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\title{RIBBON TABLEAUX,\\ HALL-LITTLEWOOD FUNCTIONS 
\\ AND UNIPOTENT VARIETIES \thanks{Partially supported
by PRC Math-Info and EEC grant n$^0$ ERBCHRXCT930400}} 

\author{Alain {\sc Lascoux}\thanks{L.I.T.P., Universit\'e
Paris 7, 2 place Jussieu, 75251 Paris cedex 05, France}, 
\rm Bernard {\sc   Leclerc}$^\dagger$ \rm and Jean-Yves
{\sc Thibon}\thanks{Institut Gaspard Monge, Universit\'e de
Marne-la-Vall\'ee, 2 rue de la Butte-Verte, 93166 Noisy-le-Grand cedex, France}}
\date{}

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\begin{document}



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\maketitle

\begin{abstract}
We introduce a new family of symmetric functions, which are defined
in terms of ribbon tableaux and generalize Hall-Littlewood functions.
We present a series of conjectures, and prove them in two special
cases. 
\end{abstract}

% ************************************************************************
%                      INTRODUCTION
% ************************************************************************

\section{Introduction}

Hall-Littlewood functions \cite{Li1}
are known to be related to a variety of topics in representation
theory, geometry and combinatorics. These symmetric functions arise in the 
character theory of finite linear groups \cite{Gr}, in the geometry of
unipotent varieties \cite{Sh1,HSh}, in particular as characteristics of
the representations of the symmetric group in their cohomology \cite{HS},
and appear to be related to the Quantum Inverse Scattering Method \cite{KR}.
From a combinatorial point of view, their description involves the deepest
aspects of the theory of Young tableaux: the multiplicative structure (plactic
monoid) and the ordered structure derived from the cyclage operation \cite{LS2,La}.
There exists also a description in terms of Kashiwara's theory of crystal bases
\cite{LLT4}.

Another kind of application of Hall-Littlewood functions is concerned
with the representation theory of the complex linear group $GL(n,\C)$.
The general setting is the following. Suppose we are given a finite dimensional
representation $V$ of $G=GL(n,\C)$. The symmetric group $\S_k$ acts
(on the right) on the tensor space $W=V^{\otimes k}$ by
$$
v_1\otimes v_2\otimes\cdots\otimes v_k\cdot\sigma =
v_{\sigma(1)}\otimes v_{\sigma(2)}\otimes\cdots\otimes v_{\sigma(k)} \ .
$$
Let $\gamma\in\S_k$ be a $k$-cycle. The eigenvalues of $\gamma$, as an
endomorphism of $W$ are $\zeta^r$, $r=0,\ldots,k-1$, where $\zeta$ is
a primitive $k$-th root of unity.

Let $W^{(i)}\subset V^{\otimes k}$ be the eigenspace of $\gamma$ corresponding
to the eigenvalue $\zeta^i$. Then, $W^{(i)}$ is a sub-$G$-module
of $W$, and an important problem is to compute its decomposition
into irreducibles from the character of $V$. In terms of symmetric functions,
this is a plethysm problem: if $F=\ch V$ is the formal character of $V$
and 
$$
\ell_k^{(i)}=\ch\left[ (\C^n)^{\otimes k} \right]^{(i)}
={1\over k}\sum_{d|k}c(i,d) p_d^{k/d}
$$
is the character of $W^{(i)}$ in the special case where $V=\C^n$ (the
basic representation of $G$), the problem is to expand the
plethysm $\ell_k^{(i)}\circ F$ as a linear combination of Schur functions
$$
\ell_k^{(i)}\circ F =
\sum_\lambda c_\lambda s_\lambda \ .
$$
The coefficients $c_\lambda$ are nonnegative integers, for which it is
rather unlikely that a closed formula could exist, and a combinatorial
interpretation of these coefficients, allowing their individual
computation, should be considered as a satisfactory solution of the problem.

This problem is solved in \cite{LLT1,LLT2}, in the case where $V$ is
a tensor product of exterior or symmetric powers of the basic representation,
\ie $V=\Lambda^{\mu_1}\C^n\otimes\cdots\otimes\Lambda^{\mu_r}\C^n$, or
$V=S^{\mu_1}\C^n\otimes\cdots\otimes S^{\mu_r}\C^n$. The answer in this case
is that the character of $W^{(i)}$ can be obtained by reducing modulo
$1-q^k$ a certain Hall-Littlewood function. The required combinatorial
interpretation is then obtained from the one of Hall-Littlewood functions.

Another case which is completely solved is when $k=2$ and $V=V_\lambda$
is an irreducible representation \cite{CL}. This involves a new version
of the Littlewood-Richardson rule, where ordinary tableaux are replaced
by domino tableaux. 
This rule can also be formulated as a property of the reduction
modulo $1-q^2$ of certain symmetric functions, also defined
in terms of domino tableaux, and depending on a parameter $q$ \cite{CL,KLLT}.

In this paper, we introduce a new family of symmetric functions
$H_\lambda^{(k)}(x;q)$, defined as generating functions of certain
sets of $k$-ribbon (or rim-hook) tableaux according to a statistic
called {\it spin}, which contains the Hall-Littlewood functions and
the domino functions as particular cases, and continue to display
the same type of behaviour, at least at the experimental level.

