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%% define your title in the usual way
\title{Cyclic sieving phenomenon on dominant maximal weights}

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\author[Young-Hun Kim, Se-jin Oh, and Young-Tak Oh]{Young-Hun Kim\thanks{\href{mailto:yhkim14@sogang.ac.kr}{yhkim14@sogang.ac.kr}. Supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (NRF--2018R1D1A1B07051048).}\addressmark{1,2},
Se-jin Oh\thanks{\href{mailto:sejin092@gmail.com}{sejin092@gmail.com}.
Supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF--2019R1A2C4069647).}\addressmark{3}, \and
Young-Tak Oh\thanks{\href{mailto:ytoh@sogang.ac.kr}{ytoh@sogang.ac.kr}.
Supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (NRF--2018R1D1A1B07051048).}\addressmark{1,4}}

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\address{
\addressmark{1}
Department of Mathematics, Sogang University, Seoul, Republic of Korea \\
\addressmark{2}
Research Institute for Basic Science, Sogang University, Seoul, Republic of Korea\\
\addressmark{3}
Department of Mathematics, Ewha Womans University, Seoul, Republic of Korea \\
\addressmark{4}
Korea Institute for Advanced Study, Seoul, Republic of Korea
}

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\abstract{
Dominant maximal weights are significant objects in the representation theory of affine Kac--Moody algebras.
We construct a (bi)cyclic sieving phenomenon on the union of dominant maximal weights for 
highest weight modules over affine Kac--Moody algebras 
in a way not depending on types, ranks and levels.
Exploiting this phenomenon, we derive closed and recursive formulae for the number of dominant maximal weights for every highest weight module and observe level-rank duality on the cardinalities. We also observe interesting interrelations among the recursive formulae of classical affine Kac--Moody algebras.
}


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\keywords{affine Kac--Moody algebra, dominant maximal weight, (bi)cyclic sieving phenomenon}


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\begin{document}

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

\section{Introduction}
Let $\g$ be an affine Kac--Moody algebra and $V(\Lambda)$ be the irreducible highest weight module with highest weight $\Lambda \in \dwt$, where $\dwt$ denotes the set of dominant integral weights. Due to Kac~\cite{Kac}, all weights of $V(\Lambda)$ are given by the disjoint union of $\delta$-strings attached to maximal weights and every maximal weight is conjugate to a unique dominant maximal weight under Weyl group action.

In \cite{Kac}, Kac established lots of fundamental properties concerned with ${\rm wt}(\Lambda)$, the set of weights of $V(\Lambda)$, using the orthogonal projection  $\bar{ \ } \colon \h^* \to \h_0^*$.
In particular, he showed that $\mx^+(\Lambda)$, the set of dominant maximal weights, is in bijection with $\ell \mathcal{C}_{{\rm af}} \cap (\overline{\Lambda}+\overline{Q})$
under this projection, thus it is finite. Here $\ell$ denotes the level of $\Lambda$ and $Q$ denotes the root lattice of $\g$.
Based on this result, there have been a number of studies on $\mx^+(\Lambda)$ (see \cite{JM,TW}).
However, in the best knowledge of the authors, approachable combinatorial models and cardinality formulae of $\mx^+(\Lambda)$ have not been available up to now except for limited cases, which motivates our study.

The {\it cyclic sieving phenomenon} was introduced by Reiner--Stanton--White in \cite{RSW}.
It was generalized and developed in various aspects including combinatorics and
representation theory (see \cite{BRS,Sagan2} for examples).


The main purpose of this extended abstract is to investigate $\mx^+(\Lambda)$ by constructing
a combinatorial model for $\mx^+(\Lambda)$ and a (bi)cyclic sieving phenomenon on this model in a way not depending on types, ranks and levels.
As applications, we derive closed and recursive formulae of $|\mx^+(\Lambda)|$ for all affine types, and observe interesting symmetries by considering $\mx^+(\Lambda)$ for all ranks and levels. We find out intriguing interrelations among the recursive formulae of various affine Kac--Moody algebras. We also realize that our symmetry for $A_n^{(1)}$ type gets along with the level-rank duality introduced by Frenkel~\cite{81Frenkel}.

For details and more results, we refer the reader to~\cite{19KOO}.

\section{Preliminaries} \label{Sec: pre}
Let $I=\{ 0,1,...,n \}$ be an index set. An \emph{affine Cartan datum} $(\cmA,P,\Pi,P^{\vee},\Pi^{\vee})$ consists of
(a) an \emph{affine Cartan matrix} $\cmA=(\cma_{ij})_{i,j \in I}$ of corank $1$,
(b) the \emph{weight lattice} $P=\bigoplus_{i=0}^{n} \Z \Lambda_i \oplus \Z \updelta$,
(c) the set of \emph{simple roots} $\Pi = \{ \affsr_i \mid i\in I \} \subset P$,
(d) the \emph{coweight lattice} $P^{\vee}= \mathrm{Hom}(P,\Z) $,
(e) the set of \emph{simple coroots} $\Pi^{\vee} = \{ h_i \mid i\in I\} \subset P^{\vee}$
subject to the condition $\inn{h_i, \affsr_j} = \cma_{ij}$ and $\langle h_j, \Lambda_i \rangle = \delta_{ij}$ for all $i,j\in I$.
Here 
$\updelta = \sum_{i = 0}^n a_i \affsr_i$ is the \emph{null root}. Let
\[c=a^\vee_0 h_0 + a^\vee_1 h_1+ \cdots + a^\vee_n h_n  \]
be the \emph{canonical central element}.
We say that a weight $\Lambda \in P$ is of {\it level} $\ell$ if $\langle c,\Lambda \rangle=\ell$.
We call the free abelian group $Q\seteq\bigoplus_{i \in I} \Z\affsr_i$ the \emph{root lattice}.
The elements of $\dwt \seteq \{\Lambda \in P \mid  \langle h_i,\Lambda \rangle \in \Z_{\ge 0},\  i \in I \}$ are called \emph{dominant integral weights}.
For a nonnegative integer $\ell$, we set
\[
\dwt_{\ell} \seteq \{\Lambda \in \dwt \mid \inn{c,\Lambda} = \ell\} \quad \text{and} \quad \pclp{\ell} := \dwt_\ell / \Z\updelta.
\]


The {\em affine Kac--Moody algebra} $\g$ associated with the affine Cartan datum $(\cmA,P,\Pi,P^{\vee},\linebreak[0]\Pi^{\vee})$ is the Lie algebra over $\C$ generated by $e_i,f_i \ ( i \in I)$ and $h \in P^{\vee}$ with defining relations.
Let $\Afin$ be the Cartan matrix associated to a finite simple Lie subalgebra $\g_0$ of $\g$.


