%  Littlewood2Jing.tex
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\long\def\bee#1\ee{\begin{equation}#1\end{equation}}
\def\be{\begin{equation}}

\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}

\def\s{\scriptstyle }
\def\a{\alpha}
\def\b{\beta}
\def\l{\lambda}
\def\e{\epsilon}
\def\L{\Lambda}
\def\N{{\mathbb N}}
\def\Z{{\mathbb Z}}
\def\A{{\mathbb A}}
\def\B{{\mathbb B}}
\def\C{{\mathbb C}}
\def\X{{\mathbb X}}
\def\Y{{\mathbb Y}}
\def\AB{ {\mathbb A}\moins {\mathbb B}}
\def\ss{\sigma}
\def\Sym{{\mathfrak Sym}}
\def\cO{{\mathcal O}}
\def\Sp{{\mathcal Sp}}

\def\bu{\scriptstyle \bullet}
\def\moins{\raise 1pt\hbox{$\s -$}}
\def\plus{\raise 1pt \hbox{ $\s +$}}

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\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\def\proof{\noindent{\it Proof.}\ }

\begin{document}
\title{Littlewood's formulas for characters of orthogonal and
  symplectic groups}
\author{
Alain Lascoux\\
{\small Institut Gaspard Monge, CNRS, Universit\'e de Marne-la-Vall\'ee}\\
{\small 77454 Marne-la-Vall\'ee Cedex, France}\\
{\small e-mail: Alain.Lascoux @ univ-mlv.fr}\\
}
\date{}
\maketitle

\begin{abstract}
Litllewood gave expansions of products of the type
$\prod 1/(1-a_ia_j)$. Several authors published generalizations involving
a small number of extra factors $\prod 1/(1-xa_i)$. We show 
that the method of Littlewood (that we express in terms of $\l$-rings) 
covers these extensions. We also express Littlewood's coefficients in terms of 
orthogonal  or symplectic Schur functions.
\end{abstract}

Given an infinite alphabet $\A=\{ a_1,a_2,a_3,\ldots\}$, Littlewood
\cite{Littlewood} gave expansions of products of the type
\bee
\prod ( 1-a_i)^{\pm 1} \prod_{i<j} (1- a_ia_j)^{\pm 1} \quad \& \quad
\prod ( 1-a_i)^{\pm 1} \prod_{i\leq j} (1- a_ia_j)^{\pm 1} \ ,
\ee
the case $\prod ( 1-a_i)^{-1}\prod_{i<j} (1- a_ia_j)^{-1}$ being due to
Schur \cite{Schur}.

By functoriality of symmetric functions, 
Littlewood's formulas imply without further computations  expansions of 
more complicated products like
\bee
\prod ( 1-a_i)^{\pm 1} \prod ( 1-b_i)^{\mp 1} 
\prod_{i\leq j} (1- a_ia_j)^{\pm 1} \prod_{i< j} (1- b_ib_j)^{\pm 1}
\prod (1-a_i b_j)^{\mp 1} 
\ee
involving a second alphabet $\B=\{ b_1,b_2,\ldots\}$.

However, Littlewood's method does not restrict to expliciting 
the above products, and allows to expand products
of the type
\bee
\prod_i \frac{1}{(1-x_1a_i)\cdots (1-x_na_i)} \prod_{i<j} \frac{1}{1-a_ia_j}
\ee
involving a finite alphabet $\X=\{ x_1,\ldots, x_n\}$,
the alphabets $\A$ and $\X$ playing now a disymmetrical role.

To obtain his formulas, Littlewood describes in fact the adjoint of
the multiplication by certain series of symmetric functions 
(see the theorem below). We shall state his results in terms
of $\l$-rings. 

The case $\X=\{x_1,x_2\}$ has been considered by Ishikawa \cite{Ishikawa},
Ishikawa and Wakayama \cite{Ishikawa_Wakayama}, Bressoud \cite{Bressoud},
 and the case $\X=\{x_1,x_2,x_3\}$ 
 by Jouhet and Zeng \cite{JouhetZeng}.

Before reinterpreting their formulas, let us recall some definitions,
for which we refer to \cite{Mac}.

