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\title[Combinatorics of generalized exponents]{Combinatorics of generalized exponents}

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\author{C\'edric Lecouvey\thanks{\href{mailto:cedric.lecouvey@lmpt.univ-tours.fr}{
			cedric.lecouvey@lmpt.univ-tours.fr}}\addressmark{1}
	\and Cristian Lenart\thanks{\href{mailto:clenart@albany.edu}{clenart@albany.edu}. The second author gratefully acknowledges the partial support from the NSF grant DMS--1362627.}\addressmark{2}
}
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\address{\addressmark{1}Facult\'e des Sciences et Techniques, Universit\'e de Tours Tours, France\\ \addressmark{2}Department of Mathematics and Statistics, State University of New York at Albany, 1400 Washington Avenue, Albany, NY 12222, U.S.A.}

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\received{\November 10 2018}

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\abstract{We give a purely combinatorial proof of the positivity of the stabilized forms
	of the generalized exponents associated to each classical root system. In finite type $C_{n}$, we
	obtain a combinatorial description of the generalized exponents based on symplectic King tableaux. We also present three applications of our combinatorial formula. Our methods are expected to extend
	to the orthogonal types.
}

\resumetitle{R\'esum\'e}
%% put your French abstract here, or comment this out if you don't have one
\resume{Pour chaque syst\`{e}me de racines de type classique, nous donnons une
	preuve combinatoire de la positivit\'{e} de la forme stabilis\'{e}e des
	exposants g\'{e}n\'{e}ralis\'{e}s. En type $C_{n}$, nous obtenons une
	description combinatoire des exposants g\'{e}n\'{e}ralis\'{e}s en termes de
	tableaux de King et en donnons trois applications. Nos m\'{e}thodes
	doivent s'\'{e}tendre aux types orthogonaux.	
}

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\keywords{Weyl characters, Lusztig $q$-analogues, representation theory, crystals, branching rules.
}

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\begin{document}
	
\maketitle

\section{Introduction}

Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$ of rank $n$ and
$G$ its corresponding Lie group.\ The group $G$ acts on the symmetric algebra
$S(\mathfrak{g})$ of $\mathfrak{g}$, and it was proved by Kostant\ \cite{kost}
that $S(\mathfrak{g})$ factors as $S(\mathfrak{g})=H(\mathfrak{g}%
)\otimes S(\mathfrak{g})^{G}$, where $H(\mathfrak{g})$ is the harmonic part of
 $S(\mathfrak{g)}$. The generalized exponents of
$\mathfrak{g}$, as defined by Kostant \cite{kost}, are the polynomials appearing as the coefficients in the
expansion of the graded character of $H(\mathfrak{g)}$ in the basis of the
Weyl characters.\ It was shown by Hesselink \cite{hes} that these polynomials
coincide, in fact, with the Lusztig $t$-analogues \cite{Lut} of zero weight multiplicities
in the irreducible finite-dimensional representations of $\mathfrak{g}$. In
particular, they have non-negative integer coefficients, because they are affine
Kazhdan--Lusztig polynomials.

For $\mathfrak{g}=\mathfrak{sl}_{n}$, the generalized exponents admit a nice
combinatorial description in terms of the Lascoux--Sch\"{u}tzenberger charge
statistic on semistandard tableaux of zero weight \cite{LSc1}. This
statistic is defined via the cyclage operation on tableaux, which is based on the Schensted
insertion scheme. This combinatorial description extends, in fact, to any
Lusztig $t$-analogue of type $A_{n-1}$, that is, possibly associated to a
nonzero weight (also called Kostka polynomials). Another
interpretation of the charge statistic in terms of crystals \cite{HK,kash} of type $A_{n-1}$
was given later by Lascoux, Leclerc and Thibon in \cite{LLT}.

Despite many efforts during the last three decades, no general combinatorial
proof of the positivity of the Lusztig $t$-analogues is known beyond type $A$.
Nevertheless such proofs have been obtained in some particular cases \cite{Ion}, \cite{Ion2}, \cite{lec} and \cite{Lec2}.

In this extended abstract, we give a combinatorial description
of the stabilized version of the generalized exponents and a proof of their
positivity by using the combinatorics of type $A_{+\infty}$ crystal graphs.
This can be regarded as a generalization of results in \cite{LLT} for the
weight zero, and in fact we were able to rederive the latter without any reference to
the charge statistic or the combinatorics of semistandard tableaux. Our
description is in terms of the so-called distinguished vertices in crystals of type
$A_{+\infty}$, but we show that these vertices are in natural bijection with
some generalizations of symplectic King tableaux, which makes the link with stable
Lusztig $t$-analogue more natural. Next, we provide a complete combinatorial
proof of the positivity of the generalized exponents in the non-stable $C_{n}$
case.\ Observe there that the non-stable case is much more involved than the
stable one, essentially because we need a combinatorial description of the non-Levi 
branching from $\mathfrak{gl}_{2n}$ to $\mathfrak{sp}_{2n}$, which is
complicated in general.\ Here one needs in a crucial way recent duality results
by Kwon \cite{kwoces,kwoldk} giving a crystal interpretation of the
previous branching and a combinatorial model relevant to its study. We strongly expect that our approach extends to the orthogonal types, once all the results of \cite{kwoldk} are available for the non-Levi orthogonal branchings. 
We refer the reader to \cite{LL} for complete proofs and more results.

Section~2 recalls the definition of
the generalized exponents. Section~3
is devoted to the combinatorial description of the stabilized form (in classical type) of the
generalized exponents in terms of distinguished tableaux, which we define and study
here. In Section~4, we give the promised combinatorial description of
the generalized exponents in type $C_{n}$ by using King tableaux \cite{King}. In Section~5, we give three applications of the description in Section~4.

%\textbf{Acknowledgment.} The second author gratefully acknowledges the partial support from the NSF grant DMS--1362627.

