwidth}}\footnotesize
{\bf Step 1.} & \footnotesize {\bf Step 2.} &\footnotesize {\bf Step 3.} & \footnotesize{\bf Step 4.} \\\hline\begin{center}
\begin{tikzpicture}[scale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]
\draw[very thick,fill=gray!35] (4,4) -- (4,3) -- (5,3) -- (5,2) -- (6,2) -- (6,1) -- (7,1) -- (7,0) -- (8,0) -- (4,0) -- (4,4);
\draw[very thick,fill=gray] (0,4) -- (1,4) -- (1,3) -- (2,3) -- (2,2) -- (3,2) -- (3,1) -- (4,1) -- (4,0) -- (4,4) -- (0,4);
\draw (0,4) -- (4,4) -- (4,0);
\draw (1,4) -- (1,3) -- (5,3) -- (5,0);
\draw (2,4) -- (2,2) -- (6,2) -- (6,0);
\draw (3,4) -- (3,1) -- (7,1) -- (7,0);
\draw (4,0) -- (8,0) -- (8,0);
\end{tikzpicture}\end{center} & \begin{center}
\begin{tikzpicture}[scale=0.55,baseline={([yshift=-.8ex]current bounding box.center)},scale=0.7]
\draw[very thick,fill=gray!35] (0,4) -- (0,3) -- (1,3) -- (1,2) -- (2,2) -- (2,1) -- (3,1) -- (3,0) -- (4,0) -- (0,0) -- (0,4);
\draw[very thick,fill=gray] (0,4) -- (1,4) -- (1,3) -- (2,3) -- (2,2) -- (3,2) -- (3,1) -- (4,1) -- (4,0) -- (4,4) -- (0,4);
\foreach \i in {0,1,2,3,4} {
\draw (0,\i) -- (4,\i);
\draw (\i,0) -- (\i,4);
}
\end{tikzpicture}\end{center}
&\begin{center}\vspace{-1.3em}
\begin{tikzpicture}
\draw (-0.1,-0.1) rectangle (1.6,1.6);
\node at (0,1.6)[anchor=south west]{$\lambda$};
\node at (1.6,1.6)[anchor=south west]{$\nu$};
\node at (0,0)[anchor=south west]{$\kappa$};
\node at (1.6,0)[anchor=south west]{$\mu$};
\node at (0.8,0.8) {\scriptsize $\filling(\lambda,\kappa,\nu,\mu)$};
\end{tikzpicture}\end{center}
&\begin{center}\scalebox{0.6}{$
\begin{pmatrix}
\cdot &\cdot &\cdot &\cdot &\cdot&\cdot\\
\cdot &\cdot &\cdot &\cdot &\cdot&\cdot\\
\cdot &\cdot &\cdot &\cdot &\cdot&\cdot\\
\cdot &\cdot &\cdot &\cdot &\cdot&\cdot\\
\cdot &\cdot &\cdot &\cdot &\cdot&\cdot\\
\cdot &\cdot &\cdot &\cdot &\cdot&\cdot
\end{pmatrix}
$}
\vspace{0.9em}\hspace{.8em}
\newcommand\n{8}
    \scalebox{0.6}{\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)},scale=0.75]
        \foreach \number in {1,...,\n}{
            \node (N-\number) at ({\number*(360/\n)}:2cm) {$\circ$};
        }

        \foreach \number in {1,...,\n}{
            \foreach \y in {1,...,\n}{
                \draw (N-\number) -- (N-\y);
            }
        }
    \end{tikzpicture}}
\end{center}
\end{tabular}
}
\end{center}
\caption{Overview of the steps in our map
\label{fig:overview_promevac}}
\end{figure}

In this section we summarize the method 
to obtain a map from highest weight words of weight zero to chord diagrams that intertwines promotion and rotation.

\begin{definition}
A \defn{chord diagram} of size $n$ is a graph with $n$ vertices depicted on a circle labelled $1,\dots, n$ in counter-clockwise orientation.
The \defn{rotation} of a chord diagram is obtained by rotating all edges clockwise by $\frac{2\pi}{n}$ around the center of the diagram.
\end{definition}

In our setting, chord diagrams are undirected graphs with possibly multiple edges between the same two vertices. We can therefore identify
a chord diagram with its \defn{adjacency matrix}, which is a symmetric $n\times n$ matrix $M=(m_{ij})_{1\leqslant i,j\leqslant n}$
with non-negative integer entries $m_{ij}$ equal to the number of edges between vertex $i$ and vertex $j$. We denote by $\rot M$ the 
adjacency matrix corresponding to the rotation of the chord diagram.