It is well known that the category $\mathcal{O}_{{\rm int}}$, consisting of integrable weight $\g$-modules, is a semisimple tensor category such that every irreducible object is isomorphic to \emph{the highest weight module $V(\Lambda)$} ($\Lambda \in P^+$).

Let ${\rm wt}(V(\Lambda))$ be the set of weights of $V(\Lambda)$.
The elements of $\mx_\g(\Lambda) := \{\mu \in {\rm wt}(V(\Lambda)) \mid \mu+\updelta \not \in \mathrm{wt}(V(\Lambda))\}$ is called \emph{maximal weights}. We set
\[
\mx_\g^+(\Lambda) \seteq \mx_\g(\Lambda) \cap P^+.
\]
We sometimes omit the subscript $\g$  for simplicity. It is well known that $\mx(\Lambda) = W \cdot \mx^+(\Lambda),$  where $W$ is the Weyl group of $\g$.

Let $\h_0$ be the vector space spanned by $\{ h_i \ | \ i \in I_0 \seteq I \setminus \{0 \} \}$. The \emph{orthogonal projection} $\bar{ \ } \colon \h^* \to \h_0^*$ and the set $\mathcal{C}_{{\rm af}}$ are introduced in \cite{Kac}.

\begin{proposition}[{\cite[Proposition 12.6]{Kac}}] \label{prop: Kac bijection}
    The map $\mu \longmapsto \overline{\mu}$ defines a bijection from $\mx^+(\Lambda)$ onto $\ell \mathcal{C}_{{\rm af}} \cap (\overline{\Lambda}+\overline{Q})$
    where $\Lambda$ is of level $\ell$.
\end{proposition}

Throughout this paper, we denote by $\bullet$ the \emph{dot product} on $\Q^n$. In addition, for $k \in \Z_{>0}$ and $m,m' \in \Z$, we write $m\equiv_k m'$ if $k \mid m-m'$, and $m \not\equiv_k m'$ otherwise.

\section{Embedding $\mx^+(\Lambda)$ into $\pclp{\ell}$}\label{sec: embedding}
In this section, we will assume that $\Lambda$ is of the form $\sum_{0 \le i \le n} p_i \Lambda_i$
because $\ellcov = \ell \mathcal{C}_{{\rm af}} \cap (\overline{\Lambda + k \updelta}+\overline{Q})$ for all $k \in \Z$.

An equivalence relation $\sim$ on $P$, defined by $\Lambda \sim \Lambda'$ if and only if $\Lambda - \Lambda' \in Q$, was introduced in~\cite[Definition 3.1]{13BFS}.
We note that it induces an equivalence relation, called the {\it sieving equivalence relation}, on $\pclp{\ell}$ defined as follows:
For $\Lambda, \Lambda' \in \pclp{\ell} \ $,
\begin{align}\label{eq: sieving relation}
\Lambda \sim \Lambda' \quad \text{if and only if} \quad \ellcov = \ell \mathcal{C}_{{\rm af}} \cap (\overline{\Lambda'}+\overline{Q}).
\end{align}


Let $\Uppi_0 \seteq \{\finsr_i \mid i \in I_0\}$ be the set of simple roots of $\g_0$ and $\upvarpi \seteq \{\finfw_i \mid i \in I_0\}$ the set of fundamental dominant weights of $\g_0$.
Let $P_0:=\mathbb Z \upvarpi$ be the weight lattice of $\g_0$ and $Q_0:=\mathbb Z \Uppi_0$ the root lattice of $\g_0$.
Then $P_0/Q_0$ is known to be a finite group, called the {\it fundamental group of $\Phi_0$ $($the set of roots of $\g_0)$.}
Its structure is well known in the literature. For instance, see \cite{Humph}.

We note that there are at most $|P_0 / Q_0|$ equivalence classes on $\pclp{\ell}$.
%In the following, we provide a complete list of representatives of very simple form.
For each type, we define a set $\DRpclp$, called the set of {\it distinguished representatives}.
Indeed it is designed so that every $\Lambda \in \DRpclp$ is of the form $(\ell - 1)\Lambda_0 + \Lambda_i$.
For instance,
\[
\DRpclp = \begin{cases}
\{(\ell - 1)\Lambda_0 + \Lambda_i \mid i = 0,1,\ldots, n \} & \text{if $\g = A_n^{(1)}$}, \\
\{(\ell - 1)\Lambda_0 + \Lambda_i \mid i = 0,1,n-1,n\} & \text{if $\g = D_n^{(1)}$}.
\end{cases}
\]
For other types, see~\cite[Table 2.2]{19KOO}.

\begin{lemma}
$\DRpclp$ is a complete set of pairwise inequivalent representatives of $\pclp{\ell}/ \hspace{-.5ex} \sim$, the set of equivalence classes of $\pclp{\ell}$ under the sieving equivalence relation.
In particular, the number of equivalence classes is given by $|P_0/Q_0|$.
\end{lemma}

For $\Lambda \in  \DRpclp$, set  $\pclp{\ell}(\Lambda):= \{ \Lambda' \in \pclp{\ell}  \; | \; \Lambda \sim \Lambda'\}$
and consider the map $\iota_\Lambda: \ellcov \ra \pclp{\ell}(\Lambda)$ defined by
\[
\sum_{1 \le i \le n} m_i \finfw_i \mapsto m_0\Lambda_0 + \sum_{1 \le i \le n} m_i\Lambda_i
\ \ (m_0 \seteq \ell - \sum_{1 \le i \le n} a^\vee_i m_i).
\]

\begin{proposition}\label{lem: ellCaf}
For $\Lambda \in  \DRpclp$, the map $\iota_\Lambda$ is a bijection.
\end{proposition}