\section{Generalized exponents}
\subsection{Background}
Let $\mathfrak{g}_{n}$ be a simple Lie algebra over $\mathbb{C}$ of rank $n$
with triangular decomposition $\mathfrak{g}_{n}\mathfrak{=}\oplus _{\alpha
	\in R_{+}}\mathfrak{g}_{\alpha }\oplus \mathfrak{h}\oplus \oplus _{\alpha
	\in R_{+}}\mathfrak{g}_{-\alpha }\,,$so that $\mathfrak{h}$ is the Cartan
subalgebra of $\mathfrak{g}_{n}$ and $R_{+}$ its set of positive roots. The
root system $R=R_{+}\sqcup (-R_{+})$ of $\mathfrak{g}_{n}$ is realized in a
real Euclidean space $E$ with inner product $(\cdot ,\cdot )$.\ For any $%
\alpha \in R,$ we write $\alpha ^{\vee }=\frac{2\alpha }{(\alpha ,\alpha )}$
for its coroot. Let $S\subset R_{+}$ be the subset of simple roots and $Q_{+}
$ the $\mathbb{Z}_{+}$-cone generated by $S$. The set $P$ of integral
weights for $\mathfrak{g}_{n}$ satisfies $(\beta ,\alpha ^{\vee })\in 
\mathbb{Z}$ for any $\beta \in P$ and $\alpha \in R$. We write $%
P_{+}=\{\beta \in P\mid (\beta ,\alpha ^{\vee })\geq 0$ for any $\alpha \in
S\}$ for the cone of dominant weights of $\mathfrak{g}_{n}$, and denote by $%
\omega _{1},\ldots ,\omega _{n}$ its fundamental weights. Let $W$ be the
Weyl group of $\mathfrak{g}_{n}$ generated by the reflections $s_{\alpha }$
with $\alpha \in S$, and write $\ell $ for the corresponding length function.

By a classical theorem due to Kostant, the graded character of the harmonic
part of the symmetric algebra $S(\mathfrak{g}_{n})$ satisfies%
\[
\mathrm{char}_{t}(H(\mathfrak{g}_{n}))=\frac{\prod_{i=1}^{n}(1-t^{d_{i}})}{%
	(1-t)^{n}}\prod_{\alpha \in R}\frac{1}{1-te^{\alpha }}%
=\prod_{i=1}^{n}(1-t^{d_{i}})\mathrm{char}_{t}(S(\mathfrak{g}_{n}))\,,
\]%
where we have $d_{i}=m_{i}+1$, for $i=1,\ldots ,n$, and $m_{1},\ldots ,m_{n}$
are the (classical) exponents of $\mathfrak{g}_{n}$. In type $A_{n}$ we have 
$m_{i}=i$, in types $B_{n}$ and $C_{n}$ $m_{i}=2i-1$ and in type $D_{n}$ $%
m_{i}=2i-1$ for $i=1,\ldots ,n-1$ with $m_{n}=n-1$. On the other hand, it is
known (see \cite{hes}) that $\mathrm{char}_{t}(H(\mathfrak{g}_{n}))$
coincides with the Hall-Littlewood polynomial $Q_{0}^{\prime }$, namely we
have 
\[
\mathrm{char}_{t}(H(\mathfrak{g}_{n}))=Q_{0}^{\prime }=\sum_{\lambda \in
	P_{+}}K_{\lambda ,0}^{\mathfrak{g}_{n}}(t)s_{\lambda }^{\mathfrak{g}_{n}}\,,
\]%
where $s_{\lambda }^{\mathfrak{g}_{n}}$ is the Weyl character associated to
the finite-dimensional irreducible representation $V(\lambda )$ of $%
\mathfrak{g}_{n}$ with highest weight $\lambda $. The polynomials $K_{\lambda ,0}^{%
	\mathfrak{g}_{n}}(t)$ are the generalized exponents of $\mathfrak{g}_{n}$,
and they coincide with the Lusztig $t$-analogues \cite{Lut} associated to the zero
weight subspaces in the representations $V(\lambda )$. The classical
exponents $m_{1},\ldots ,m_{n}$ correspond to the adjoint representation of $%
\mathfrak{g}_{n}$, namely we have $K_{\widetilde{\alpha },0}^{\mathfrak{g}%
	_{n}}(t)=\sum_{i=1}^{n}t^{m_{i}}\,,$where $\widetilde{\alpha }$ is the
highest root in $R_{+}$.

\subsection{Classical types}
In classical types, $\mathrm{char}_{t}(S(\mathfrak{g}_{n}))$ is easy to
compute. Let $\mathcal{P}_{n}$ be the set of partitions with at most $n$
parts, and $\mathcal{P}$ the set of all partitions. The rank of the
partition $\gamma$ is defined as the sum of its parts, and is denoted by $%
\left\vert \gamma\right\vert$.

In type $A_{n-1}$, we start from the Cauchy identity%
\[
\prod_{1\leq i,j\leq n}\frac{1}{1-tx_{i}y_{j}}=\sum_{\gamma\in\mathcal{P}%
	_{n}}t^{\left\vert \gamma\right\vert }s_{\gamma}(x)s_{\gamma}(y). 
\]
Here $s_{\nu}(x)$ stands for the ordinary Schur function in the variables $%
x_{1},\ldots,x_{n}$. By setting $y_{i}=\frac{1}{x_{i}}$ for any $i=1,\ldots,n$, and by
considering the images of the symmetric polynomials in $R^{A_{n-1}}=\mathrm{%
	Sym}[x_{1},\ldots,x_{n}]/(x_{1}\cdots x_{n}-1)$, we get 
\begin{align}
\mathrm{char}_{t}(S(\mathfrak{sl}_{n}))&=(1-t)\sum_{\gamma\in\mathcal{P}%
	_{n}}t^{\left\vert \gamma\right\vert
}s_{\gamma}(x)s_{\gamma}(x^{-1})=(1-t)\sum_{\gamma\in\mathcal{P}%
	_{n}}t^{\left\vert \gamma\right\vert }s_{\gamma}s_{\gamma^{\ast}}=
\label{charSA} \\
&=(1-t)\sum_{\gamma\in\mathcal{P}_{n}}t^{\left\vert \gamma\right\vert
}\sum_{\lambda\in\mathcal{P}_{n-1}}c_{\gamma,\gamma^{\ast}}^{\lambda}s_{%
	\lambda}(x)\,.  \nonumber
\end{align}
Here $\gamma^{\ast}=-w_{\circ}(\gamma)$, where $w_{\circ}$ is the
permutation of maximal length in $S_{n}$ and the $c_{\gamma,\gamma^{\ast}}^{\lambda}$'s 
are the Littlewood-Richardson coefficients. We also use the same notation for a
symmetric polynomial and its image in $R^{A_{n-1}}$.