Given two alphabets $\A$, $\B$, the complete functions $S_j(\AB)$ 
are defined by the generating series (with $z$ an extra variable)
\bee
  \ss_z(\A-\B):=  \prod_{b\in \B} (1-zb) /\prod_{a\in \A}  (1-za)  
\ee

Given  $\l=(\l_1,\l_2,\ldots, \l_r) \in  \N^r $,
 the Schur function $S_\l(\AB)$ is
$$ S_\l(\AB) = \Bigl| S_{\l_{i+j-i}}(\AB) \Bigr|_{1\leq i,j \leq r}  \ .$$
When $\l$ is weakly decreasing, it is called a {\it partition}.
In that case, following Frobenius, one uses another description of
partitions, by decomposing their diagram  into its diagonal hooks, 
and writing $\l =(\a\, | \, \b)$ \cite{Mac} p.3.

On the space $\Sym(\A)$ of  symmetric functions in $\A$, 
there exists a scalar product
$(\, ,\, )_\A$, and Schur functions indexed by partitions 
constitute  an orthormal
basis with respect to it~:
\bee  \label{eq:Scal1}
 \bigl(\, S_\l(\A) ,\, S_\l(\A)\bigr)_\A  =1  \qquad \& \qquad
  \bigl(\, S_\l(\A) ,\, S_\nu(\A)\bigr)_\A  =0 \quad \l\neq \nu \ .
\ee 

Another way to state (\ref{eq:Scal1}) is to assert the existence
of a \emph{reproducing kernel}
$$ K(\A,\, \B):= \ss_1(\A \B) = \prod_{a\in\A,\, b\in\B} (1-ab)^{-1} $$
which satisfies 
\bee  \label{eq:Kernel}
 \forall f\in \Sym  \ ,\
       \bigl(\, K(\A,\, \B) ,\, f(\A) \bigr)_\A = f(\B)  \ ,
\ee
because the Cauchy formula states that
\bee \label{eq:Cauchy}
  K(\A,\, \B) =\sum_\l S_\l(\A)\, S_\l(\B)  \ ,
\ee
sum over all partitions.

Given any symmetric function $f$, the adjoint $D(f;\, \bu)$ of
the multiplication by $f$ is defined by
\bee
\forall f,\, g,\, h\in \Sym  \ ,\
  \bigl(\, D(f;g)\,  ,\, h\bigr)  = (g \, ,\, fh)
\ee



\vskip 1cm

The formulas 
\bee
  \ss_1(\L^2(\A)) = \sum S_\mu(\A)   \ ,\ 
           \text{all $\mu$ with columns of even lengths}
\ee
and 
\bee
  \ss_1(\A+\L^2(\A)) = \sum S_\l(\A) \ ,\ \text{all partitions $\l$}
\ee
are equivalent, because $\L^2(\A+1) = \A+\L^2(\A)$. Combinatorially,
considering the letter $1$ to be bigger than all the letters in $\A$,
it is equivalent to enumerate all tableaux in $\A$, or all tableaux 
in the alphabet $\A+1:=\A\cup \{1\}$ with columns of even length
(we write unions of alphabets with a `+`).
Indeed, the $1$'s occupy the horizontal strip such that the inner
shape $\mu$ has columns of even lengths.

The computations of Littlewood in chapter XI of his book
\cite{Littlewood} can be rephrased 
in the $\l$-rings paradigm as follows~: 
\begin{theorem}[Littlewood]
 Given  two alphabets $\A$, $\X$, then one has the identities
\begin{eqnarray}
 \label{eq:Litt1}
  \ss_1(\A\X + \L^2(\A)) &=& \sum_\mu  S_\mu(\A)\, 
                        D\bigl(\ss_1(\L^2);S_\mu)\bigr)(\X)  \\
\label{eq:Litt2}
  \ss_1(\A\X + S^2(\A)) &=& \sum_\mu  S_\mu(\A)\, 
                      D\bigl(\ss_1(S^2);S_\mu)\bigr)(\X)    \\
\label{eq:Litt3}
  \ss_1(\A\X - \L^2(\A)) &=& \sum_\mu   S_\mu(\A)\, 
                       D\bigl(\ss_1(-\L^2);S_\mu)\bigr)(\X) \\
  \label{eq:Litt4}
  \ss_1(\A\X - S^2(\A))  &=&  \sum_\mu  S_\mu(\A)\, 
          D\bigl(\ss_1(-S^2);S_\mu)\bigr)(\X)  \ ,
\end{eqnarray} 
sum over all partitions $\mu$. 
\end{theorem}

