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\begin{document}
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\title{The Pragacz identity and a new algorithm for
Littlewood--Richardson coefficients}
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\author{ }
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\date{ }
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\maketitle
\vskip 1cm
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\begin{abstract}
Recently an identity relating the combinatorial definition of a
supersymmetric S-function to a Weyl type formula was proved by
Pragacz~\cite{Pragacz}. In the present paper we show how this
identity gives rise to a new algorithm for Littlewood--Richardson
coefficients, which is easy to implement. We discuss the
present algorithm and compare it with some classical
algorithms for the calculation of Littlewood--Richardson
coefficients.
\end{abstract}
\vskip 1cm
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\section{Introduction}  \label{sec-intro}
The Littlewood--Richardson rule~\cite{LR} gives a method for computing
the coefficients in the expansion of the product of two Schur
functions (or S-functions) as a sum of Schur functions. Let
$\mu$ and $\nu$ be two partitions, with $s_\mu$ and $s_\nu$ the
corresponding  S-functions, then
\beq
s_\mu\,s_\nu = \sum_\la c^\la_{\mu\nu}\,s_\la,
\label{lr-coeff}
\eeq
where $c^\la_{\mu\nu}$ are the Littlewood--Richardson
coefficients~\cite{Macdonald}. The Littlewood--Richardson rule
gives in principal an algorithm to calculate $c^\la_{\mu\nu}$
for given $\la$, $\mu$ and $\nu$, but is most
appropriate to the solution of the
following problem~: given two partitions $\mu$ and
$\nu$, determine all nonzero coefficients $c^\la_{\mu\nu}$.
Often, the problem is the reverse~: given a partition $\la$,
determine all nonzero coefficients $c^\la_{\mu\nu}$. In
Section~\ref{sec-applications} we shall give some examples of
applications where this ``reverse problem'' arises.

Although the Littlewood--Richardson rule can be used to solve
this reverse problem, the new algorithm introduced here seems
to be particularly suitable for this question. This new algorithm
arises from an identity relating the combinatorial definition of
a supersymmetric S-function to a Weyl type formula. The
Weyl type formula was discovered independently by Van der Jeugt
{\sl et al}~\cite{VHKT} and by Serge'ev~(see~\ref{Pragacz}),
who communicated the formula to Pragacz. The latter was the
first to publish a proof of the identity~\ref{Pragacz}.

\section{S-functions and supersymmetric S-functions}
\label{sec-S-functions}
\setcounter{equation}{0}

Let $\la = (\la_1,\ldots,\la_p,0,0,\ldots)$
be a partition of the non-negative
integer $N$, with $\la_1\geq\ldots\geq\la_p>0$ and $\sum_i \la_i=N$.
The number $p=l(\la)$ is called the length of $\la$.
The Young diagram $F^\la$ of shape $\la$ is the set of left-adjusted
rows of squares with $\la_i$ squares (or boxes) in the $i$th row
(reading from top to bottom). For example, the Young diagram of
$(5,2,1,1)$ is given by~:
\beq
F^\la =\quad \vcenter
 {\offinterlineskip
 \halign{&\mystrut\vrule#&\hbox to 11.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan{11}\hrulefill\cr
   & && && && && &\cr
   \multispan{11}\hrulefill\cr
   & && &\cr
   \multispan5\hrulefill\cr
   & &\cr
   \multispan3\hrulefill\cr
   & &\cr
   \multispan3\hrulefill\cr
   }}
\eeq
The partition $\la'$ conjugate to $\la$ corresponds to the
transposed diagram of $\la$, i.e.~to the diagram obtained by
reflecting $F^\la$ in the main diagonal. For example,
the conjugate of $(5,2,1,1)$ is $(4,2,1,1,1)$~:
\beq
F^{\la'} =\quad \vcenter
 {\offinterlineskip
 \halign{&\mystrut\vrule#&\hbox to 11.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan9\hrulefill\cr
   & && && && &\cr
   \multispan9\hrulefill\cr
   & && &\cr
   \multispan5\hrulefill\cr
   & &\cr
   \multispan3\hrulefill\cr
   & &\cr
   \multispan3\hrulefill\cr
   & &\cr
   \multispan3\hrulefill\cr
   }}
\eeq
Denote by ${\cal S}(x)$ the ring of symmetric functions in $m$
independent variables $x_1,\ldots,x_m$~\ref{Macdonald}. The
S-functions $s_\la$ form a $\Zah$-basis of ${\cal S}(x)$. There
are various ways to define the S-functions. Let $\al$ be
an $m$-tuple of non-negative integers,
and denote the monomial $x_1^{\al_1}\ldots x_m^{\al_m}$
by $x^\al$. The skew-symmetric polynomial $a_\al$ is defined to
be
\beq
a_\al = \sum_{w\in S_m} \ep(w)\;w(x^\al) = \hbox{det}(x_i^{\al_j}),
\label{skew-poly}
\eeq
where $S_m$ is the symmetric group  and $\ep(w)$ is the
{\sl signature} of $w$. For the element
$\de = \de_m = (m-1,m-2,\ldots,1,0)$, $a_\de$ becomes the Vandermonde
determinant~:
\beq
a_\de = a_{\de_m} = \hbox{det}(x_i^{m-j}) = \prod_{1\leq i<j\leq m}(x_i-x_j).
\label{Vandermonde}
\eeq
Then the S-function $s_\al$ is defined to be~:
\beq
s_\al = s_\al(x) = s_\al(x_1,\ldots,x_m) = a_{\al+\de}/a_\de.
\label{S-function}
\eeq
It follows from the definition that every S-function $s_\al$ is
equal to $\pm s_\la$ for some partition $\la$, or equal to zero.
The definition~(\ref{S-function}) can be called a Weyl type formula,
since this is the form of Weyl's character formula for
the Lie algebra $gl(m)$.