For any positive integer $m$, define $\mathcal{P}_{m}^{(2)}$ as the set of
partitions of the form $2\kappa$ with $\kappa\in\mathcal{P}_{m}$, and $%
\mathcal{P}_{m}^{(1,1)}$ as the subset of $\mathcal{P}_{m}$ containing the
partitions of the form $(2\kappa)^{\prime}$ with $\kappa\in\mathcal{P}$.
Moreover, we denote by $s_{\lambda}^{\mathfrak{so}_{2n+1}}$, $s_{\lambda}^{%
	\mathfrak{sp}_{2n}}$, and $s_{\lambda}^{\mathfrak{so}_{2n}}$ the irreducible
characters corresponding to the highest weight $\lambda$, for the Lie
algebras of types $B_n$, $C_n$, and $D_n$, respectively.

In type $B_{n}$, we start from the Littlewood identity \cite{Li} 
\[
\prod_{1\leq i<j\leq2n+1}\frac{1}{1-ty_{i}y_{j}}=\sum_{\nu\in\mathcal{P}%
	_{2n+1}^{(1,1)}}t^{\left\vert \nu\right\vert /2}s_{\nu}(y)\,, 
\]
and we specialize $y_{2n+1}=1,y_{2i-1}=x_{i}$, and $y_{2i}=\frac{1}{x_{i}}$,
for any $i=1,\ldots,n$.\ This gives 
\[
\mathrm{char}_{t}(S(\mathfrak{so}_{2n+1}))=\sum_{\nu\in\mathcal{P}%
	_{2n+1}^{(1,1)}}t^{\left\vert \nu\right\vert /2}\sum_{\lambda\in \mathcal{P}%
	_{n}}c_{\nu}^{\lambda}(\mathfrak{so}_{2n+1})s_{\lambda }^{\mathfrak{so}%
	_{2n+1}}\,,
\]
where $c_{\nu}^{\lambda}(\mathfrak{so}_{2n+1})$ is the branching coefficient
corresponding to the restriction from $\mathfrak{gl}_{2n+1}$ to $\mathfrak{so%
}_{2n+1}$. Similarly, we get: 
\begin{align*}
\mathrm{char}_{t}(S(\mathfrak{sp}_{2n})) & =\sum_{\nu\in\mathcal{P}%
	_{2n}^{(2)}}t^{\left\vert \nu\right\vert /2}\sum_{\lambda\in\mathcal{P}%
	_{n}}c_{\nu}^{\lambda}(\mathfrak{sp}_{2n})s_{\lambda}^{\mathfrak{sp}%
	_{2n}}\;\;\;\text{ and} \\
\mathrm{char}_{t}(S(\mathfrak{so}_{2n})) & =\sum_{\nu\in\mathcal{P}%
	_{2n}^{(1,1)}}t^{\left\vert \nu\right\vert /2}\sum_{\lambda\in\mathcal{P}%
	_{n}}c_{\nu}^{\lambda}(O_{2n})s_{\lambda}^{O_{2n}}\,.
\end{align*}
Note that here we considered the character $s_{\lambda}^{O_{2n}}$ of the $%
O(2n)$-module $V^{O(2n)}(\lambda)$ \linebreak parametrized by the partition $\lambda$.

\begin{proposition}
	\label{br-rules} We have the following identities.
	
	\begin{enumerate}
		\item In type $A_{n-1}$, for any $\lambda \in \mathcal{P}_{n-1}$, we have $%
		\frac{K_{\lambda ,0}^{\mathfrak{sl}_{n}}(t)}{\prod_{i=1}^{n}(1-t^{i})}%
		=\sum_{\gamma \in \mathcal{P}_{n}}t^{\left\vert \gamma \right\vert
		}c_{\gamma ,\gamma ^{\ast }}^{\lambda }\,.$
		
		\item In type $B_{n},$ for any $\lambda \in \mathcal{P}_{n}$, we have $\frac{%
			K_{\lambda ,0}^{\mathfrak{so}_{2n+1}}(t)}{\prod_{i=1}^{n}(1-t^{2i})}%
		=\sum_{\nu \in \mathcal{P}_{2n+1}^{(1,1)}}t^{\left\vert \nu \right\vert
			/2}c_{\nu }^{\lambda }(\mathfrak{so}_{2n+1})\,.$
		
		\item In type $C_{n}$, for any $\lambda \in \mathcal{P}_{n}$, we have $\frac{%
			K_{\lambda ,0}^{\mathfrak{sp}_{2n}}(t)}{\prod_{i=1}^{n}(1-t^{2i})}=\sum_{\nu
			\in \mathcal{P}_{2n}^{(2)}}t^{\left\vert \nu \right\vert /2}c_{\nu
		}^{\lambda }(\mathfrak{sp}_{2n})\,.$
		
		\item In type $D_{n}$, for any $\lambda \in \mathcal{P}_{n}$, we have $\frac{%
			K_{\lambda ,0}^{O(2n)}(t)}{(1-t^{n})\prod_{i=1}^{n-1}(1-t^{2i})}=\sum_{\nu
			\in \mathcal{P}_{2n}^{(1,1)}}t^{\left\vert \nu \right\vert /2}c_{\nu
		}^{\lambda }(O_{2n})\,.$
	\end{enumerate}
\end{proposition}

Observe that in the previous Assertion 1, the factor $(1-t)$ in (\ref{charSA}) gives the missing 
\textquotedblleft $ d_{i}=1$\textquotedblright\ in type $A_{n-1}$ and in Assertion 2, the partition $\lambda $ can 
have an odd rank. 