\proof All four generating series factorize~:
$\ss_1(\A\X + \L^2(\A))= \ss_1(\A\X) \ss_1( \L^2(\A))$. Since
 $\ss_1(\A\X)$ is a reproducing kernel, one has for any $\mu$~:
\begin{eqnarray*}
 \bigl(\, \ss_1(\A\X + \L^2(\A))\,  ,\, S_\mu(\A) \bigr)_\A &=&
  \bigl(\, \ss_1(\A\X)\,  ,\, D(\ss_1(\L^2);S_\mu)(\A) \bigr)_\A \\
  &=&  D(\ss_1(\L^2);S_\mu)(\X) \ .
\end{eqnarray*}
and this proves the first identity.

The same proof would work for  $\ss_1(\A\X)\, f(\A)$,
$f$ being any  symmetric function.  Indeed property (\ref{eq:Kernel}) entails 
\bee
  \ss_1(\A\X)\, f(\A) = \sum_\mu  S_\mu(\A)\,  D(f;S_\mu)(\X) \ ,
\ee 
and formulas (\ref{eq:Litt2}, \ref{eq:Litt3}, \ref{eq:Litt4}) are a
 special case of it.                                           \hfill QED

\medskip  Let us mention that other expansions are possible. Indeed,
\begin{eqnarray*}
 S^2(\A \pm\X) &=& S^2(\A) \pm S^1(\A)S^1(\X) + S^2(\pm \X)  \\
   \L^2(\A \pm\X) &=& \L^2(\A) \pm \L^1(\A)\L^1(\X) +  \L^2(\pm \X) \ .
\end{eqnarray*}
It implies, for example, that 
\begin{multline}  \label{eq:Lit11}
  \ss_1(\A\X + \L^2(\A)) = \ss_1(- \L^2(\X))\,   \ss_1( \L^2(\A+\X)) \\
   = \ss_1(- \L^2(\X))\, \sum\nolimits_\nu S_\nu(\A+\X)     \ ,
\end{multline}
sum over all partitions $\nu$ with columns of even lengths. 
To go further, 
one would decompose $S_\nu(\A+\X)$ in the basis of Schur functions in $\A$, and 
use in general Littlewood-Richardson's rule to collect functions in $\X$. 
But in the case where $\X$ is of small cardinality, the expansion can be
written directly. For example (\cite{Bressoud}, th.IV), 
if $\X=x_1+x_2$, equation (\ref{eq:Lit11}) can be rewritten as 
\bee
 \ss_1\bigl((x_1+x_2)\A + \L^2(\A)\bigr) 
          = (1-x_1x_2) \sum\nolimits_\nu S_\nu(\A+x_1+x_2)
\ee
and is easy to expand in terms of the $S_\mu(\A)$, interpreting it as the sum
of all Young tableaux with even columns, on an alphabet with last two
letters $x_1,\, x_2$.

\medskip 
Let us show in more details how to make explicit the coefficients in
Littlewood's identities,  for $card(\X)=n=1,2,3$, recovering the related 
results of \cite{Bressoud,Ishikawa,Ishikawa_Wakayama,JouhetZeng}. 
We shall treat only the case of  $\ss_1(\X\A +\L^2\A)$, the other being
similar.

When $n=1$, $\X=\{x_1\}$.
Then $D(S_\nu;S_\mu)(\X)$ is non zero iff the diagram of
  $\mu/\nu$ is an horizontal strip. Therefore, given any partition  $\mu$,
there is one and only one partition $\nu$ with even columns such that 
$D(S_\nu;S_\mu)(\X)$ is non zero. Equation (\ref{eq:Litt1}) specializes to
\bee
 \ss_1( x_1 \A + \L^2(\A)) = \sum_\mu x_1^{odd(\mu)}\, S_\mu(\A) \ ,
\ee
where $odd(\mu)$ is the number of columns of $\mu$ of odd length.