Note that $\left( \ellcov  \right) \cap  \left( \ell\mathcal{C}_{{\rm af}} \cap (\overline{\Lambda'}+\overline{Q}) \right)=\emptyset$ if $\Lambda \nsim \Lambda'$, which yields a bijection
\begin{align}\label{eq: bijection between ellcov and pclp}
\displaystyle \bigsqcup_{\Lambda \in \DRpclp}\ellcov \overset{1-1}{\longleftrightarrow} \bigsqcup_{\Lambda \in \DRpclp} \pclp{\ell} (\Lambda)=\pclp{\ell} \ .
\end{align}

Now, we introduce a simple description of $ \pclp{\ell}(\Lambda)$ for each $\Lambda \in  \DRpclp$. To do that, we need preparations:
Let $\congN \seteq |P_0 / Q_0|$ and $\hat{\congN} \seteq \max\{  |g|
\; | \; g \in  P_0 / Q_0 \}$. For a subset $S \subset \Z^n$, set
$\redr(S): = \{ \mathbf{\overline{s}}\subset (\Z_{\congN})^n \;  | \; \mathbf{s} \in S\}$, where $\mathbf{\overline{s}} = \mathbf{s} + (\congN\Z)^n$.

\begin{definition}\label{def: sieving}
Let $\g$ be an affine Kac--Moody algebra. We call a subset $S \subset \Z^n$ a {\it root-sieving set}
if, for all $\bx \in P_0$, (1) $\bx \in Q_0$ if and only if $\mathbf{s} \bullet [\bx]_{\upvarpi} \equiv_{\congN} 0$ for all $\mathbf{s} \in S$,
(2) the set $\redr(S)\subset (\Z_{\congN})^n$ is $\mathbb
Z_{\hat{\congN}}$-linearly independent, and (3) $|\redr(S)|=|S|.$ 
In this case, the elements of $S$ are called \emph{root-sieving vectors} of $S$.
\end{definition}


\begin{convention}\label{conv: Sieving} 
{\rm (1)} We choose a special root sieving set, denoted by $\svs$, as follows:
\[
\svs = \begin{cases}
\{\svv=(1,2,\ldots,n)\} & \text{ if } \g=A_n^{(1)}, \\
\{\svv=(2,0,2,0,\ldots,0,2,1,3)\} & \text{ if } \g=D_n^{(1)}(n\equiv_2 1), \\
\{\svv^{(1)}=(0,0,\ldots,0,2,2), \svv^{(2)}=(2,0,2,0,\ldots,2,0,2,0)\} & \text{ if } \g=D_n^{(1)}(n\equiv_2 0).
\end{cases}
\]
For other types, see \cite[Table 2.4]{19KOO}.

\noindent
{\rm (2)} For a root sieving vector $\svv=(s_1,s_2,\ldots,s_n)$, we denote $(0,s_1,s_2,\ldots,s_n)$ by $\wtsvv$.
\end{convention}


With the root sieving sets $\svs$ given in \cref{conv: Sieving}, we define a new statistics $\evS$, called the \emph{$\svs$-evaluation}, by
\begin{align}\label{eq: S-evaluation}
\evS: \pclp{\ell} \ra \Z_{\ge 0 }^k, \quad \sum_{0\le i \le n} m_i \Lambda_i \mapsto \left( \wtsvv^{(k)} \bullet \bm \right)_{k =1 \text{ or } 1,2}
\end{align}
where $\bm = (m_0,m_1,\ldots,m_n)$.


\begin{theorem}\label{thm: eq rel and sieving}
Let $\svs$ be the set given in \cref{conv: Sieving}.
For any $\Lambda \in \DRpclp$, we have
\begin{align}\label{eq: pis ell Lambda}
\pclp{\ell}(\Lambda) = \left\{ \Lambda' \in \pclp{\ell} \; \middle| \; \evS(\Lambda') \equiv_{\congN} \evS(\Lambda)   \right\}.
\end{align}
\end{theorem}
Note that \cref{thm: eq rel and sieving} does not depend on the choice of a root-sieving set.

\section{Sagan's action and generalization}\label{sec: Sagan action and general}
From this section, we will investigate the structure and enumeration of $\pclp{\ell}(\Lambda)$ for all $\Lambda \in \DR(\pclp{\ell})$ in a viewpoint of (bi)cyclic sieving phenomena (\cite{RSW}). In order to do this, we give a suitable (bi)cyclic group action on $\pclp{\ell}$ by generalizing Sagan's action in \cite{Sagan1}. For details of Sagan's action and our generalization, see~\cite[Section 3]{19KOO}.

Throughout this section, we assume that $d,k$ are positive integers and $\ell$ is a nonnegative integer.
Given a $kd$-tuple $\bm = (m_0,m_1,\ldots,m_{kd-1}) \in \Z_{\ge 0}^{kd}$,
we set $\bm[j;d]:=\sum_{0 \le t \le d-1} m_{jd +t}$ for $0\le j \le k-1$.
Also, given a $k$-tuple $\bnu=(\nu_0, \nu_1,\ldots, \nu_{k-1}) \in \Z^k_{> 0}$, we set
\begin{align*}
\bM_\ell (d;\bnu) := \left\{ \bm = (m_0,m_1,\ldots,m_{kd-1}) \in \Z_{\ge 0}^{kd} \; \middle| \;
\sum_{0 \le j \le k-1} \nu_j \bm[j;d]= \ell
\right\}.
\end{align*}

To each $\bm = (m_0,m_1,\ldots,m_{kd-1}) \in {\bM_\ell} (d;\bnu)$ we associate a word $\mathbf{w}(\bm;d;\bnu)$ with entries in $\{0,\nu_0,\nu_1,\ldots,\nu_{k-1}\}$ produced by the following algorithm:

\begin{algorithm}\label{Algorithm} Assume we have a $kd$-tuple $\bm= (m_0,m_1,\ldots,m_{kd-1}) \in \bM_\ell (d;\bnu)$.\\[.5ex]
(${\mathbf A}$1) Set $\mathbf{w}$ to be the empty word and $j = 0, t = 0$. Go to $(\mathbf{A}$2).\\[.5ex]
(${\mathbf A}$2) Set $\mathbf{w}$ to be the word obtained by concatenating $m_{jd + t}$ $\nu_{j}$'s at the right of $\mathbf{w}$. If $j = k-1$ and $t = d-1$, return $\mathbf{w}$ and terminate the algorithm.
Otherwise, go to $(\mathbf{A}$3).\\[.5ex]
(${\mathbf A}$3) Set $\mathbf{w}$ to be the word obtained by concatenating $0$ at the right. Go to $(\mathbf{A}$4).\\[.5ex]
(${\mathbf A}$4) If $t \neq d-1$ then set $t = t + 1$ and go to $(\mathbf{A}$2). If $t = d-1$ set $j= j+1$ and $t = 0$, and go to $(\mathbf{A}$2).
\end{algorithm}
Set $\wordset_\ell(d;\bnu) \seteq \{\mathbf{w}(\bm;d;\bnu) \mid \bm \in \bM_\ell (d;\bnu)  \}$.
Let $\Uppsi : {\bM_\ell} (d;\bnu) \ra \wordset_\ell(d;\bnu)$ be a map defined by $\Uppsi(\bm) = \mathbf{w}(\bm;d;\bnu)$.
\begin{lemma}\label{lem: injective}
The map $\Uppsi$ is a bijection.
\end{lemma}