For type $D_{n}$, the dominant weights are not necessarily partitions,
whereas this is the case in Assertion 4 of the previous proposition. So here
we have in fact to write 
\[
K_{\lambda,0}^{O(2n)}(t)=K_{\omega(\lambda),0}^{\mathfrak{so}%
	_{2n}}(t)=K_{\iota(\omega(\lambda)),0}^{\mathfrak{so}_{2n}}(t) 
\]
for any partition $\lambda\in\mathcal{P}_{n}\setminus\mathcal{P}_{n-1}$ and $%
\omega(\lambda)$.

We have then by a theorem of Lascoux and Sch\"{u}tzenberger \cite{LSc1} 
\[
K_{\lambda ,0}^{\mathfrak{sl}_{n}}(t)=\sum_{T\in SST(\lambda )_{0}}t^{%
	\mathrm{ch}_{n}(T)}\,,
\]%
where $SST(\lambda )_{0}$ is the set of semistandard tableaux labeled by
letters of $\{1<\cdots <n\}$ of weight $\mu =(a,\ldots ,a)=0$ (i.e. each
letter $i$ appear $a$ times in $T$) where $a=\left\vert \lambda \right\vert
/n$, and $\mathrm{ch}_{n}(T)$ is the charge statistic evaluated on $T$.\
Recall that this charge statistic is defined by rather involved
combinatorial operation such as cyclage on tableaux.

\subsection{Stable versions}

\label{Section_Stable}When the ranks of the classical root systems
considered go to infinity, the previous relations simplify. In particular,
for $n$ sufficiently large, we have 
\[
c_{\nu }^{\lambda }(\mathfrak{so}_{2n+1})=\sum_{\delta \in \mathcal{P}%
}c_{\lambda ,2\delta }^{\nu }\,,\;\;\;c_{\nu }^{\lambda }(\mathfrak{sp}%
_{2n})=\sum_{\delta \in \mathcal{P}}c_{\lambda ,(2\delta )^{\prime }}^{\nu
}\,,\;\;\text{ and }\;\;c_{\nu }^{\lambda }(\mathfrak{so}_{2n})=\sum_{\delta
	\in \mathcal{P}}c_{\lambda ,2\delta }^{\nu }\,.
\]%
Observe that, for $\mathfrak{g}=\mathfrak{so}_{2n+1}$, this implies in
particular that $c_{\nu }^{\lambda }(\mathfrak{so}_{2n+1})=0$ when the ranks
of $\lambda $ and $\nu $ do not have the same parity, which is false in
general. Thus when $\left\vert \lambda \right\vert $ is even,$\,$we get the
relations%
\begin{align*}
\frac{K_{\lambda ,0}^{B_{\infty }}(t)}{\prod_{i=1}^{\infty }(1-t^{2i})}& =%
\frac{K_{\lambda ,0}^{D_{\infty }}(t)}{\prod_{i=1}^{\infty }(1-t^{2i})}%
=\sum_{\nu \in \mathcal{P}^{(1,1)}}\sum_{\delta \in \mathcal{P}%
	^{(2)}}t^{\left\vert \nu \right\vert /2}c_{\lambda ,\delta }^{\nu }\text{ in
	type }B_{\infty },D_{\infty }, \\
\frac{K_{\lambda ,0}^{C_{\infty }}(t)}{\prod_{i=1}^{\infty }(1-t^{2i})}&
=\sum_{\nu \in \mathcal{P}^{(2)}}\sum_{\delta \in \mathcal{P}%
	^{(1,1)}}t^{\left\vert \nu \right\vert /2}c_{\lambda ,\delta }^{\nu }\;\;\;%
\text{ in type }C_{\infty }\,.
\end{align*}%
In particular, this gives 
\begin{equation}
K_{\lambda ,0}^{B_{\infty }}(t)=K_{\lambda ,0}^{D_{\infty }}(t)\;\;\text{
	and }\;\;K_{\lambda ,0}^{B_{\infty }}(t)=K_{\lambda ^{\prime },0}^{C_{\infty
}}(t)\,.  \label{StableRedC}
\end{equation}%
All these stabilized forms are in fact formal power series in $t$ equal to
zero when the rank of $\lambda $ is odd.

\section{Stabilized generalized exponents and crystal graphs of type $%
	A_{+\infty}$}

\subsection{Crystal of type $A_{+\infty}$}

\label{sub_crystA} We refer to \cite{HK,kash} for complements on Kashiwara's crystals, including the standard notation. Recall that crystals of type $A_{+\infty }$ are those
associated to the infinite Dynkin diagram%
\[
\overset{1}{\circ }-\overset{2}{\circ }-\overset{3}{\circ }\cdot \cdot \cdot 
\]%
The partitions label the dominant weights of $\mathfrak{sl}_{+\infty }$. If
we denote by $(\omega _{i})_{\geq 1}$ the sequence of fundamental weights of 
$\mathfrak{sl}_{+\infty }$, we have for any partition $\lambda \in \mathcal{P%
}$,  $\lambda =\sum_{i}a_{i}\omega _{i}\,,$ where $a_{i}$ is the number of
columns with height $i$ in the Young diagram of $\lambda $.

To each partition $\lambda $ corresponds the crystal $B(\lambda )$ of the
irreducible infinite-dimensional representation of $\mathfrak{sl}_{+\infty }$
parametrized by $\lambda $. A classical model for $B(\lambda )$ is that of
semistandard tableaux of shape $\lambda $ on the infinite alphabet $\mathbb{Z%
}_{>0}=\{1<2<3<\cdots \}$. Given $b\in B(\lambda )$, we define $\boldsymbol{%
	\varepsilon }(b)=\sum_{i=1}^{+\infty }\varepsilon _{i}(b)\omega _{i}$ and $%
\boldsymbol{\varphi }(b)=\sum_{i=1}^{+\infty }\varphi _{i}(b)\omega _{i}\,,$ %
where both sums are in fact finite. The weight of $b\in B(\lambda )$ then
verifies $\mathrm{wt}(b)=\boldsymbol{\varphi }(b)-\boldsymbol{\varepsilon }%
(b)$.

\subsection{Combinatorial preliminaries}

In the sequel we consider the order $\leq$ on $\mathcal{P}$ such that $%
\lambda\leq\mu$ if and only if $\mu-\lambda\in P_{+}^{\infty}$, that is, $%
\mu-\lambda$ decomposes in the basis of the $\omega_{i}$'s with non-negative
integer coefficients.