Another formula for $s_\la$ is the determinantal formula in terms
of elementary symmetric functions $e_r$ or complete symmetric functions
$h_r$ ($r\geq 0$), defined by means of the generating functions~:
\begin{eqnarray}
\sum_r e_r t^r & = & \prod_i (1+x_it), \\
\sum_r h_r t^r & = & \prod_i (1-x_it)^{-1}.
\end{eqnarray}
Then~\cite{Macdonald}~:
\beq
s_\la = \hbox{det}(h_{\la_i-i+j}) = \hbox{det}(e_{\la_i'-i+j}).
\label{S-det-form}
\eeq

Finally, there is a combinatorial definition for $s_\la$. A
generalised Young tableau $T$ of shape $\la$ is a numbering of
the boxes of $F^\la$ with elements of $\{1,\ldots,m\}$ such that
the numbers  increase strictly down columns and increase weakly
from left to right across rows. For example, let $m=4$~:
\beq
T =\quad \vcenter
 {\offinterlineskip
 \halign{&\mystrut\vrule#&\hbox to 11.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan{11}\hrulefill\cr
   &1&&1&&2&&2&&4&\cr
   \multispan{11}\hrulefill\cr
   &2&&3&\cr
   \multispan5\hrulefill\cr
   &3&\cr
   \multispan3\hrulefill\cr
   &4&\cr
   \multispan3\hrulefill\cr
   }}
\eeq
Then $x^T = x_1^{k_1}\cdots x_m^{k_m}$, where $k_i$ is the number
of entries $i$ in the tableau $T$, and
\beq
s_\la = \sum_T x^T,
\label{S-comb-form}
\eeq
where the summation is over all generalised Young tableau $T$ of
shape $\la$.

In the definition of supersymmetric S-functions, the notion of
skew S-functions $s_{\la/\mu}$ is required. We say $\mu\leq\la$ if
$\mu_i\leq\la_i$ for every $i$. For $\mu\leq\la$, the skew Young
diagram $F^{\la/\mu}$ of shape $\la-\mu$ is the diagram $F^\la$ with
the boxes corresponding to $F^\mu$ removed. For example, if
$\la=(5,3,2,1)$ and $\mu=(2,1,1)$ then
\beq
F^{\la/\mu} =\quad \vcenter
 {\offinterlineskip
 \halign{&\mystrut\vrule#&\hbox to 11.6pt{\hss$\scriptstyle#$\hss}\cr
   \omit&\omit&\omit&\omit&\multispan7\hrulefill\cr
   \omit& &\omit& && && && &\cr
   \omit&\omit&\multispan9\hrulefill\cr
   \omit& && && &\cr
   \omit&\omit&\multispan5\hrulefill\cr
   \omit& && &\cr
   \multispan5\hrulefill\cr
   & &\cr
   \multispan3\hrulefill\cr
   }}
\eeq
A generalised Young tableau of shape $\la-\mu$ is a numbering of the boxes
of $F^{\la/\mu}$ with $\{1,\ldots,m\}$, again such that the numbers
are strictly increasing down columns and weakly
increasing along rows of $F^{\la/\mu}$. Then,
\beq
s_{\la/\mu} = \sum_T x^T,
\eeq
where the summation is over all generalised Young tableau of shape
$F^{\la/\mu}$. There exists a determinantal formula for
$s_{\la/\mu}$~\cite[p.~40]{Macdonald}, and an expansion in terms
of ordinary S-functions~:
\beq
s_{\la/\mu} = \sum_{\nu} c^\la_{\mu\nu} s_\nu,
\eeq
where $c^\la_{\mu\nu}$ are the Littlewood--Richardson coefficients
appearing in the expansion of the product of two
S-functions~(\ref{lr-coeff}).
\vskip 1cm