Take now $n=2$. A skew diagram $\mu/\nu$  gives a non-zero function
 \mbox{$S_{\mu/\nu}(x_1\!\plus x_2)$} 
iff it is made of horizontal strips of height
1 and 2. Decompose the partition into blocks ${\mathcal B}_k$, $k=1,2,\ldots$
of columns of the same length $k$ (denoting $\ell_k$ its width). 
If $k$ is odd, then the restriction of $\mu/\nu$ to this block must be an
horizontal strip of length $\ell_k$, giving a factor $S_{\ell_k}(x_1+x_2)$.
If $k$ is even, the restriction of $\mu/\nu$ is a $2\times i$ rectangle,
with $0\leq i\leq \ell_k$. The figure shows the restriction of a strip
to two consecutive blocks~:
\def\bs{\blacksquare}
$$ \begin{array}{c}
  \hspace{30 pt} \overbrace{\hspace{30 pt}}^i \hspace{30 pt}\\[-2pt]
 \begin{smallmatrix}
  \square &\cdots &\square &\bs &\cdots &\bs \\
  \square &\cdots &\square &\bs &\cdots &\bs  &\bs &\cdots &\bs \\
\square &\cdots &\square &\square &\cdots &\square &\square&\cdots &\square\\
 \vdots &      &\vdots   &\vdots &        &\vdots  &\vdots &      &\vdots\\
\square &\cdots &\square &\square &\cdots &\square &\square&\cdots &\square
\end{smallmatrix}  \\[-8pt]
\underbrace{\hspace{60 pt}}_{\ell_{2r}}
\underbrace{\hspace{30 pt}}_{\ell_{2r-1}}
\end{array}
$$


The sum over all possible $i$ gives a
factor $S_{\ell_k}(1+x_1x_2)$. Moreover, there is no interference between
the different blocks, and therefore, one gets that the coefficient of
$S_\mu(\A)$ in $\ss_1\bigl( (x_1\!\plus x_2) \A + \L^2(\A)\bigr)$ is 
\bee
  D(\ss_1(\L^2);S_\mu)(x_1+x_2) =
\prod_{k=0}^\infty  S_{\ell_{2k}}(1+x_1x_2)\, S_{\ell_{2k+1}}(x_1+x_2)  \ .
\ee

The case $n=3$ is treated by Jouhet and Zeng \cite{JouhetZeng}. 
The independent blocks
are now the blocks ${\mathcal B}_k$ of columns of length $2k\plus 1,\, 2k$.

There are two possible configurations for those $\mu/\nu$ giving a non-zero
contribution~:
$$ \begin{array}{c}
  \hspace{30 pt} \overbrace{\hspace{28 pt}}^i \hspace{30 pt}\\[-2pt]
 \begin{smallmatrix}
  \bs &\cdots &\bs    &\bs &\cdots &\bs \\
  \square &\cdots &\square &\bs &\cdots &\bs  &\bs &\cdots &\bs \\
\square &\cdots &\square &\bs &\cdots &\bs  &\bs &\cdots &\bs \\
\square &\cdots &\square &\square &\cdots &\square &\square&\cdots
                                                         &\square\\[-2pt]
 \vdots &      &\vdots   &\vdots &        &\vdots  &\vdots &      &\vdots\\
\square &\cdots &\square &\square &\cdots &\square &\square&\cdots &\square
\end{smallmatrix}  \\[-8pt]
\underbrace{\hspace{60 pt}}_{p}
\underbrace{\hspace{30 pt}}_{q}
\end{array} \qquad,\quad p=\ell_{2k+1},\ q=\ell_{2k}, 
 \quad i\geq 1 $$
The sum over all $i=1,\ldots,\, p$ gives
$$ x_1x_2x_3 S_{p-1}(1+x_1x_2x_3) S_q(\L^2(x_1+x_2+x_3)) \ . $$