The partitions in $\mathcal{P}^{(2)}$ (resp. in $\mathcal{P}^{(1,1)}$) are
those which can be tiled with horizontal (resp. vertical) dominoes.\
Equivalently, a partition $\kappa$ belongs to $\mathcal{P}^{(2)}$ (resp. $%
\mathcal{P}^{(1,1)}$) if and only if the number of columns (resp. rows) of
fixed height (resp. length) is even. So 
\[
\kappa\in\mathcal{P}^{(2)}\;\;\Longleftrightarrow\;\;\kappa=\sum_{i}2a_{i}%
\omega _{i}\;\;\text{ and }\;\;\kappa\in\mathcal{P}^{(1,1)}\;\;%
\Longleftrightarrow\;\;\kappa=\sum _{i}a_{i}\omega_{2i}\,. 
\]
Set $\mathcal{P}^{\boxplus}=\mathcal{P}^{(2)}\cap\mathcal{P}^{(1,1)}$.\ It
follows that 
\[
\kappa\in\mathcal{P}^{\boxplus}\;\;\Longleftrightarrow\;\;\kappa=%
\sum_{i}2a_{i}\omega_{2i}\,,
\]
that is, $\lambda$ decomposes in terms of the fundamental weights $%
\omega_{2i}$ with even coefficients. In the general case of a partition $%
\kappa\in\mathcal{P}$ written as $\kappa=\sum_{i}a_{i}\omega_{i}$ we define%
\[
\kappa_{\boxplus}=\sum_{i}(a_{2i}-(a_{2i}\;\mathrm{mod}\;2))\omega_{2i}\;\;%
\text{ and }\;\;\kappa^{\boxplus}=\kappa-\kappa_{\boxplus}=\sum_{i}a_{2i+1}%
\omega_{2i+1}+\sum_{i}(a_{2i}\;\mathrm{mod}\;2)\omega_{2i}\,. 
\]
So $\kappa_{\boxplus}$ and $\kappa^{\boxplus}$ are partitions and $%
\kappa_{\boxplus}\in\mathcal{P}^{\boxplus}$.


We denote by $P_{(2)}^{\infty }$ and $P_{(1,1)}^{\infty }$ the sublattices
of $P=\oplus _{i\geq 1}\mathbb{Z\omega }_{i}$ defined by $P_{(2)}^{\infty
}=\oplus _{i\geq 1}2\mathbb{Z\omega }_{i}$ and $P_{(1,1)}^{\infty }=\oplus
_{i\geq 1}\mathbb{Z\omega }_{2i}\,.$ Observe that $P_{(2)}^{\infty }\cap 
\mathcal{P}=\mathcal{P}_{(2)}$ and $P_{(1,1)}^{\infty }\cap \mathcal{P}=%
\mathcal{P}_{(1,1)}$. We have also $P_{\boxplus }^{\infty }=P_{(2)}^{\infty
}\cap P_{(1,1)}^{\infty }=\oplus _{i\geq 1}2\mathbb{Z\omega }_{2i}$ and $%
\mathcal{P}^{\boxplus }=\mathcal{P}\cap P_{(2)}^{\infty }\cap
P_{(1,1)}^{\infty }$. We define the order $\leq _{\boxplus }$ on $\mathcal{P}$
by $\lambda \leq _{\boxplus }\mu \Longleftrightarrow \mu -\lambda \in 
\mathcal{P}^{\boxplus }$.

\subsection{A combinatorial description of the series $K_{\protect\lambda%
		,0}^{C_\infty}(t)$}

\label{subsec_Cinfinite}

\begin{definition}
	\label{def6.2} Consider a partition $\mu $. A vertex $b\in B(\lambda )$ is
	called $\mu $-distinguished if there exists $(\nu ,\delta )\in \mathcal{P}%
	^{(2)}\times \mathcal{P}^{(1,1)}$ such that $\boldsymbol{\varphi }(b)=\nu
	-\mu $ and $\boldsymbol{\varepsilon }(b)=\delta -\mu .$
\end{definition}

\begin{definition}
	\label{def6.3} Let $D(\lambda)$ be the set of all vertices in $B(\lambda)$
	which are $\mu $-distinguished for at least a partition $\mu$.
\end{definition}

Clearly, if $b$ is $\mu$-distinguished, then $b$ is $(\mu+\kappa )$%
-distinguished for any $\kappa\in\mathcal{P}^{\boxplus}$ (change $%
(\nu,\delta)\in\mathcal{P}^{(2)}\times\mathcal{P}^{(1,1)}$ to $(\nu
+\kappa,\delta+\kappa)\in\mathcal{P}^{(2)}\times\mathcal{P}^{(1,1)}$). For
any $b\in D(\lambda)$, set 
\[
S_{b}=\{\mu\in\mathcal{P}\mid b\text{ is }\mu\text{-distinguished}\}\,. 
\]

\begin{lemma}
	\label{lem6.4} The set $S_{b}$ has the form $S_{b}=\mu _{b}+\mathcal{P}%
	^{\boxplus }$ and $\mu _{b}$ is minimal for $\leq _{\boxplus }$ such that $b$
	is $\mu _{b}$-distinguished. Moreover, for any $\mu \in S_{b}$, we have $\mu
	_{b}=\mu ^{\boxplus }.$
\end{lemma}

The following proposition makes more explicit the structure of the
distinguished tableaux.

\begin{proposition}
	\label{Prop_Char_Distin}Let $b$ be a vertex of $B(\lambda )$ with $\lambda
	\in \mathcal{P}$. Then $b$ is distinguished if and only if $\varepsilon
	_{i}(b)=0$ for any odd $i$ and $\varphi _{i}(b)$ is even for any odd $i.$
	Moreover, we then have $\mu _{b}=\sum_{i}(\varphi _{2i}(b)\;\mathrm{mod}%
	\;2)\omega _{2i}=:\varphi (b)\;\mathrm{mod}\;2$\thinspace .
\end{proposition}

\begin{theorem}
	\label{thm6.7} We have $K_{\lambda ,0}^{C_{\infty }}(t)=\sum_{b\in D(\lambda
		)}t^{\left\vert \boldsymbol{\varphi }(b)+\mu _{b}\right\vert /2}\,.$
\end{theorem}