Let us now introduce some of the notions of supersymmetric S-functions.
The ring of doubly symmetric polynomials in $x=(x_1,\ldots,x_m)$ and
$y=(y_1,\ldots,y_n)$ is ${\cal S}(x,y) =
{\cal S}(x)\otimes_{\Zah}{\cal S}(y)$. An element $p\in{\cal S}(x,y)$
has the {\sl calcellation property} if it satisfies the following~:
when the substitution $x_1=t$, $y_1=-t$ is made in $p$, the resulting
polynomial is independent of $t$. We denote ${\cal S}(x/y)$ the subring
of ${\cal S}(x,y)$ consisting of the elements satisfying the cancellation
property. The elements of ${\cal S}(x/y)$ are called supersymmetric
functions.

Next, we shall introduce a combinatorial definition for {\sl supersymmetric
S-functions} $s_\la(x/y)$; the proof that they are indeed elements of
${\cal S}(x/y)$ follows later. Given a partition $\la$, a supertableau
$S$ of shape $\la$ is a numbering of the boxes of $F^\la$ with elements
from $\{1,\ldots,m\}$ and $\{1',\ldots,n'\}$ such that the unprimed
numbers form a generalised Young tableau $T$ of shape $\mu$ for some $\mu\leq\la$,
and such that the primed numbers form the conjugate of a generalised
Young tableau of shape $\la-\mu$. For example, for $m=n=3$~:
\beq
S =\quad \vcenter
 {\offinterlineskip
 \halign{&\mystrut\vrule#&\hbox to 11.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan{13}\hrulefill\cr
   &1&&1&&2&&3&&1'&&3'&\cr
   \multispan{13}\hrulefill\cr
   &2&&3&&3&&1'&&2'&&3'&\cr
   \multispan{13}\hrulefill\cr
   &3&&2'&&3'&\cr
   \multispan7\hrulefill\cr
   &1'&&3'&\cr
   \multispan5\hrulefill\cr
   &1'&\cr
   \multispan3\hrulefill\cr
   }}
\eeq
Let $k_i$ be the number of entries $i$ and $l_j$ be the number of
entries $j'$ is $S$. Then $(x/y)^S = x_1^{k_1}\cdots x_m^{k_m}
y_1^{l_1}\cdots y_n^{l_n}$, and one defines~:
\beq
s_\la(x/y) = \sum_S (x/y)^S,
\eeq
where the summation is over all supertableau $S$ of shape $\la$.
It follows from this definition and ? that
\beq
s_\la(x/y) = \sum_\mu s_\mu(x) \, s_{(\la/\mu)'}(y),
\eeq
or using ? and $c^{\la'}_{\mu'\nu'} = c^\la_{\mu\nu}$~:
\beq
s_\la(x/y) = \sum_{\mu,\nu} c^\la_{\mu\nu} s_\mu(x)\,s_{\nu'}(y).
\eeq
Note that the definition ? implies that the funtion $s_\la(x/y)$
is zero unless $\la_{m+1}\leq n$.