$$   \begin{array}{c}
 \hspace{75 pt} \overbrace{\hspace{28 pt}}^j\\[-8pt]
  \begin{smallmatrix}
  \bs    &\bs &\cdots &\bs \\
\square &  \square &\cdots &\square &\square &\cdots &\square  &\bs &\cdots &\bs \\
\square &\square &\cdots &\square &\square &\cdots &\square &\bs &\cdots &\bs \\
\square &\square &\cdots &\square &\square&\cdots &\square&\square&\cdots
&\square\\[-2pt]
\vdots   &\vdots &        &\vdots  &\vdots &      &\vdots &\vdots &
&\vdots\\
\square &\square &\cdots &\square &\square&\cdots &\square &\square&\cdots &\square
\end{smallmatrix}  \\[-8pt]
\underbrace{\hspace{40 pt}}_{p}
\underbrace{\hspace{60 pt}}_{q}
\end{array}\ ,\ 
i=0,\ 0\leq j\leq q \ . $$

The sum over all $j=0,\ldots,\, q$ is
$$ S_p(x_1+x_2+x_3)\, S_q(1+\L^2(\X))  \ .$$

For example, for $\mu=[7,7,4]$, the possible configurations, together with 
their factors, are~:
\begin{gather*} 
\begin{array}{c}
  \begin{smallmatrix}  \bs &\bs &\bs &\bs \\ 
     \square &\square &\square &\bs &\bs &\bs &\bs\\
     \square &\square &\square &\bs &\bs &\bs &\bs
  \end{smallmatrix}\\[5pt]   S_{411}(\X) \ S_{33}(\X) 
\end{array}  
+\begin{array}{c}
  \begin{smallmatrix}  \bs &\bs &\bs &\bs \\
 \square &\square &\bs &\bs &\bs &\bs &\bs\\
\square &\square &\bs &\bs &\bs &\bs &\bs
  \end{smallmatrix}\\[5pt]   S_{422}(\X) \ S_{33}(\X)
\end{array}
+\begin{array}{c}
  \begin{smallmatrix}  \bs &\bs &\bs &\bs \\
    \square &\bs &\bs &\bs &\bs &\bs &\bs\\
    \square &\bs &\bs &\bs &\bs &\bs &\bs
  \end{smallmatrix}\\[5pt]   S_{433}(\X) \ S_{33}(\X)
\end{array}
+\begin{array}{c}
  \begin{smallmatrix}  \bs &\bs &\bs &\bs \\
    \bs &\bs &\bs &\bs &\bs &\bs &\bs\\
    \bs &\bs &\bs &\bs &\bs &\bs &\bs
  \end{smallmatrix}\\[5pt]   S_{444}(\X) \ S_{33}(\X)
\end{array} \\
+\begin{array}{c}
  \begin{smallmatrix}  \bs &\bs &\bs &\bs \\
     \square &\square &\square &\square &\square &\square&\square\\
     \square &\square &\square &\square &\square &\square&\square 
  \end{smallmatrix}\\[5pt]   S_{4}(\X) \ S_{00}(\X)
\end{array}
+\begin{array}{c}
  \begin{smallmatrix}  \bs &\bs &\bs &\bs \\
     \square &\square &\square &\square &\square &\square&\bs\\
     \square &\square &\square &\square &\square &\square&\bs
  \end{smallmatrix}\\[5pt]   S_{4}(\X) \ S_{11}(\X)
\end{array}
+\begin{array}{c}
  \begin{smallmatrix}  \bs &\bs &\bs &\bs \\
     \square &\square &\square &\square &\square &\bs&\bs\\
     \square &\square &\square &\square &\square &\bs&\bs
  \end{smallmatrix}\\[5pt]   S_{4}(\X) \ S_{22}(\X)
\end{array}
+\begin{array}{c}
  \begin{smallmatrix}  \bs &\bs &\bs &\bs \\
     \square &\square &\square &\square &\bs &\bs&\bs\\
     \square &\square &\square &\square &\bs &\bs &\bs
  \end{smallmatrix}\\[5pt]   S_{4}(\X) \ S_{33}(\X)
\end{array}   \ .
\end{gather*}

In summary, the contribution of the block  ${\mathcal B}_k$ is
\bee
 c({\mathcal B}_k)= \sum_{i=0}^p S_{p-i}(\X) S_{qq}(\X) + S_p\X) \Bigl(
 1+S_{11}(\X) +\cdots +  S_{nn}(\X) \Bigr)
\ee
and the coefficient of
$S_\mu(\A)$ in $\ss_1\bigl( (x_1\!\plus x_2\!\plus x_3) \A + \L^2(\A)\bigr)$ 
is the product of all the factors $c({\mathcal B}_k)$ given by the
decomposition of $\mu$ into blocks. This expression is equivalent to
formula 8 of Jouhet and Zeng. 
 