\subsection{Distinguished tableaux and zero weight King type tableaux}\label{dist-tab}

To see that the distinguished tableaux we introduced previously are in
natural bijection with zero weight tableaux very close to King tableaux, consider the sets $T_{C_{\infty}}(\lambda)$ of semistandard
tableaux of shape $\lambda$ on the infinite ordered alphabet $\{1<\overline
{1}<2<\overline{2}<\cdots\}$. There will be no condition on the position of
the barred letters here, contrary to the definition of King tableaux. Recall the notation of \cref{subsec_Cinfinite}. For any distinguished
vertex $b$ in $D(\lambda)$, set
$
\boldsymbol{\theta}(b)=\boldsymbol{\varphi}(b)+\mu_{b}\,,%
$
and let $\theta_{j}(b)$ be the coefficient of $\omega_j$ in the expansion of $\boldsymbol{\theta}(b)$. Since $\boldsymbol{\theta}(b)$ is a dominant weight for
$\mathfrak{sl}_{\infty}$, it can be regarded as a partition. Recall also that
$\left\vert \lambda\right\vert$ is even, says $\left\vert \lambda\right\vert =2\ell$. In the sequel of this section, we
shall assume that $B(\lambda)$ is realized as the set of semistandard tableaux on
the infinite ordered alphabet $\mathbb{Z}_{>0}$.\ For any integer $i\geq1$, a
reverse lattice skew tableau on $\{2i-1,2i\}$ is a semistandard filling of a
skew Young diagram with columns of height at most $2$ by letters $2i-1$ and
$2i$ whose Japanese reading is a lattice word (i.e., in each left
factor the number of letters $2i$ is less or equal to that of letters $2i-1$).

\begin{example}{\rm 
Assume $i=2$. Then
\begin{equation}%
\begin{tabular}
[c]{lllllllllllllll}\cline{12-12}\cline{12-15}
&  &  &  &  &  &  &  &  &  &  & \multicolumn{1}{|l}{$3$} &
\multicolumn{1}{|l}{$3$} & \multicolumn{1}{|l}{$3$} & \multicolumn{1}{|l|}{$3$%
}\\\cline{8-12}\cline{8-15}
&  &  &  &  &  &  & \multicolumn{1}{|l}{$3$} & \multicolumn{1}{|l}{$3$} &
\multicolumn{1}{|l}{$4$} & \multicolumn{1}{|l}{$4$} & \multicolumn{1}{|l}{$4$}
& \multicolumn{1}{|l}{} &  & \\\cline{4-4}\cline{4-6}\cline{8-12}
&  &  & \multicolumn{1}{|l}{$3$} & \multicolumn{1}{|l}{$3$} &
\multicolumn{1}{|l}{$4$} & \multicolumn{1}{|l}{} &  &  &  &  &  &  &  &
\\\cline{1-4}\cline{1-6}%
\multicolumn{1}{|l}{$3$} & \multicolumn{1}{|l}{$3$} & \multicolumn{1}{|l}{$4$}
& \multicolumn{1}{|l}{$4$} & \multicolumn{1}{|l}{} &  &  &  &  &  &  &  &  &
& \\\cline{1-4}%
\end{tabular}
\label{skewtab}%
\end{equation}
is a reverse lattice skew tableau on $\{3,4\}$.}
\end{example}

The following proposition is a reformulation of \cref{Prop_Char_Distin}.

\begin{proposition}
\label{Prop_Char_Distin_T}A semistandard tableau $T$ of shape $\lambda$ is
distinguished if and only if for any integer $i\geq1$, the skew tableau
obtained by keeping only the letters $2i-1$ and $2i$ in $T$ is a reverse
lattice tableau, and the rows of $\boldsymbol{\theta}(T)$ have even lengths.
\end{proposition}

We now explain the correspondence between distinguished
tableaux and zero weight King type tableaux. Observe that a tableau $T\ $in $T_{C_{\infty}}(\lambda)$ of weight zero is a
juxtaposition of skew tableaux of weight $0$ on $\{i,\overline{\imath}\}$ obtained
by keeping only the letters $i$ and $\overline{\imath}$. So to obtain a bijection
between the set of distinguished tableaux of shape $\lambda$ and the subset 
$T_{C_{\infty}}^{0}(\lambda)\subset T_{C_{\infty}}(\lambda)$ of zero weight
tableaux, it suffices to describe a bijection between the set of reverse
lattice tableaux on $\{2i-1,2i\}$ of given shape and weight in $2\omega
_{i}\mathbb{Z}_{\geq0},$ and the set of skew tableaux on $\{i,\overline{\imath}\}$
with weight $0$. Now recall that we have the structure of a $U_{q}%
(\mathfrak{sl}_{2})$-crystal on the set of all skew semistandard tableaux of
fixed skew shape both on $\{2i-1,2i\}$ and $\{i,\overline{\imath}\}$. By replacing
each letter $2i-1$ by $i$ and each letter $2i$ by $\overline{\imath}$, we get a
crystal isomorphism $f$.\ The distinguished tableaux correspond to the highest
weight vertices of weight in $2\omega_{i}\mathbb{Z}_{\geq0}$ for the
$\{2i-1,2i\}$-structure, whereas the tableaux of weight $0$ give the vertices
of weight $0$ in the $\{i,\overline{\imath}\}$-crystal structure.\ By observing
that only $U_{q}(\mathfrak{sl}_{2})$-crystals with highest weight in
$2\omega_{i}\mathbb{Z}_{\geq0}$ admit a vertex of weight $0$, which is then
unique, we obtain that the map $\mathcal{C}$ which associates to each zero
weight vertex in the $\{i,\overline{\imath}\}$-crystal structure its highest weight
vertex in the $\{2i-1,2i\}$-crystal structure is the bijection we need. More
precisely, the map $\mathcal{C}$ (resp. its inverse) is obtained as usual: we start by encoding in the
reading of each $\{i,\overline{\imath}\}$-tableau (resp. of each $\{2i-1,2i\}$%
-tableau) the letters $i$ by $+$ and the letters $\overline{\imath}$ by $-$ (resp.
the letters $2i-1$ by $+$ and the letters $2i-1$ by $-$), and next by recursively
deleting all the factors $+-$, thus obtaining a reduced word of the form $-^{m}%
+^{m}$ (resp. $+^{2m}$).\ It then suffices to change the $m$ letters $\overline{\imath}$
corresponding to the $m$ surviving symbols $-$ into $i$ and to apply the
isomorphism $f^{-1}$ (resp. change $m$ letters $2i-1$
corresponding to the rightmost $m$ surviving symbols $+$ into $2i$ and apply the
isomorphism $f$).