\begin{lemm}
The generating function for the functions $s_\la(x/y)$ is given by
\beq
G(x,y,z) = {\prod (1+y_iz_a) \over \prod (1-x_jz_b)}
  = \sum_\la s_\la(x/y)\,s_\la(z).
\eeq
\end{lemm}
\noindent {\sl Proof.} Using the well known expansions~\cite{Macdonald}
\begin{eqnarray}
\prod(1+y_iz_a) & = & \sum_\nu s_{\nu'}(y)s_\nu(z), \\
\prod(1-x_jz_b)^{-1} & = & \sum_\mu s_\mu(x) s_\mu(z),
\end{eqnarray}
and the Littlewood--Richardson product for the S-functions in $z$,
one finds that
\beq
G(x,y,z) = \sum_{\la,\mu,\nu} c^\la_{\mu\nu} s_\mu(x) s_{\nu'}(y) s_\la(z),
\eeq
which gives rise to ? using ?. \mybox

Note that by splitting up the variables $(z) = (z_1,\ldots,z_p)$
into $(z)=(u,v)=(u_1,\ldots,u_q$, $v_1,\ldots,v_r)$ and using
lemma~? twice, one obtains
\begin{eqnarray}
G(x,y,z)& = & {\prod(1+y_iu_a) \over \prod(1-x_ju_b)}
              {\prod(1+y_iv_a) \over \prod(1-x_jv_b)}  \\
 & = & \sum_\mu s_\mu(x/y)s_\mu(u) \sum_\nu s_\nu(x/y) s_\nu(v).
\end{eqnarray}
But $s_\la(z) = s_\la(u,v) = \sum_{\mu,\nu} c^\la_{\mu\nu} s_\mu(u)s_\nu(v)$,
hence ? leads to~:
\beq
s_\mu(x/y) s_\nu(x/y) = \sum_\la c^\la_{\mu\nu} s_\la(x/y),
\eeq
which means that the supersymmetric S-functions obey the same
product rule as ordinary S-functions.

It can be seen from ? that the functions $s_\la(x/y)$
are doubly symmetric polynomials.
The fact that they also satisfy the cancellation property
follows immediately from the generating function ?. This
justifies the name supersymmetric S-functions. In fact, Stembridge
proved the following theorem~:
\begin{theo}[Stembridge]
The set of $s_\la(x/y)$ form a $\Zah$-basis of ${\cal S}(x/y)$.
\end{theo}

Stembridge's proof made use of the determinantal form of supersymmetric
S-functions, which we shall introduce now.
One defines the supersymmetric functions $h_r(x/y)$ ($r\geq 0$)
as follows in terms of homogeneous and elementary symmetric functions~:
\beq
h_r(x/y) = \sum_k h_{r-k}(x) e_k(y).
\eeq
It is easy to verify from ? that $s_r(x/y) = h_r(x/y)$, and
one can deduce from ? that the generating function for $h_r(x/y)$
is given by
\beq
\prod(1+x_jt) \prod(1-y_it)^{-1} = \sum_r h_r(x/y) t^r.
\eeq
The following lemma gives a determinantal formula for the supersymmetric
S-function~:
\begin{lemm}
\beq
s_\la(x/y) = \hbox{det} \left(h_{\la_i-i+j}(x/y)\right).
\eeq
\end{lemm}
\noindent {\sl Proof.} Since $s_\la(x) = \hbox{det}\left(h_{\la_i-i+j}(x)
\right)$ and the functions $h_r(x/y)=s_r(x/y)$ satisfy the same
product rule as $h_r(x)=s_r(x)$ (see ?) the statement follows
immediately.\mybox

So far we have given a combinatorial and determinantal form of
the supersymmetric function. In the next section we shall study
a Weyl type formula for $s_\la(x/y)$, first proven by Pragacz.

\section{The Pragacz identity}
\label{sec-Pragacz}
\setcounter{equation}{0}

It was shown by Berele and Regev~\cite{BB} that, just as the
functions $s_\la(x)$ are the characters of simple modules of
the Lie algebra $gl(m)$, the supersymmetric S-functions are
related to characters of simple modules of the Lie superalgebra
$gl(m/n)$. In fact, they show that a covariant
tensor module can be labelled by a partition $\la$ with
$\la_{m+1}\leq n$, and that the character of this module is given by
$s_\la(x/y)$. Unfortunately, unlike the Lie algebra case, the
covariant tensor modules of $gl(m/n)$ do not encompass all
finite-dimensional simple modules of $gl(m/n)$. In an effort
to find characters for all simple $gl(m/n)$ modules, several
attempts were made to generalise a character formula of the
Weyl type due to Kac, and appropriate to yet another class of
modules, the so-called {\sl typical} modules of $gl(m/n)$. For
a review of these attempts, we refer to~\cite{VHKT}. In their
analysis, Van der Jeugt {\sl et al} realised that one particular
Weyl type formula was correct for the class of covariant tensor
modules, and hence should coincide with the expression $s_\la(x/y)$.
Independently, Serge'ev discovered the same identity and communicated
it to Pragacz, who gave a simple proof.