\medskip
Littlewood's expansions were motivated by the description of characters
of orthogonal and symplectic groups. In turn, these characters allow to
expand Littlewood's formulas (\ref{eq:Litt3}),  (\ref{eq:Litt4}).  

Following \cite{Littlewood}, p.240,  using a second alphabet $\X$, 
define the {\it orthogonal Schur function} $\cO_\l(\A)$ by the generating
series~:
\bee
\prod_{i\leq j} (1-a_ia_j) \prod_i\prod_j (1-a_ix_j)^{-1} =
 \sum_\l \cO_\l(\X)\, S_\l(\A)  \ ,
\ee
that is,
\bee
  \ss_1\bigl( \A\X -S^2(\A) \bigr)
        = \sum_\l \cO_\l(\A)\, S_\l(\X)  \ .
\ee
Formula (\ref{eq:Litt4}) can now be rewritten~:
\bee
  \cO_\l = D\bigl( \ss_1(-S^2)\, ; S_\l\bigr) 
\ee 
and the inverse formula is 
\bee
  S_\l = subs\Bigl(S=\cO, D( \ss_1(S^2)\, ; S_\l)\Bigr) 
\ee
which means that in that last case, 
one uses the operator $D\bigl( \ss_1(S^2)\, ;\,  \bu\bigr)$
as a formal operator on a vector space, the basis of which is indexed by 
partitions (and the action being obtained by identifying this basis with
Schur functions).

For example, Littlewood (p. 241) gives 
(writing in the expansion of $\ss_1(-S^2)$ 
and $\ss_1(S^2)$ only the terms which give
a non-zero contribution)~:
$$ \cO_{332} = D(S_0 \moins S_2\plus S_{31}\moins S_{33}\, ;\, S_{332})= 
   S_{332} -S_{332/2} +S_{332/31} -S_2   $$
and the identity  
$$ D( S_0\plus S_2\plus S_{22}\plus S_{222}\, ;\, S_{332}) =
 S_{332} + S_{33}+S_{321} + S_{211} +S_{31} +S_{11} $$
implies, by formal change of $S$ into $\cO$~:  
$$ S_{332} = \cO_{332} + \cO_{33}+S_{321} + \cO_{211} +\cO_{31} +\cO_{11} \ . $$

Ishikawa and Wakayama \cite{Ishikawa_Wakayama} give the expansion of
  $ \ss_1( (x_1+x_2)\A- S^2(\A))$, with $x_1x_2=1$. 
It means computing the specialization of orthogonal Schur functions for 
such $\X$. In fact the complete functions (resp. power sums) of $\X$ are
Tchebychef polynomials of the first and second kind in the variable
$(x_1+x_2)/2$.  Ishikawa and Wakayama characterize those partitions
for which $\cO_\l(\X) \neq 0$ in terms of their Frobenius decomposition,
and obtains that in that case $\pm \cO_\l(\X)$ is equal to  
a Tchebychef polynomial of the second kind. 

Using symmetrizing operators, one easily expands 
$\ss_1(-S^2)$ as~: 
\begin{eqnarray}
  \ss_1(-S^2) &=& \sum \pm S_{\e_1,\e_2,\e_3,\ldots}   =
   S_{000\ldots} -S_{200\ldots} -S_{040\ldots} +S_{240\ldots} +\cdots \\
    &=& S_0 -S_2 +S_{31} -S_{33} + \cdots   \ .
\end{eqnarray}
In the first summation, Schur functions are indexed by vectors 
$[\e_1,\e_2,\e_3,\ldots]$ such that  $\e_i\in \{0,2i\}$,
and not by partitions, and the sign alternates with the parity of 
the number of non-zero components.