Now we define a $C_d = \inn{\siggen_d}$-action on $\wordset_\ell(d;\bnu)$.
First, we break $\mathbf{w} = w_1 w_2\ldots w_{u}$ into subwords of length $d$ as many as possible as follows:
\begin{align*}
\mathbf{w} & = w^1 \ | \ w^2 \ | \ \cdots \ | \ w^t \ | \ w_{td+1}\cdots w_u,
\end{align*}
where $t = \lfloor u/d \rfloor$ and $w^j=w_{(j-1)d+1}w_{(j-1)d+2} \cdots w_{jd}$ for $1 \le j \le t$.
Note that $\siggen_d$ acts on each
subword $w^j$ by cyclic shift, i.e., $\siggen_d \cdot w^j \seteq w_{jd} w_{(j-1)d+1} w_{(j-1)d+2} \cdots w_{jd-1}$.
Assume that $j_0$ is the smallest integer such that $ \siggen_d \cdot w^j_0 \ne w^j_0$.
Then we set
\begin{align}\label{eq: def of hsig}
\siggen_d \sqact \mathbf{w} \seteq w^1 \ | \ w^2 \ | \ \cdots \ | \ w^{{j_0}-1} \ | \ \siggen_d \cdot w^{j_0} \ | \ w^{j_0+1} \ | \ \cdots \ | \ w^t \ | \ w_{td+1}\cdots w_{u}.
\end{align}
If there is no such $j_0$, we set $ \siggen_d \sqact \mathbf{w} \seteq \mathbf{w}$.

\begin{theorem}\label{thm: hsigaction}
For any $\bnu = (\nu_0, \nu_1,\ldots, \nu_{k-1}) \in \Z^k_{> 0}$, the action defined as above is indeed a $C_d$-action on $\wordset_\ell(d;\bnu).$
\end{theorem}

Now we define a $C_d$-action on $\bM_\ell (d;\bnu)$
by transporting the $C_d$-action $\sqact$ on $\wordset_\ell(d;\bnu)$ via the bijection $\Uppsi$, that is,
\begin{align}\label{eq: Sagact}
\siggen_d \sqact \bm \seteq \Uppsi^{-1}( \siggen_d \sqact \Uppsi(\bm))\quad \text{for all $\bm \in \bM_\ell (d;\bnu)$}.
\end{align}

\begin{remark}\label{rem: hsig remark}
Suppose that $C_d$ acts on $\bM_\ell(d;\bnu)$ as in~\eqref{eq: Sagact}.
Then, for any $r \in \Z_{>0}$, $\bM_\ell(d;\bnu)$ is also equipped with a $C_{rd}$-action $\sqact_d$ given by
\begin{align}\label{eq: hsigaction multi}
\siggen_{rd} \sqact_d \bm \seteq \siggen_d \sqact \bm.
\end{align}
\end{remark}

Let us generalize the above setting a little further. Let $d, k, k',r \in \Z_{} > 0$ and $\ell \in \Z_{\ge 0}$. For $\bnu=(\nu_0, \nu_1,\ldots, \nu_{k-1}) \in \Z^k_{> 0}$ and $\bnu'=(\nu'_0, \nu'_1,\ldots, \nu'_{k'-1}) \in \Z^{k'}_{> 0}$, set
\begin{align}\label{eq: def of M}
    \bM_\ell(rd,d;\bnu,\bnu') \seteq \left\{ \bm \in \Z_{\ge 0}^{krd + k'd} \; \middle| \;
    \sum_{0 \le j \le k-1} \nu_j \bm[j;rd] + \sum_{0 \le j \le k'-1} \nu_j'\bm[kr + j;d] =
    \ell \right\}.
\end{align}
Using the actions given in~\eqref{eq: Sagact} and~\eqref{eq: hsigaction multi}, we define a new $C_{rd}$-action, denoted by $\sqact_{rd,d}$, on $\bM_\ell(rd,d;\bnu,\bnu')$ as follows:
Given $\bm \in \bM_\ell(rd,d;\bnu,\bnu')$, we break it into
$\bm_{\le krd-1} \seteq (m_0,m_1,\ldots,m_{krd - 1}) \in \bM_l(rd;\bnu)$ and $\bm_{\ge krd} \seteq (m_{kd}, m_{kd+1}, \ldots, m_{krd+k'd-1}) \in \bM_{l'}(d;\bnu')$, where $\ell = l + l'$.
Now, we define
\begin{align}\label{eq: hsigaction double}
\siggen_{rd} \sqact_{rd,d} \bm \seteq \begin{cases}
(\siggen_{rd} \sqact \bm_{\le krd-1})* \bm_{\ge krd}  & \text{if $\siggen_{rd} \sqact \bm_{\le krd-1} \neq \bm_{\le krd-1}$,} \\
\bm_{\le krd-1}* (\siggen_{rd} \sqact_d \bm_{\ge krd}) & \text{otherwise,}
\end{cases}
\end{align}
where $\bm * \bm'$ is the tuple obtained by concatenating $\bm$ and $\bm'$.