Let us first describe the formula itself. Let $\la$ be a partition
with $\la_{m+1}\leq n$. Consider the Young diagram $F^\la$, let
$F^\ka$ be the part of $F^\la$ that falls within the $m\times n$
rectangle, and let $F^\ta$, resp.~$F^\et$, be the remaining part
to the right, resp.~underneath this rectangle. This is illustrated,
for $m=5$, $n=8$ and $\la= $(12 9 9 6 3 3 2 2 1 1),
as follows~:
\beq
F^\la =\quad \vcenter
 {\offinterlineskip
 \halign{&\smallstrut\vrule#&\hbox to 7.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan{25}\hrulefill\cr
   & && && && && && && && && && && && &\cr
   \multispan{25}\hrulefill\cr
   & && && && && && && && && &\cr
   \multispan{19}\hrulefill\cr
   & && && && && && && && && &\cr
   \multispan{19}\hrulefill\cr
   & && && && && && && &\omit& &width0.2pt&\cr
   \multispan{13}\hrulefill&\cr
   & && && && &\omit& &\omit& &\omit& &\omit& &width0.2pt&\cr
   \multispan7\hrulefill&\multispan{10}\myrulefill\cr
   &  &&  &&  &\cr
   \multispan7\hrulefill\cr
   & && &\cr
   \multispan5\hrulefill\cr
   & && &\cr
   \multispan5\hrulefill\cr
   & &\cr
   \multispan3\hrulefill\cr
   & &\cr
   \multispan3\hrulefill\cr
   }}
\qquad \hbox{hence} \qquad \matrix{%
 \ka & = & (8,8,8,6,3)\hfil \cr
 \ta & = & (4,1,1) \hfil \cr
 \et & = & (3,2,2,1,1) \hfil \cr}
\eeq
Then, the Weyl type formula for $s_\la(x/y)$ is given by
\begin{eqnarray}
s_\la(x/y) & = & \prod_{i<j}(x_i-x_j)^{-1} \prod_{k<l}(y_k-y_l)^{-1}\times\\
& &\sum_{w\in S_m\times S_n} \ep(w)\,w\Bigl\{x^{\ta+\de_m}
 y^{\et'+\de_n}\,\prod_{(i,j)\in F^\ka}(x_i+y_j)\Bigr\},
\end{eqnarray}
where $(i,j)\in F^\ka$ if and only if the box with row-index $i$
(read from left to right) and column-index $j$ (read from top
to bottom) belongs to $F^\ka$.

Note that this is indeed a Weyl type formula, the first part ?
being Weyl's denominator for the Lie algebra $gl(m)\oplus gl(n)$, and
the second part consisting of a sum over the Weyl group of $gl(m)\oplus
gl(n)$.
It is easy to verify that the right hand side of ? is indeed
doubly symmetric; the first step in Pragacz's proof is to show that
it also satisfies the cancellation property. Then Theorem~? implies that
the right hand side of ? can be written as a $\Zah$-linear combination
of $s_\si(x/y)$ functions. Next Pragacz shows that the coefficient
of $s_\la(x/y)$ is 1, and uses an inductive argument to show that
there is only one term in the linear combination. We shall not
rewrite Pragacz's proof here, but refer the reader to his paper.