It is remarkable that the operator $D\bigl(\ss_1(-S^2); \bullet  \bigr)$
can also be written as an enumeration of vectors~:
\bee
   S_\l \to \cO_\l = D\bigl(\ss_1(-S^2),S_\l\bigr) = \sum_{\e_1,\e_2,\ldots}  
       \pm S_{\l/[\e_1,\e_2,\ldots]} 
\ee
(recall that, for any two integral vectors $\l$, $\nu$, 
$S_{\l/\nu}$ stands for the determinant $\bigl| S_{\l_i+j-i-\nu_j}  \bigr|$).

Indeed, the fact that $D(S_\nu; S_\l)= S_{\l/\nu}$ for two partitions,
extends, by reordering the rows of the determinantal expression of
$S_\nu$ and the columns of $S_{\l/\nu}$, to the case where $\nu$
is not a partition, i.e. is not decreasing.

Thus $\cO_{332}$ can also be written 
\begin{multline*}
 \cO_{332} =  S_{332/000} - S_{332/200} -S_{332/040} + S_{332/240}   \\
= \left| \begin{smallmatrix}  S_3 &S_4 &S_5\\ S_2 & S_3 & S_4\\
        S_0 & S_1 &S_2 \end{smallmatrix} \right|
 - \left| \begin{smallmatrix}  S_1 &S_4 &S_5\\ S_0 & S_3 & S_4\\
        0 & S_1 &S_2 \end{smallmatrix} \right|
- \left| \begin{smallmatrix}  S_3 &S_0 &S_5\\ S_2 & 0 & S_4\\
        S_0 & 0 &S_2 \end{smallmatrix} \right|
+ \left| \begin{smallmatrix}  S_1 & S_0 &S_5\\ S_0 & 0 & S_4\\
        0 & 0 &S_2 \end{smallmatrix} \right|  \ .
\end{multline*}

By multilinearity of determinants, one can write the above sum as a single 
determinant, due to Weyl
(\cite{Weyl}, th.7.9.A),  that we display for the order 3~:
\bee  \cO_{\l_1 \l_2\l_3} =
\begin{vmatrix}
 S_{\l_1} -S_{\l_1-2} & S_{\l_1+1} -S_{\l_1+1-4} & S_{\l_1+2} -S_{\l_1+2-6}\, \\
\, S_{\l_2-1} -S_{\l_2-1-2} & S_{\l_2} -S_{\l_2-4} & S_{\l_2+1} -S_{\l_2+1-6}\\
 S_{\l_3-2} -S_{\l_3-2} & S_{\l_3-1} -S_{\l_3-1-4} & S_{\l_3} -S_{\l_3-6}
\end{vmatrix}  
\ee
For example, $\cO_{332}$, again, is equal to 
$$ \begin{vmatrix}   S_3 -S_1 & S_4-S_0 & S_5-0 \\
                    S_2-S_0 &  S_3-0  & S_4-0 \\
                    S_0-0     & S_1 -0  & S_2-0
 \end{vmatrix}  $$

We shall refer to \cite{Koike} for determinantal expressions of orthogonal
or symplectic characters. The case of ``nearly rectangular shape''
leads to an interesting combinatorics obtained by Krattenthaler 
\cite{Krattenthaler}.   

\medskip
Similarly, define the {\it symplectic Schur function} 
$\Sp_\l$ by
\bee
  \Sp_\l  = D(\ss_1(-\L^2)\, ; S_\l)   \ ,
\ee
but this time, it will be evaluated into the alphabet\\  
$ \Y := \{ x_1,1/x_1, x_2,1/x_2, \ldots, x_n, 1/x_n\}$ instead of $\X$. 

The identity 
\bee 
  \ss_1(-\L^2) = \sum  S_{\e_0, \e_1,\e_2,\e_3,\ldots} \ ,
\ee
sum over  all  $\e_i\in \{0,2i\}$, with $\e_0=0$,
extends, as before, to 
\bee
 \Sp_\l  = D(\ss_1(-\L^2)\, ; S_\l) = 
  \sum_{\e_1,\e_2,\ldots} S_{\l/[0, \e_1,\e_2,\ldots]} \ ,
\ee
and this sum can be written as a single determinant
due to Weyl (\cite{Weyl}, th.7.8.E)~:    
\bee \begin{vmatrix}
 S_{\l_1} & S_{\l_1+1} -S_{\l_1+1-2}     & S_{\l_1+2} -S_{\l_1+2-4}\, \\
 \, S_{\l_2-1} & S_{\l_2} -S_{\l_2 -2}  & S_{\l_2+1} -S_{\l_2+1-4}\\ 
 S_{\l_3-2}  & S_{\l_3-1} -S_{\l_3-1-2} & S_{\l_3} -S_{\l_3-4}
\end{vmatrix}
\ee