\section{(Bi)cyclic sieving phenomena on $\pclp{\ell}$} \label{Sec: CSP1}
The {\it cyclic sieving phenomenon} was introduced by Reiner--Stanton--White in \cite{RSW}.
Let $X$ be  a finite set, with an action of a cyclic group $C$ of order $m$. Elements within a $C$-orbit share the same stabilizer subgroup, whose cardinality is called the \emph{stabilizer-order} for the orbit.
Let $X(q)$ be a polynomial in $q$ with nonnegative integer coefficients.
For $d \in \Z_{>0}$, let $\omega_d$ be a $d$th primitive root of the unity.
We say that $(X, C, X(q))$ exhibits the  cyclic sieving phenomenon if, for all $c\in C$, we have
$$
|X^c| = X(\omega_{o(c)}),
$$
where $o(c)$ is the order of $c$ and $X^c$ is the fixed point set under the action of $c$.
Note that this condition is equivalent to the following congruence:
\begin{align*}
X(q)\equiv \sum_{0 \le i \le m-1}  b_i  q^i \pmod {q^m-1},
\end{align*}
where $b_i$ counts the number of $C$-orbits on $X$ for which the stabilizer-order divides $i$.

A generalization of the cyclic sieving phenomenon, called the bicyclic sieving phenomenon, was introduced in~\cite[Section 3]{BRS}. Let $X$ be a finite set with a permutation action of a finite \emph{bicyclic group}, that is, a product $C_m \times C_{m'}$ for some $m,m' \in \Z_{>0}$.
Fix embeddings $\omega : C_m  \ra \C^{\times}$ and $\omega': C_{m'}\ra \C^{\times}$ into the complex roots of unity.
Let $X(q_1,q_2) \in \Z_{\ge 0 }[q_1,q_2]$.
We say that the triple $(X,C_m\times C_{m'},X(q_1,q_2))$ exhibits the \emph{bicyclic sieving phenomenon} if for all $(c,c') \in C_m \times C_{m'}$, we have
\[
X(\omega(c), \omega'(c')) = \left| \left\{ x \in X \; \middle| \; (c,c')\,x = x \right\} \right|.
\]
This condition is equivalent to the following congruence:
\[X(q_1,q_2) \equiv \sum_{ 0 \le j_1 <m, \; 0 \le j_2 < m' } b(j_1,j_2)q_1^{j_1}q_2^{j_2} \pmod{q_1^{m} - 1, q_2^{m'} -1} \]
where $b(j_1,j_2)$ is the number of orbits of $C_m \times C_{m'}$ on $X$ satisfying certain conditions (see~\cite[Proposition 3.1]{BRS}).

Now, let us introduce the triple for the (bi)cyclic sieving phenomenon on $\pclp{\ell}$.
First, we let $X \seteq \pclp{\ell} = \bigsqcup_{\Lambda \in \DRpclp} \pclp{\ell} (\Lambda)$.

To define a (bi)cyclic group action, we note that the symmetric group $\Sym_{[0,n]}$ over the set $\{0,1,\ldots, n\}$ acts on $\pclp{\ell}$ by permuting indices of coefficients, that is,
\[
\sigma \cdot \sum_{0 \le i \le n} m_i\Lambda_i = \sum_{0 \le i \le n} m_{\sigma(i)}\Lambda_i \quad \text{for $\sigma \in \Sym_{[0,n]}$}.
\]
We also note that if $\g \ne D_n^{(1)} (n \equiv_2 0) $  then $P_0/Q_0 \simeq C_{\congN}$, where $C_{\congN}$ is a cyclic group of order $\congN$.
For   $\g \ne D_n^{(1)}(n\equiv_2 0)$,   we take an appropriate $\upsigma \in \Sym_{[0,n]}$ of order $\congN$. For instance, we let
\[
\upsigma = \begin{cases}
(0,1,\ldots,n)& \text{if $\g =A_{n}^{(1)}$},\\
(0,n,1,n-1)(2,3)(4,5)\cdots(n-3,n-2) & \text{if $\g = D_{n}^{(1)}(n\equiv_2 1)$}.
\end{cases}
\]
Indeed $\congN=n+1$ if $\g =A_{n}^{(1)}$ and $\congN=4$ if $\g = D_{n}^{(1)}(n\equiv_2 1)$.
For other types, see~\cite[Table 4.1]{19KOO}.
Now, we define a $C_\congN = \inn{\sigma_\congN}$-action on $\pclp{\ell}$ by
\begin{align}\label{eq: CSP action}
\sigma_\congN \cdot \sum_{0 \le i \le n} m_i \Lambda_i \seteq \sum_{0 \le i \le n} m_{\upsigma(i)} \Lambda_i \quad \text{for any $\sum_{0 \le i \le n} m_i \Lambda_i \in \pclp{\ell}$.}
\end{align}

For $\g =D_n^{(1)}(n\equiv_2 0)$, we let
\[
\upsig^{(1)} = (0,n)(1,n-1) \in \Sym_{[0,n]} \quad \text{and}\quad  \upsig^{(2)} = (0,1)(2,3)\cdots(n-4,n-3)(n-1,n) \in \Sym_{[0,n]}.
\]
Note that $\upsig^{(1)}$ and $\upsig^{(2)}$ commute to each other in $\Sym_{[0,n]}$, so $\langle \upsig^{(1)}, \upsig^{(2)} \rangle \simeq C_2 \times C_2$. We define a $C_2 \times C_2 = \inn{\siggen_2} \times \inn{\siggen_2}$-action on $\pclp{\ell}$ by ($e$ denotes the identity of $C_2$)
\begin{align}\label{eq: C2 times C2 action}
(\siggen_2,e) \cdot \sum_{0 \le i \le n} m_i \Lambda_i \seteq \sum_{0 \le i \le n} m_{\upsig^{(1)}(i)} \Lambda_{i} \quad \text{and} \quad (e,\siggen_2) \cdot \sum_{0 \le i \le n} m_i \Lambda_i \seteq \sum_{0 \le i \le n} m_{\upsig^{(2)}(i)} \Lambda_{i}.
\end{align}

Finally, to define $X(q)$ or $X(q_1,q_2)$, let us consider its generating function as follows: For $\g \neq D_n^{(1)}(n\equiv_2 0)$,
let $\widetilde{\svv} = (s_0,s_1,s_2,\ldots,s_n)$. For $\ell \in \Z_{\ge 0}$, we define $X(q) \seteq \pclp{\ell}(q)$ by
\begin{align}\label{eq: geometric form}
\sum_{\ell \ge 0} \pclp{\ell}(q)t^\ell  \seteq \prod_{0 \le i \le n} \dfrac{1}{1-q^{s_i}t^{a_i^\vee}}.
\end{align}
where $\svv$ is the root-sieving vector given in \cref{conv: Sieving}. Then we have
\begin{align*}
\pclp{\ell}(q) &= \sum_{i \ge 0} \left| \left\{\Lambda \in \pclp{\ell} \; \middle| \; \evS(\Lambda) = i
\right\} \right|q^i \equiv \sum_{\Lambda \in  \DRpclp } \left|\pclp{\ell} (\Lambda) \right|q^{ \evS(\Lambda)}   \pmod{q^\congN-1}. % \quad   \text{(by)}.
\end{align*}