\section{The new expression for $c^\la_{\mu\nu}$}
\label{sec-new-lr} \setcounter{equation}{0}

Let $\la=(\la_1,\ldots,\la_p)$ be a given partition.
We say that $R$ is a subset of $F^\la$,
denoted by $R\subset F^\la$, if $R$ is a subset of the positions of
$F^\la$, i.e.~if
\beq
R \subset \{(1,1),(1,2),\ldots,(1,\la_1),(2,1),\ldots,(2,\la_2),\ldots
            ,(p,\la_p)\}.
\eeq
A subset $R$ of $F^\la$ can be indicated in the Young diagram by
marking the positions of $F^\la$ corresponding to $R$. For
example:
\beq
\matrix{
 \la & = & (5,3,2,1) \hfill \cr
 R   & = & \{(1,1),(1,4),(2,3),(3,2)\}\hfill \cr}
\qquad
\vcenter
{\offinterlineskip
 \halign{&\mystrut\vrule#&\hbox to 11.6pt{\hss$\scriptstyle#$\hss}\cr
   \multispan{11}\hrulefill\cr
   &\times&& && &&\times&& &\cr
   \multispan{11}\hrulefill\cr
   & && &&\times&\cr
   \multispan7\hrulefill\cr
   & &&\times&\cr
   \multispan5\hrulefill\cr
   & &\cr
   \multispan3\hrulefill\cr
   }}
\eeq
Given any $R\subset F^\la$, we define the row sum $\rh^R$, resp.~the
column sum $\ga^R$ as follows~:
\beq
\rh^R_k=\#\{(i,j)\in R|i=k\},\quad\hbox{resp.}\quad
\ga^R_k=\#\{(i,j)\in R|j=k\}.
\eeq
In example~?, $\rh^R=(2,1,1)$ and $\ga^R=(1,1,1,1)$.

For the given partition $\la$, choose $m$ and $n$ such that
$m=\la'_1$ and $n=\la_1$, in other words, such that $F^\la$ fits
exactly inside the $m\times n$ box.

\begin{prop}
\beq
s_\la(x/y) = \sum_{R\in F^\la} s_{\la-\rh^R}(x)\,s_{\ga^R}(y).
\eeq
\end{prop}

\noindent {\sl Proof.} We can use ? with $\ka=\la$ and $\ta=\et=0$.
Then, using the abbreviations
\beq
\De_x = \prod_{i<j}(x_i-x_j)\qquad\hbox{and}\qquad
\De_y = \prod_{i<j}(y_i-y_j),
\eeq
we find~:
\begin{eqnarray}
s_\la(x/y) & = & \De_x^{-1}\De_y^{-1}
     \sum_{w\in S_m\times S_n} \ep(w)\,w\Bigl\{x^{\de_m}y^{\de_n}
     \prod_{(i,j)\in F^\la} (x_i+y_j)\Bigr\}  \\
& = & \De_x^{-1}\De_y^{-1}
     \sum_{w\in S_m\times S_n} \ep(w)\,w\Bigl\{x^{\la+\de_m}y^{\de_n}
     \prod_{(i,j)\in F^\la} (1+x_i^{-1}y_j)\Bigr\}  \\
& = & \De_x^{-1}\De_y^{-1}
     \sum_{w\in S_m\times S_n} \ep(w)\,w\Bigl\{x^{\la+\de_m}y^{\de_n}
     \sum_{R\subset F^\la} x^{-\rh^R}y^{\ga^R} \Bigr\}.
\end{eqnarray}
Using the decomposition $w=\si\times\pi$ with $\si\in S_m$,
$\pi\in S_n$ and $\ep(w)=\ep(\si)\ep(\pi)$, this leads to~:
\begin{eqnarray}
s_\la(x/y) & = & \sum_{R\in F^\la} \biggl\{
                 \De_x^{-1}\sum_{\si\in S_m}\ep(\si)
                 \si\bigl(x^{\la-\rh^R+\de_m}\bigr)\quad
                 \De_y^{-1}\sum_{\pi\in S_n}\ep(\pi)
                 \pi\bigl(y^{\ga^R+\de_n}\bigr) \biggr\}\\
 & = & \sum_{R\in F^\la} s_{\la-\rh^R}(x)\,s_{\ga^R}(y).
\end{eqnarray}
\mybox

Note that the indices for the S-functions in~? are not
necessarily partitions, but using~? one sees that
\beq
s_\al = \ep(\si) s_{\si\cdot\al},\quad\forall\si\in S,
\qquad\hbox{where }\si\cdot\al = \si(\al+\de)-\de.
\eeq
Hence either such an S-function vanishes, or else it is equal
to $\pm s_\mu$ with $\mu$ a partition.