We now outline the idea to construct a rotation and promotion intertwining map for oscillating tableaux and $r$-fans of Dyck paths. A visual guideline 
is given in \cref{fig:overview_promevac}.

\begin{construction}\label{construction:tableau_to_chord}
The construction is given as follows:
\begin{description}[nosep]
\item[Step 1]  Iteratively calculate promotion of a highest weight word of weight zero and length $n$ using Lenart's schema \eqref{eq:local_commutor} a total of $n$ times.
\item[Step 2] Group the results into a square grid, called the \defn{promotion matrix}.
\item[Step 3] Fill the cells of the square grid with certain non-negative integers according to a filling rule $\filling$ that only depends on the four corners
of the cells in the schema~\eqref{eq:local_commutor}.
\item[Step 4] Regard the filling as the adjacency matrix of a graph, which is the chord diagram.
\end{description}
\end{construction}

\begin{definition}
\label{definition.fillingfans}
The \defn{filling rule} for oscillating tableaux and fans of Dyck paths, with cells labelled as in Step 3 of \cref{fig:overview_promevac}, is
$\filling(\lambda,\kappa,\nu,\mu) = \text{number of negative entries in $\kappa + \nu-\lambda$}$.
\end{definition}

\begin{remark}
For oscillating tableaux at most one negative entry can occur, so that $\filling(\lambda,\kappa,\nu,\mu)$ is either 0 or 1.
Note that the filling rules are new even in the case of oscillating tableaux as the proofs
in~\cite{PfannererRubeyWestbury.2020} did not follow this construction.
\end{remark}

\begin{definition}
Denote by $\Mosc$ (resp. $\Mfan$) the function that maps an $r$-symplectic oscillating tableau (resp. $r$-fan of Dyck paths) of length $n$ to an 
$n\times n$ adjacency matrix using \cref{construction:tableau_to_chord} and the filling rule in \cref{definition.fillingfans}.
\end{definition}

\begin{proposition}
\label{proposition:rotation_promotion}
The map $\mathsf{M}_X$ for $X\in \{O,F\}$ intertwines promotion and rotation, that is
$\mathsf{M}_X \circ \mathsf{pr} = \rot \circ \mathsf{M}_X$.
\end{proposition}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Fomin growth diagrams}
\label{section.fomin growth}
Generally speaking, a \defn{Fomin growth diagram} is a means to bijectively map sequences of partitions satisfying certain constraints to non-negative 
integer fillings of a Ferrers shape, drawn in French notation~\cite{Fomin.1986,Krattenthaler.2006}. To map a $0/1$ filling of a Ferrers shape to a sequence 
of partitions, we iteratively label all corners of the cells of the shape with partitions by certain local forward rules, F1-F6~\cite[4]{Krattenthaler.2006}. 
Conversely, given partitions labelling the right and top corners of a cell, the local backwards rules B1-B6 determine the last partition and the $0/1$ 
filling of the cell~\cite[5]{Krattenthaler.2006}. 

To map a non-negative integer filling of a Ferrers shape to a sequence of partitions, we produce a ``blow up" construction of the original shape
for the Burge variant which contains south-east chains of $1$'s, as done by~\cite[Section 4.4]{Krattenthaler.2006}. If a cell is filled with a positive entry $m$, we replace the cell with an $m\times m$ grid of cells with $1$'s along the diagonal.
If there exist several nonzero entries in one column or row, we arrange the grids of cells so that the $1$'s form a south-east
chain in each column and row.

Since the filling of the blow up growth diagram consists of $1$'s and $0$'s, we now apply the forward local rules F1-F6 to obtain the partition labels of all corners. 
We then ``shrink back" the labelled blow up growth diagram to obtain a labelling of the original Ferrers diagram by only using the partitions labelling the 
intersections that occurred in the original Ferrers diagram. The labelling from the blow up construction is given explicitly by the local forward rules 
F$^4$0 - F$^4$2 and local backward rules B$^4$0 - B$^4$2 \cite[22]{Krattenthaler.2006}. 