\begin{example}\cellsize=3.1ex
{\rm 
The skew tableau of weight $0$ on $\{2,\overline{2}\}$ corresponding to
\rm{(\ref{skewtab})} is
\[%
\tableau{&&&&&&&&&&&2&2&2&\overline{2}\\
&&&&&&&2&\overline{2}&\overline{2}&\overline{2}&\overline{2}\\
&&&2&2&\overline{2}\\
2&2&\overline{2}&\overline{2}}
%\begin{tabular}
%[c]{lllllllllllllll}\cline{12-12}\cline{12-15}
%&  &  &  &  &  &  &  &  &  &  & \multicolumn{1}{|l}{$2$} &
%\multicolumn{1}{|l}{$2$} & \multicolumn{1}{|l}{$2$} &
%\multicolumn{1}{|l|}{$\overline{2}$}\\\cline{8-12}\cline{8-15}
%&  &  &  &  &  &  & \multicolumn{1}{|l}{$2$} & \multicolumn{1}{|l}{$\overline
%{2}$} & \multicolumn{1}{|l}{$\overline{2}$} & \multicolumn{1}{|l}{$\overline
%2}$} & \multicolumn{1}{|l}{$\overline{2}$} & \multicolumn{1}{|l}{} &  &
%\\\cline{4-4}\cline{4-6}\cline{8-12}
%&  &  & \multicolumn{1}{|l}{$2$} & \multicolumn{1}{|l}{$2$} &
%\multicolumn{1}{|l}{$\overline{2}$} & \multicolumn{1}{|l}{} &  &  &  &  &  &
%&  & \\\cline{1-4}\cline{1-6}%
%\multicolumn{1}{|l}{$2$} & \multicolumn{1}{|l}{$2$} &
%\multicolumn{1}{|l}{$\overline{2}$} & \multicolumn{1}{|l}{$\overline{2}$} &
%\multicolumn{1}{|l}{} &  &  &  &  &  &  &  &  &  & \\\cline{1-4}%
%\end{tabular}
\]}
\cellsize=2.5ex
\end{example}

In the sequel, we shall abuse the notation and identify the two crystal
structures corresponding up to the isomorphism $f$.

\begin{example}\label{ex-11}\cellsize=3.1ex
{\rm 
Assume $\lambda=(1,1)$. Then we get
\[
T_{C_{\infty}}^0(\lambda)=\left\{\tableau{k\\ \overline{k}}
%\begin{tabular}
%[c]{|c|}\hline
%$k$\\\hline
%$\overline{k}$\\\hline
%\end{tabular}
\mid k\in\mathbb{Z}_{\geq1}\right\}  \;\;\text{ and }\;\;K_{C_{\infty}}^0(\lambda
)=\left\{\tableau{k\\ \overline{k}}
%\begin{tabular}
%[c]{|c|}\hline
%$k$\\\hline
%$\overline{k}$\\\hline
%\end{tabular}
\mid k\in\mathbb{Z}_{\geq2}\right\}  .
\]
This gives%
\[
H\left(\tableau{k\\ \overline{k}}
%\begin{tabular}
%[c]{|c|}\hline
%$k$\\\hline
%$\overline{k}$\\\hline
%\end{tabular}
\right)  =%
\begin{tabular}
[c]{|c|}\hline
$2k-1$\\\hline
$2k$\\\hline
\end{tabular}
\;\;\text{ and }\;\;\boldsymbol{\varphi}\left(
\begin{tabular}
[c]{|c|}\hline
$2k-1$\\\hline
$2k$\\\hline
\end{tabular}
\right)  =\omega_{2k}\text{ for any }k\geq1.
\]
Therefore
\[
\boldsymbol{\theta}\left(
\begin{tabular}
[c]{|c|}\hline
$2k-1$\\\hline
$2k$\\\hline
\end{tabular}
\right)  =2\omega_{2k}\text{ for any }k\geq1\text{.}%
\]
Finally $K_{\lambda,0}^{C_{\infty}}(t)=\sum_{k\geq1}t^{2k}=\frac{t^{2}}{1-t^{2}}$.}
\end{example}



\section{Type $C_n$ generalized exponents via the Kwon model}\label{genexpc}

In this section, we refine the results in  \cref{dist-tab} to the finite type $C_n$, based on Kwon's model for the corresponding branching coefficients \cite{kwoces,kwoldk}. We also need to use a combinatorial map realizing the conjugation symmetry of Littlewood-Richardson coefficients. It turns out that Kwon's model, the version of the conjugation symmetry map used here, and the distinguished tableaux in \cref{subsec_Cinfinite} fit together in a beautiful way. This allows us to express the related statistic in terms of a natural combinatorial labeling of the vertices of weight 0 in the corresponding type $C_n$ crystal of highest weight $\lambda$, namely the corresponding tableaux due to King \cite{King}.


\begin{theorem}
\label{kfking} We have
\[
K_{\lambda,0}^{C_{n}}(t)=\sum_{T\in K_{C_{n}}^{0}(\lambda)}t^{\mathrm{ch}
_{C_{n}}(L(T))}\,,
\]
where
\[
\mathrm{ch}_{C_{n}}(L(T))=\sum_{i=1}^{2n-1}(2n-i)\left\lceil \frac
{\varepsilon_{i}(L(T))}{2}\right\rceil \,.
\]

\end{theorem}

\begin{remarks}\label{remkn}{\rm 
(1) There does not seem to be a simple way to express the related statistic above directly in terms of $T$. However, the map $T\mapsto L(T)$ is a simple one. 