\begin{coro}
\beq
c^\la_{\mu\nu} = \sum_{{\scriptstyle \si\in S_m}\atop{\scriptstyle
 \pi\in S_n}} \ep(\si)\ep(\pi)\,f^\la_{\si\cdot\mu,\pi\cdot\nu'},
\eeq
where $f^\la_{\al\be}$ is the number of ways of removing boxes
from $F^\la$ such that the column sum of the removal is equal to
$\be$ and such that the row sum is equal to $\la-\al$.
\end{coro}

\noindent {\sl Proof.} The identity
\beq
\sum_{\mu,\nu} c^\la_{\mu\nu} s_\mu(x)\, s_{\nu'}(y) =
\sum_{R\in F^\la}s_{\la-\rh^R}(x)\, s_{\ga^R}(y)
\eeq
follows from ? and ?, and implies the statement. \mybox

Although at first sight ? does not seem to be a very efficient way to calculate
a particular $c^\la_{\mu\nu}$, it turns out that using the
identity~? is a rather elegant way to calculate the whole list
of nonzero $c^\la_{\mu\nu}$ for all $\mu$ and $\nu$. In fact it
gives rise to a very straightforward algorithm, easy to implement,
and relatively fast to run on a computer.

In the next section we shall discuss this algorithm, its implementation,
and compare with the ``reverse'' Littlewood--Richardson rule.
In section~\ref{sec-applications} we shall discuss some
problems where the algorithm can be used.

\section{The algorithm}
\label{sec-algo}
\setcounter{equation}{0}

\section{Some applications}
\label{sec-applications}
\setcounter{equation}{0}

We shall give two examples of problems where the whole list of
nonzero $c^\la_{\mu\nu}$ for given $\la$ is needed, rather than
any particular coefficient.

The first is a rather obvious example, and can be anticipated from
the previous sections. Since the characters of simple $gl(m)$ modules
are given by ordinary S-functions, it follows from~? that the
branching rule $gl(m/n)\rightarrow gl(m)\oplus gl(n)$ is given by
\beq
\{\la\}\rightarrow\sum_{\mu,\nu}c^\la_{\mu\nu}\{\mu\}\{\nu'\},
\eeq
where $\{\la\}$, with $\la$ a partition satisfying $\la_{m+1}\leq n$.\,
is the label for a covariant tensor module of
$gl(m/n)$, and $\{\mu\}$, resp.~$\{\nu\}$, is the label for a
simple module of $gl(m)$, resp.~$gl(n)$. Note that certain
modification rules need to be taken into account in the
right hand side of~?.

The second example does not involve any Lie superalgebras, but
deals with the reduction of a simple module of the Lie algebra
$gl(m+n)$ to the direct sum of simple modules of its Lie subalgebra
$gl(m)\oplus gl(n)$. This time the branching rule is given by
\beq
\{\la\}\rightarrow\sum_{\mu,\nu}c^\la_{\mu\nu}\{\mu\}\{\nu\},
\eeq
which is almost identical to~?, except that no conjugation of
$\nu$ is required. In~?, $\{\la\}$, $\{\mu\}$ and $\{\nu\}$ are
the labels of simple modules of $gl(m+n)$, $gl(m)$ and $gl(n)$
respectively. Again, modification rules apply.

%
\def\rfr#1#2#3#4#5#6 %AUTHOR YEAR JOURNAL VOLUME PAGES TITLE
{#1, ``#6,''\ {\sl #3}\ {\bf #4}, #5 (#2)}
%
\begin{thebibliography}{99}
\bibitem{LR}
\rfr{D.E.~Littlewood and A.R.~Richardson}{1934}{Philos.\
Trans.\ Roy.\ Soc.\ London Ser.~A}{233}{49--141}{Group characters
and algebra}
\bibitem{Macdonald}
I.G.~Macdonald, ``Symmetric functions and Hall polynomials'',
Oxford University Press, Oxford (1979)
\bibitem{Pragacz}
P.~Pragacz, ``Algebro--geometric applications of Schur S- and
Q-polynomials'', in {\sl S\'eminaire d'Alg\`ebre Paul Dubreil
et Marie-Paule Malliavin, Proceedings}, Lecture Notes in Mathematics,
Springer Verlag, Berlin (to be published)
\bibitem{VHKT}
J.~Van der Jeugt, J.~W.~B.~Hughes, R.~C.~King and
J.~Thierry-Mieg, ``Character formulae for irreducible modules of the Lie
superalgebra $sl(m/n)$,'' to be submitted (1989)
\end{thebibliography}
%
\end{document}