Lovers of $\l$-rings will prefer the expression
\bee  \Sp_\l =  S_\l(\Y-0, \Y-2, \Y-4, \ldots)   :=
 \Bigl| S_{\l_i+j-i}(\Y -2j+2)  \Bigr|   \ ,
\ee
with, for any alphabet $\Y$, any number $k$,
$$ \ss_z(\Y-k) := \ss_z(\Y-1-\cdots -1)= (1-z)^k\, \ss_z(\Y) \ .$$

Indeed, because 
$$ S_i(\Y-k) = S_i(\Y) - \binom{k}{1} S_{i-1}(\Y)  
   +\binom{k}{2} S_{i-2}(\Y)  -\cdots \pm S_{i-k}(\Y)  \ , $$
expanding each term of the determinant 
$\Bigl| S_{\l_i+j-i}(\Y -2j+2)  \Bigr|$, one obtains 
Weyl's determinant by eliminating in each column multiples of
 the columns on its left. 

In particular, when  $x_1, \ldots, x_n$ are specialized to 1, 
the determinant specializes to a determinant of binomial coefficients
\bee 
\left|\,  \binom{2n+\l_i-i-j+1}{\l_i+j-i}\, \right|_{1\leq i,j\leq n}
\ee
which is one of the many expressions of the dimension 
of a representation of  ${\mathfrak Sp}_{2n}$.

Let us mention that the interpretation of the orthogonal and symplectic
Schur functions as characters depends on the parity of the number of
variables. See Proctor \cite{Proctor} for the `\lq\lq odd symplectic groups''.
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\bibitem{Ishikawa}  M. Ishikawa
\emph{  Minor summation formula of Pfaffians and Schur functions identities},
 FPSAC 95 (Formal Power Series and Algebraic Combinatorics),
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(1995) 379--386.

\bibitem{Ishikawa_Wakayama} M. Ishikawa, M. Wakayama.
 \emph{Applications of Minor-Summation Formula II.
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136-157.

\bibitem{JouhetZeng} F. Jouhet, J. Zeng.
\emph{Some new identities for Schur functions }
preprint (2001).

\bibitem{Koike}  K. Koike, I. Terada.
\emph{Young diagrammatic methods for the representation theory of the
classical groups of type $B_n$, $C_n$, $D_n$}, 
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\bibitem{Krattenthaler} C. Krattenthaler.
\emph{Identities for classical group characters of nearly rectangular
shape}, J. Algebra {\bf 209} (1998) 1--64. 
  

\bibitem{Littlewood} {D.E. Littlewood}.
 \emph{The theory of group characters},
  Oxford University Press (1950).

\bibitem{Mac}
I.G. Macdonald. \emph{Symmetric functions and Hall polynomials}, Clarendon
Press, second edition, Oxford, (1995).

\bibitem{Proctor} R. Proctor. \emph{Interconnections between symplectic and
orthogonal characters}, Contemp. Math. {\bf 88} (1989) 145--162.

\bibitem{Schur} I. Schur. \emph{Gesammelte Abhandlungen}, vol III, p.456,
Springer (1973). 

\bibitem{Weyl} H. Weyl. \emph{The classical groups},
      Princeton Univ. Press (1939).

\end{thebibliography}

\end{document}
\vskip 15mm
{\obeylines
\hfill          C.N.R.S., Institut Gaspard Monge   
\hfill          Universit\'e de Marne-la-Vall\'ee, 
\hfill          5 Bd Descartes, Champs sur Marne,  
\hfill          77454 Marne La Vall\'ee Cedex 2 FRANCE 
\hfill \tt        Alain.Lascoux@univ-mlv.fr 
\hfill          http://phalanstere.univ-mlv.fr/$\sim$al 
}


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