For $\g = D_n^{(1)}(n\equiv_2 0)$, we let
$\fks^{(1)} \seteq \frac{1}{2}\wtsvv^{(1)}$ and $\fks^{(2)} \seteq \frac{1}{2}\wtsvv^{(2)}$ and
define $X(q_1,q_2) \seteq \pclp{\ell}(q_1,q_2)$ for $\ell \in \Z_{\ge 0}$ by
$$\sum_{\ell \ge 0} \pclp{\ell}(q_1,q_2)t^\ell \seteq \prod_{0 \le i \le n} \dfrac{1}{1-q_1^{\fks^{(1)}_i}q_2^{\fks^{(2)}_i}t^{a_i^\vee}}.$$

For $t = 1,2$, let $\ev_{\fks^{(t)}}: \pclp{\ell} \ra \Z_{\ge 0 }$ be a map defined by $\ev_{\fks^{(t)}}\left(\sum_{0 \le i \le n} m_i \Lambda_i \right) =  \fks^{(t)} \bullet \mathbf{m}$. Then we have
\begin{align}\label{eq: Dnotcong_n40}
\pclp{\ell}(q_1,q_2) & \equiv \sum_{\Lambda \in \DRpclp} \left|\pclp{\ell} (\Lambda)\right| q_1^{\ev_{\fks^{(1)}}(\Lambda)} q_2^{\ev_{\fks^{(2)}}(\Lambda)} \pmod{q_1^{2} - 1, q_2^{2} - 1}.
\end{align}


\begin{theorem}\label{thm: CSP}
For $\g  \ne D_n^{(1)}(n \equiv_2 0)$,   the triple $\left(\pclp{\ell}, C_{\congN}, \pclp{\ell}(q) \right)$
exhibits the cyclic sieving phenomenon under the $C_\congN$-action given in~\eqref{eq: CSP action}.
\end{theorem}

For $\g = A_{n}^{(1)}$, \cref{thm: CSP} follows from the classical result~\cite[Theorem 1.1 (a)]{RSW} of cyclic sieving phenomena.
For other types, it can be proved by using the action defined in~\eqref{eq: hsigaction double}.
We here only deal with $D_n^{(1)}(n\equiv_{2}1)$ type since the method of proof for each type is essentially same.

For $m \in \Z_{\ge 0}$ and $k \in \Z_{>0}$, we denote by $m^k$ the sequence $m, m, \ldots, m$ consisting of $k$ $m$'s.
Moreover, we let $\eta \seteq \frac{n-3}{2}$ and
\[
\pclp{\ell}(q) \equiv \sum_{0 \le i \le 3} b_i q^i \pmod{q^4 - 1}.
\]
\begin{lemma}\label{lem: Dn1 hsig and Cact}
    Under the $C_4$-action on $\pclp{\ell}$ given in~\eqref{eq: CSP action} and the $C_4$-action on $\bM_\ell(4,2;(1),(2^\eta))$ given in \eqref{eq: hsigaction double}, we have
 \[ \pclp{\ell}(q)\vert_{q=i} = \big| \big( \pclp{\ell} \big)^{C_4} \big| = \big| \bM_\ell(4,2;(1),(2^\eta))^{C_4} \big| = b_0 - b_2 \quad \text{and} \quad b_1 =b_3.\]
\end{lemma}
\begin{lemma}\label{lem: Dn1 hsig' and Cact^2}
Under the $C_4$-action on $\pclp{\ell}$ given in~\eqref{eq: CSP action} and the $C_2$-action on $\bM_\ell(2,1;(1^2),(2^{2\eta}))$ given in \eqref{eq: hsigaction double}, we have
\[ \pclp{\ell}(q)\vert_{q=-1} = \big| \big( \pclp{\ell} \big)^{\siggen_4^2} \big| = \big|  \bM_\ell(2,1;(1^2),(2^{2\eta}))^{C_2} \big| = b_0 - b_1 + b_2 - b_3.\]
\end{lemma}

\begin{proof}[\underline{Proof of \cref{thm: CSP} for $\g = D_n^{(1)}(n\equiv_2 1)$}]
Let $\zeta_4$ be a $4$th primitive root of unity. We will see that
\[\big|\big( \pclp{\ell} \big)^{\siggen_4^j} \big| = \pclp{\ell}(\zeta_4^j) \quad \text{for $j = 0,1,2,3$.}\]
When $j = 0$, since $\big|\big( \pclp{\ell} \big)\big| =  \pclp{\ell}(1)$, it is trivial. For the case $j \in \{1,3\}$, note that
$\left( \pclp{\ell} \right)^{C_4} = \left( \pclp{\ell} \right)^{\siggen_4^j}$ and $\pclp{\ell}(\zeta_4^j) =  b_0 + b_1\zeta^j_4 - b_2 - b_3\zeta^j_4$.
Thus, by \cref{lem: Dn1 hsig and Cact}, we have
\[\big| \big( \pclp{\ell} \big)^{C_4} \big| =\big|  \bM_\ell(4,2;(1),(2^\eta))^{C_4} \big| = b_0 - b_2 = \pclp{\ell}(\zeta_4^j). \]
For the case $j = 2$, note that
$\pclp{\ell}(-1) = b_0 - b_1 + b_2 - b_3$.
Thus, by \cref{lem: Dn1 hsig' and Cact^2}, we have
\[\big| \big( \pclp{\ell} \big)^{\siggen_4^2} \big| = \big|  \bM_\ell(2,1;(1^2),(2^{2\eta}))^{C_2} \big| = b_0 - b_1 + b_2 - b_3 = \pclp{\ell}(-1).\qedhere\]
\end{proof}