\begin{construction}
\label{construction.Burgegrowthrule}
Let $\fan = (\emptyset =\mu^0,\mu^1,\dots,\mu^{n}=\emptyset)$ be an $r$-fan of Dyck paths. The associated triangular growth diagram is
the Ferrers shape $(n-1, n-2, \ldots, 2, 1, 0)$. Label the cells according to the following specification:
\begin{enumerate}[nosep]
\item Label the north-east corners of the cells on the main diagonal from the top-left to the bottom-right with the partitions in $\fan$.
\item For each $i\in \{0,\dots,n-1\}$ label the corner on the first subdiagonal adjacent to the labels $\mu^{i}$ and $\mu^{i+1}$ with the partition
$\mu^{i} \cap \mu^{i+1}$.
\item Use backwards rules Burge B$^4$0-B$^4$2 to obtain all other labels and cell fillings.
\end{enumerate}
\end{construction}
For oscillating tableaux, \cref{construction.Burgegrowthrule} reduces to rules B1-B6 and F1-F6.

We denote by $\Gfan(\fan)$ (resp. $\Gosc$) the symmetric $n\times n$ matrix one obtains from the filling of the growth diagram by putting zeros in the
unfilled cells and along the diagonal and completing this to a symmetric matrix.
%
Starting from a filling of a growth diagram one obtains the $r$-fan by filling the cells of a growth diagram, setting all vectors on
corners on the bottom and left border of the diagram to be the empty partition and applying the forwards growth rules Burge F$^4$0-F$^42$
(resp. F1-F6).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Main results}
\label{section.main results}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Results for oscillating tableaux and $r$-fans of Dyck paths}

Our main result states that the fillings of the growth diagrams (\cref{construction.Burgegrowthrule}) and the fillings of the 
promotion matrices (\cref{definition.fillingfans}) coincide. For oscillating tableaux this was not stated explicitly 
in~\cite{PfannererRubeyWestbury.2020}, but can be deduced from the proofs.

\begin{theorem}\label{thm:fans-main}
For an oscillating tableau of weight zero $\osc$ and an $r$-fan of Dyck paths $\fan$, we have $\Gosc(\osc)=\Mosc(\osc)$ and 
$\Gfan(\fan) = \Mfan(\fan)$. In particular, the maps $\Mosc$ and $\Mfan$ are injective.
\end{theorem}

\begin{proposition}
\label{proposition.promotion fillings}
For an $r$-fan of Dyck paths $\fan$, 
$ \fantoosc \circ \mathsf{pr}_{\spincrystal}(\fan) = \mathsf{pr}^r_{\cboxcrystal} \circ \fantoosc (\fan)$.
In addition, $\Mfan(\fan) = \mathsf{blocksum}_{r}(\Mosc(\fantoosc(\fan)))$, where $\mathsf{blocksum}_{r}(\Mosc(\fantoosc(\fan)))$ is the matrix obtained by 
replacing $r \times r$ blocks in $\Mosc(\fantoosc(\fan))$ with the sum of their entries.
\end{proposition}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Cyclic sieving}
\label{section.cyclic sieving}

The cyclic sieving phenomenon was introduced by Reiner, Stanton and White~\cite{RSW.2004}.
\begin{definition}
Let $X$ be a finite set and $C$ be a cyclic group generated by $c$ acting on $X$.
Let $\zeta \in \mathbb{C}$ be a $|C|^{th}$ primitive root of unity and $f(q) \in \mathbb{Z}[q]$ be a polynomial in $q$.
Then the triple $(X,C,f)$ exhibits the \defn{cyclic sieving phenomenon} if for all $d\geqslant 0$ we have that the size of the fixed point set
of $c^d$ (denoted $X^{c^d}$) satisfies $|X^{c^d}| = f(\zeta^d)$.
\end{definition}

We state cyclic sieving phenomena for the promotion action on oscillating tableaux and fans of Dyck paths.
For this we need the \defn{local energy function} (see for example~\cite{OSS.2003})
$H \colon \mathcal{B} \otimes \mathcal{B} \to \mathbb{Z}$.
The local energy function is constant on connected components.