(2) \cref{kfking} gives a statistic for computing the Kostka--Foulkes polynomial on King tableaux, rather than on the Kashiwara--Nakashima (KN). A natural question is whether the statistic above can be translated to the KN tableaux and moreover if it is related to the charge statistic constructed in \cite{lec} (which conjecturally computes the Kostka--Foulkes polynomials). This question is addressed in \cite{GT}.
}
\end{remarks}

We will now continue \cref{ex-11}.

\begin{example}\cellsize=3.1ex
{\rm 
Assume $\lambda=(1,1)$ in type $C_n$. Then we get
\[
K_{C_{n}}^0(\lambda
)=\left\{\tableau{k\\ \overline{k}}
%\begin{tabular}
%[c]{|c|}\hline
%$k$\\\hline
%$\overline{k}$\\\hline
%\end{tabular}
\mid k=2,\ldots,n\right\}  .
\]
This gives%
\[
L\left(\tableau{k\\ \overline{k}}
%\begin{tabular}
%[c]{|c|}\hline
%$k$\\\hline
%$\overline{k}$\\\hline
%\end{tabular}the weyl
\right)  =%
\begin{tabular}
[c]{|c|}\hline
$2k-1$\\\hline
$2k$\\\hline
\end{tabular}
\;\;\text{ and }\;\;\boldsymbol{\varepsilon}^*\left(
\begin{tabular}
[c]{|c|}\hline
$2k-1$\\\hline
$2k$\\\hline
\end{tabular}
\right)  =\omega_{2(n-k+1)}\text{ for any }k=2,\ldots,n.
\]
Therefore
\[
\boldsymbol{\theta}_n^*\left(
\begin{tabular}
[c]{|c|}\hline
$2k-1$\\\hline
$2k$\\\hline
\end{tabular}
\right)  =2\omega_{2(n-k+1)}\text{ for any }k=2,\ldots,n\text{.}%
\]
Finally%
$[K_{\lambda,0}^{C_{n}}(t)=\sum_{k=1}^{n-1}t^{2k}=\frac{t^{2}-t^{2n}}{1-t^{2}}\,.
$
}
\end{example}

\section{Three applications of \cref{kfking}}
\subsection{Growth of generalized exponents}

First we analyze the growth of the generalized exponents of type $C_{n}$ with respect to the rank $n$. The (weight $0$) symplectic King tableaux of type $C_{n}$ embed into those of type $C_{n+1}$ by changing the entries $k,\overline{k}$ to $k+1,\overline{k+1}$, for all $k$, respectively. Moreover, it is easy to see that this map preserves the statistic in \cref{kfking}. So we obtain the following result, which to our knowledge is new. 

\begin{theorem}
For any integer $n$ and any partition $\lambda$ with at most $n$ parts, we have
$K_{\lambda,0}^{C_{n+1}}(t)-K_{\lambda,0}^{C_{n}}(t)\in\mathbb{Z}_{\geq0}[t]$.
\end{theorem}

\subsection{Reducing a type $C$ generalized exponent to one of type $A$} We now prove a conjecture of the first author \cite{lec}. This conjecture is the first step in the construction of the type $C_{n}$ charge statistic in \cite{lec}, and proves the conjecture that this charge computes the corresponding Kostka--Foulkes polynomials in the case of column shapes; see \cref{remkn}~(2). We now label the Dynkin diagram of type $C_n$ such that the special node is $n$. Consider the fundamental weight $\omega_{2p}$, where $p\in\{1,\ldots,\lfloor n/2\rfloor\}$. All the zero weight vertices in the crystal $B(\omega_{2p})$ belong to the same type $A_{n-1}$ component, which has highest weight $\gamma_p:=\varepsilon_1+\ldots+\varepsilon_p-\varepsilon_{n-p+1}-\ldots-\varepsilon_n$, where $\varepsilon_i$ are the coordinate vectors in ${\mathbb R}^n$. In type $A_{n-1}$, this weight corresponds to the partition $(1^{n-2p},2^p)$. 

\begin{theorem}\label{ac} We have $K^{C_n}_{\omega_{2p},0}(t)=K_{\gamma_p,0}^{A_{n-1}}(t^2)\,.$
\end{theorem}



\begin{remark}{\rm 
\cref{ac} also permits to establish the conjecture of \cite{lec} for
Lusztig $t$-analogues associated to any fundamental weight.}  
\end{remark}

\subsection{The smallest power of $t$ in $K^{C_n}_{\lambda,0}(t)$}\label{S:smallestpower}

The largest power of $t$ in $K^{C_n}_{\lambda,0}(t)$ is well-known to be $\langle\lambda,\rho^\vee\rangle$, where  $\rho^\vee$ is half the sum of the positive coroots. Furthermore, it is also known that the smallest power is greater or equal to $|\lambda|/2$. See \cite{Lec2}. As the third application of our formula for $K^{C_n}_{\lambda,0}(t)$, we will determine this smallest power. Let $\lambda\in{\mathcal P}_n$ be such that $|\lambda|$ is even, and write $\lambda=\sum_{i=1}^n a_i\,\omega_{n+1-i}$. Define
\[s_k:=\sum_{i=1}^k a_i\,,\;\;\;\;\;b_i:=\casethree{a_i+1}{\mbox{$a_i$ odd and $s_i$ odd}}{a_i-1}{\mbox{$a_i$ odd and $s_i$ even}}{a_i}{\mbox{$a_i$ even}}\]
Also let $s_0:=0$ and $S:=s_n$.

\begin{theorem}\label{minpower} The smallest power of $t$ in $K^{C_n}_{\lambda,0}(t)$, for $|\lambda|$ even, is
\begin{equation}\label{expminpower}\frac{1}{2}\sum_{i=1}^n (n+1-i) b_i=\frac{|\lambda|}{2}+\frac{1}{2}\sum_{i\,:\,a_i\:\rm{odd}}(-1)^{s_i-1}(n+1-i)\,.\end{equation}
\end{theorem}

%\bibliographystyle{plain}
%
%\bibliography{kostka-foulkes}

\printbibliography

\end{document}



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%
%\end{thebibliography}

%\end{document}