For $\g = D_n^{(1)} (n \equiv_2 0)$, we obtain the bicyclic sieving phenomenon in a similar way as above.
\begin{theorem}\label{eq: Dn1 even biCSP}
For $\g = D_n^{(1)} (n \equiv_2 0)$, the triple
$\left( \pclp{\ell} \ ,C_2\times C_2 \ , \pclp{\ell}(q_1,q_2) \right)$ exhibits the bicyclic sieving phenomenon, under the $C_2 \times C_2$-action given in~\eqref{eq: C2 times C2 action}.
\end{theorem}

\section{Formulae on the number of maximal dominant weights} \label{Sec: application}

Using the (bi)cyclic sieving phenomenon, we derive closed formulae for $|\mx^+(\Lambda)|$ in terms of binomial coefficients.
In particular, for affine Kac--Moody algebras of classical type, we show that our closed formulae for $|\mx^+(\Lambda)|$ can be written as a sum of binomial coefficients. For instance, when $\Lambda \in \DRpclp$, we have
\begin{align}
\big|\mx_{A_n^{(1)}}^+(\Lambda)\big| & = \sum_{d \mid (n+1,\ell,i)} \dfrac{d}{(n+1)+\ell} \sum_{d'|(\frac{n+1}{d},\frac{\ell}{d})} \upmu(d') \matr{((n+1)+\ell)/dd'}{\ell/dd'},\label{eq: closed formulae}  \\
 \big|\mx_{B_n^{(1)}}^+(\Lambda)\big| & = \dfrac{1}{2}\left(\pclp{\ell}(1) + (-1)^{\delta_{in}}\pclp{\ell}(-1)\right)= \matr{n+\floor{\frac{\ell-\delta_{in}}{2}}}{n}+\matr{n+\floor{\frac{\ell-1-\delta_{in}}{2}}}{n}.
 \nonumber
\end{align}
Here $\upmu$ denotes the classical M\"obius function (see \cite[Table 2.2, Table 6.2]{19KOO} for more details).

From our closed formulae, we find recursive formulae for $|\mx^+(\Lambda)|$ which can be described as triangular arrays. For instance, let us consider the case $\g = A_{2n}^{(2)}$.
Define $T^{A^{(2)}_{\even}}_0:\Z_{\ge 0} \times \Z_{\ge 0 } \ra \Z_{\ge 0 }$ by
$T^{A^{(2)}_{\even}}_0(n,i) = 1 ~ (n \ge 0, i =0,1)$, $T^{A^{(2)}_{\even}}_0(0,\ell) = 1 ~ (\ell \ge 2) $ and
\begin{align}\label{eq: recursive A2n2}
T^{A^{(2)}_{\even}}_0(n,\ell) = T^{A^{(2)}_{\even}}_0(n,\ell-2) + T^{A^{(2)}_{\even}}_0(n-1,\ell).
\end{align}
Then for all $n \in \Z_{\ge 2},~\ell \in \Z_{\ge 0 }$, we have $T^{A^{(2)}_{\even}}_0(n,\ell) = |\mx^+_{A_{2n}^{(2)}}(\ell \Lambda_0)| = \binom{n+\floor{\frac{\ell}{2}}}{n}$.  In particular, as a triangular array, $T^{A^{(2)}_{\even}}_0$ can be described as follows:

\fontsize{8.5}{8.5}\selectfont
\begin{equation*}
\begin{tikzpicture}
\node[] {
$\begin{array}{ccccccccccccccccccccccccc}
&&&&&&&\scriptstyle{n = 0} && \scriptstyle{\ell = 0}\\
&&&&&&\scriptstyle{n = 1}&&1 && \scriptstyle{\ell = 1} \\
&&&&&\scriptstyle{n = 2}&&1&&1&& \scriptstyle{\ell = 2} \\
&&&&\scriptstyle{n = 3}&&\mycirc{1}&&1&&1&& \scriptstyle{\ell = 3} \\
&&&\scriptstyle{n = 4}&&1&&1&&\mycirc{2}&&1&& \scriptstyle{\ell = 4} \\
&&\scriptstyle{n = 5}&&1&&1&&\mydoublecirc{3}&&2&&1&& \scriptstyle{\ell = 5} \\
&\scriptstyle{n = 6}&&1&&1&&4&&3&&3&&1&& \scriptstyle{\ell = 6} \\
\scriptstyle{n = 7}&&1&&1&&5&&\mybox{4}&&6&&3&&1&& \scriptstyle{\ell = 7} \\
&1&&1&&6&&5&&\mybox{10}&&6&&4&&1 \\
\udots&&\udots&&\udots&&\udots&&\vdots&&\ddots&&\ddots&&\ddots&&\ddots &.
\end{array}$
};
\draw (-6.6,-1.2) -- (-0.1,1.9);
\draw (6.4,-1.2) -- (-0.1,1.9);
\end{tikzpicture}
\end{equation*} \fontsize{12}{12}\selectfont

It is interesting to note that this array coincides with Pascal's triangle with duplicated diagonals, i.e., $T^{A^{(2)}_{\even}}_0(n,2\ell) = T^{A^{(2)}_{\even}}_0(n,2\ell + 1) = \binom{n + \ell}{\ell}$ for $n, \ell \ge 0$, which appears in~\cite[A065941]{OEIS} in a totally different context from ours. %  whose row sums give Fibonacci sequence.
The recursive condition~\eqref{eq: recursive A2n2} says if we add the $\mycirc{1}$ and the $\mycirc{2}$ then we get $\mydoublecirc{3}$.

For another example, let us consider the case $\g = B_n^{(1)}$. Define $T^{B^{(1)}}_0:\Z_{\ge 0} \times \Z_{\ge 0 } \ra \Z_{\ge 0 }$ by
$T^{B^{(1)}}_0(n,0) = 1 ~ (n \ge 1)$, $T^{B^{(1)}}_0(n,1) = 2 ~ (n \ge 1)$, $T^{B^{(1)}}_0(0,\ell) = 2 ~ (\ell \ge 0)$ and
\[
T^{B^{(1)}}_0(n,\ell) = T^{B^{(1)}}_0(n,\ell-2) + T^{B^{(1)}}_0(n-1,\ell) ~(n\ge 1,~\ell \ge 2).
\]
Then for all $n \in \Z_{\ge 3},~\ell \in \Z_{\ge 0 }$, we have $T^{B^{(1)}}_0(n,\ell) = |\mx^+_{B_n^{(1)}}(\ell \Lambda_0)|$.  In particular, as a triangular array, $T_0^{B^{(1)}}$ can be described as follows:


\fontsize{8.5}{8.5}\selectfont
\begin{equation*}
\begin{tikzpicture}
\node[] {
	$\begin{array}{ccccccccccccccccccccccccc}
	&&&&&&&\scriptstyle{n = 0} && \scriptstyle{\ell = 0}\\
	&&&&&&\scriptstyle{n = 1}&&2 && \scriptstyle{\ell = 1} \\
	&&&&&\scriptstyle{n = 2}&&1&&2&& \scriptstyle{\ell = 2} \\
	&&&&\scriptstyle{n = 3}&&1&&2&&2&& \scriptstyle{\ell = 3} \\
	&&&\scriptstyle{n = 4}&&1&&2&&3&&2&& \scriptstyle{\ell = 4} \\
	&&\scriptstyle{n = 5}&&1&&2&&4&&4&&2&& \scriptstyle{\ell = 5} \\
	&\scriptstyle{n = 6}&&1&&2&&5&&6&&5&&2&& \scriptstyle{\ell = 6} \\
	\scriptstyle{n = 7}&&1&&2&&6&&8&&9&&6&&2&& \scriptstyle{\ell = 7} \\
	&1&&2&&7&&10&&\mydoublebox{14}&&12&&7&&2 \\
	\udots&&\udots&&\udots&&\udots&&\vdots&&\ddots&&\ddots&&\ddots&&\ddots &.
	\end{array}$
};
\draw (-6.6,-1.1) -- (-0.1,1.7);
\draw (6.4,-1.1) -- (-0.1,1.7);
\end{tikzpicture}
\end{equation*} \fontsize{12}{12}\selectfont

This array is the triangular array obtained by removing the left boundary from the triangular array in \cite[A129714]{OEIS} whose row sums are the Fibonacci numbers.

We note that $T^{B^{(1)}}_0(n,\ell)$ can be obtained from $T^{A^{(2)}_{\even}}_0$ as follows:
\[
T^{B^{(1)}}_0(n,\ell) =  T^{A^{(2)}_{\even}}_0(n,\ell) + T^{A^{(2)}_{\even}}_0(n,\ell-1) \quad\text{ for $n,\ell \ge 1$}.
\]
For instance, if we add $\mybox{4}$, $\mybox{10}$ in $T^{A^{(2)}_{\even}}_0$ then we get $\mydoublebox{14}$ in $T^{B^{(1)}}_0(n,\ell)$.


\begin{remark}
Interestingly,  all triangular arrays for classical affine type except for untwisted affine $C$-type can be constructed by {\it boundary conditions} and the triangular array of twisted affine  even $A$-type. The triangular arrays for affine $C$-type (\cite[A034851]{OEIS}) can be constructed by boundary conditions and Pascal triangle  \cite[Appendix A]{19KOO}.
\end{remark}

On the other hand, we obtain an interesting duality between level and rank from our closed and recursive formulae. For instance, in case where $\g = A_n^{(1)}$, the closed formula in~\eqref{eq: closed formulae} implies that
for $n \ge 1$ and $\ell >1$, if $(n+1,\ell,i) = (\ell, n+1,j)$ for some $0 \le i,j \le \min(n,\ell)$, then
\begin{align}\label{eq: An level-rank duality}
\big|\mx_{A_n^{(1)}}^+((\ell-1)\Lambda_0 + \Lambda_i)\big| = \big|\mx_{A_{\ell-1}^{(1)}}^+(n\Lambda_0 + \Lambda_j)\big|,
\end{align}
i.e., exchanging $n+1$ with $\ell$ preserves the number of dominant maximal weights.

Let us deal with the relation between our duality and Frenkel's duality in \cite{81Frenkel}. For a residue $i$ modulo $n$, let $\Lambda_i^{(n)}$ denote the $i$th fundamental weight of $A_{n}^{(1)}$. For $\Lambda = \sum_{i=0}^n m_i\Lambda_i^{(n)} \in \pclp{\ell}$, let $\Lambda'$ be the dominant integral weight of $A_{\ell-1}^{(1)}$ defined by
\[ \Lambda' =  \sum_{i=0}^{n} \Lambda_{m_i + m_{i+1} + \cdots + m_n}^{(\ell-1)} \in \pclp{n + 1}.\]
With this setting, Frenkel found the following beautiful duality between the $q$-specialized characters of $V(\Lambda)$ and $V(\Lambda')$:
\[ \dim_q(V(\Lambda)) \prod_{k = 0}^\infty \dfrac{1}{1 - q^{(n+1)k}} = \dim_q(V(\Lambda')) \prod_{k=0}^\infty \dfrac{1}{1-q^{\ell k}}.\]
Here $\dim_q(V)$ denotes the $q$-specialized character of $V$ (see \cite[Theorem 2.3]{81Frenkel} or \cite[Section 4.4]{T}).
Based on~\eqref{eq: An level-rank duality}, we have
\[ \big|\mx_{A_n^{(1)}}^+(\Lambda)\big| = \big|\mx_{A_{\ell-1}^{(1)}}^+(\Lambda')\big|. \]

In case where $\g = B_n^{(1)}$, the closed formula in~\eqref{eq: closed formulae} implies the following symmetries:
\begin{enumerate}
\item[{\rm (1)}] For $n \ge 3$, $\ell \ge 7$ and $\ell \equiv_2 1$, we have
\begin{align*}
\big|\mx_{B_n^{(1)}}^+(\ell\Lambda_0)\big| = \big|\mx_{B_{(\ell-1)/2}^{(1)}}^+((2n+1)\Lambda_0)\big|,
\end{align*}
\noindent i.e., exchanging $n$ with $(\ell-1)/2$ preserves the number of dominant maximal weights.
\item[{\rm (2)}] For $n \ge 3$, $\ell \ge 8$ and $\ell \equiv_2 0$, we have
\begin{align*}
\big|\mx_{B_n^{(1)}}^+((\ell-1)\Lambda_0+\Lambda_n)\big| = \big|\mx_{B_{\ell/2-1}^{(1)}}^+((2n+1)\Lambda_0+\Lambda_{\ell/2 -1 })\big|,
\end{align*}
\noindent i.e., exchanging $n$ with $\ell/2-1$ preserves the number of dominant maximal weights.
\end{enumerate}
%\acknowledgements{}

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