For the crystal $\mathcal{C}_\square$ of type $C_r$, using the ordering $1<2<\cdots<r<\overline{r}< \cdots < \overline{2}<\overline{1}$, we have 
that $H(a\otimes b)=0$ if $a\leqslant b$ and $H(a\otimes b)=1$ if $a>b$.
The classical highest weight elements in $\mathcal{B}_{\mathsf{spin}}^{\mathsf{af}} \otimes \mathcal{B}_{\mathsf{spin}}^{\mathsf{af}}$
are $(\epsilon_1,\ldots,\epsilon_r) \otimes (+,\ldots,+)$ with $\epsilon_i=+$ for $1\leqslant i \leqslant k$ and $\epsilon_i=-$ for $k< i\leqslant r$
for some $0\leqslant k \leqslant r$. Denoting by $m(\epsilon_1,\ldots,\epsilon_r)$ the number of $-$ in the $\epsilon_i$, we have
$H((\epsilon_1,\ldots,\epsilon_r) \otimes (+,\ldots,+)) = \Big\lfloor \frac{m(\epsilon_1,\ldots,\epsilon_r)+1}{2} \Big\rfloor$.
The \defn{energy function} $E \colon \mathcal{B}^{\otimes N} \to \mathbb{Z}$
is defined for $b_1\otimes \cdots \otimes b_N \in \mathcal{B}^{\otimes N}$ as
$E(b_1 \otimes \cdots \otimes b_N) = \sum_{i=1}^{N-1} i H(b_i \otimes b_{i+1})$.
Define a polynomial in $q$ using the energy function for highest weight elements in $\mathcal{B}^{\otimes n}$ of weight zero
\[
	f_{n,r}(q) = q^{c_{n,r}} \sum_{\substack{b \in \mathcal{B}^{\otimes n}\\ \mathsf{wt}(b)=0, e_i(b)=0 \text{ for } 1\leqslant i \leqslant r}} q^{E(b)},
\]
where $r$ is the rank of the root system and $c_{n,r}$ is a constant depending on the type.

\begin{theorem}
\label{theorem.CSP}
Let $X$ be the set of highest weight elements in $\mathcal{B}^{\otimes n}$ of weight zero, where $\mathcal{B}$ is minuscule.
Then $(X,C_{n},f_{n,r}(q))$ exhibits the cyclic sieving phenomenon, where $C_{n}$ is the cyclic group of order $n$ given by the action of
promotion $\mathsf{pr}$ on $\mathcal{B}^{\otimes n}$.
\end{theorem}

Note that $\mathcal{C}_\square$ and $\mathcal{B}_{\mathsf{spin}}$ are minuscule, and hence \cref{theorem.CSP}
gives a cyclic sieving phenomenon for oscillating tableaux and fans of Dyck paths. See also~\cite{FK.2014,Westbury.2016}.
For the type $A$ vector representation, the energy function is the major index and \cref{theorem.CSP} relates to results in~\cite{Rhoades.2010}.

Recall that highest weight elements of weight zero in $\mathcal{B}_{\mathsf{spin}}^{\otimes 2n}$ of type $B_r$
are in bijection with $r$-fans of Dyck paths of length $2n$. Denote by $D_{n}^{(r)}$ the set of all $r$-fans of Dyck paths of length $2n$.
We have $|D_n^{(r)}|=\prod_{1 \leqslant i \leqslant j \leqslant n-1} \frac{i+j+2r}{i+j}$, see~\cite{Krattenthaler.1995}.
Define the $q$-analogue as
$g_{n,r}(q) = \prod_{1 \leqslant i \leqslant j \leqslant n-1} \frac{[i+j+2r]_q}{[i+j]_q}$ where $[m]_q = 1+q+q^2+\cdots+q^{m-1}$.
The following conjecture is equivalent to~\cite[Conjecture 5.9]{Hopkins.2020} on plane partitions and root posets.

\begin{conjecture}
The triple $(D_{n}^{(r)}, C_{2n}, g_{n,r}(q))$ exhibits the cyclic sieving phenomenon, where $C_{2n}$ is the cyclic group of order $2n$ that acts on
$D_{n}^{(r)}$ by applying promotion.
\end{conjecture}

\begin{example}
We have $q^{-4} f_{2,2} (q)= g_{2,2}(q) = q^4 + q^2 + 1$ and 
\[
\begin{split}
	g_{3,2}(q) &= q^{12} + q^{10} + q^9 + 2q^8 + q^7 + 2q^6 + q^5 + 2q^4 + q^3 + q^2 + 1,\\
	q^{-6} f_{3,2}(q) &= q^{10} + q^9 + 2q^8 + q^7 + 3q^6 + q^5 + 2q^4 + q^3 + q^2 + 1.
\end{split}
\]
\end{example}

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\end{document}