We present a series of conjectures on these new functions, and prove
them in the two extreme cases: for the shortest possible ribbons
(dominoes), and for sufficiently long ones (the stable case).

The methods used for proving these two cases are quite different.
In the domino case, the proofs are entierely combinatorial, and
do not seem to be generalizable to other cases. This is probably
due to the fact that the hyperoctahedral group is the only
group in the series $\S_n[C_k]$, $k\ge 2$  (wreath products of a cyclic
group by a symmetric group) which is also a Weyl group.
In the stable case, the proofs rely upon the cell decompositions
of unipotent varieties \cite{Sh1,HSh}. We believe that the intermediate
cases may also be proved by similar techniques, in terms of the geometry
of other subvarieties of flag manifolds.

































\section{Hall-Littlewood functions and unipotent varieties}\label{HLUV}

Our notations for symmetric functions will be essentially those of the book \cite{Mcd},
to which the reader is referred for more details.


Let $X=\{x_1,x_2,\ldots \}$ be an infinite set of indeterminates and
$\sym$ be the ring of symmetric functions in $X$, with coefficients
in $\C(q)$, $q$ being another indeterminate. In what follows, the
scalar product $\<\ ,\ \>$ on $\sym$ will always be the standard one,
for which Schur functions form an orthonormal basis. We denote
by $Q'_\mu(X;q)$ the image of the Hall-Littlewood function $Q_\mu(X;q)$
by the ring homomorphism $p_k\mapsto (1-q^k)^{-1}p_k$. That is, $(Q'_\mu)$
is the adjoint basis of the basis $(P_\mu)$ for the standard scalar product,
and in $\lambda$-ring notation, $Q'_\mu(X;q)=Q(X/(1-q);q)$. In the Schur basis,
%
\begin{equation}
Q'_\mu(X;q)=\sum_\lambda K_{\lambda\mu}(q)s_\lambda(X)
\end{equation}
%
where the $K_{\lambda\mu}(q)$ are the Kostka-Foulkes polynomials.
The polynomial $K_{\lambda\mu}(q)$ is the generating function
of a statistic $c$ called ${\it charge}$ on the set $\tab(\lambda,\mu)$ of Young
tableaux of shape $\lambda$ and weight $\mu$
%
\begin{equation}
K_{\lambda\mu}(q)=\sum_{\t\in\tab(\lambda,\mu)}q^{c(\t)} \ .
\end{equation}
%
We shall also need the $\tQ$-functions, defined by
%
\begin{equation}
\tQ_\mu(X;q) = \sum_\lambda \tK_{\lambda\mu}(q)s_\lambda(X)
=q^{n(\mu)}Q'_\mu(X;q^{-1}) \ .
\end{equation}
%
The polynomial $\tK_{\lambda\mu}(q)$ is the generating function
of the complementary statistic $\cc(\t) = n(\mu)-c(\t)$, which is called
{\it cocharge}. The operation of {\it cyclage} endows $\tab(\lambda,\mu)$
with the structure of a rank poset, in which the rank of a tableau is
equal to its cocharge (see \cite{La}).

When the parameter $q$ is interpreted as the cardinality of a
finite field $\GF_q$, it is known that $\tK_{\lambda\mu}(q)$ is equal
to the value $\chi^\lambda(u)$ of the unipotent character $\chi^\lambda$
of $G=GL_n(\GF_q)$ on a unipotent element $u$ with Jordan canonical form
specified by the partition $\mu$ (see \cite{Lu2}).

In this specialization, the coefficients
%
\begin{equation}
\tG_{\nu\mu}(q) = \<h_\nu\, ,\,\tQ_\mu\>
\end{equation}
%
of the $\tQ$-functions on the basis of monomial symmetric functions are
also the values of certain characters of $G$ on unipotent classes. Let
$P_\nu$ denote a parabolic subgroup of type $\nu$ of $G$, for example
the group of upper block triangular matrices with diagonal blocks of sizes
$\nu_1,\ldots,\nu_r$, and consider the permutation representation of
$G$ over $\C[G/P_\nu]$. The value $\xi^\nu(g)$ of the character $\xi^\nu$ of this
representation on an element $g\in G$ is equal to the number of fixed
points of $g$ on $ G/P_\nu$. Then, it can be shown that, for a unipotent
$u$ of type $\mu$,
%
\begin{equation}
\xi^\nu(u)=\tG_{\nu\mu}(q) \ .
\end{equation}
%
The factor set $G/P_\nu$ can be identified with the variety $\F_\nu$ of
$\nu$-flags in $V=\GF_q^n$
$$
V_{\nu_1}\subset V_{\nu_1+\nu_2}\subset\ldots\subset V_{\nu_1+\ldots \nu_r}=V
$$
where $\dim V_i = i$. Thus, $\tG_{\nu\mu}(q)$ is equal to the number
of $\GF_q$-rational points of the algebraic variety $\F_\nu^u$ of fixed
points of $u$ in $\F_\nu$.

It has been shown by N. Shimomura (\cite{Sh1}, see also 
 \cite{HSh}) that the corresponding complex variety $\F_\nu^u[\C]$
admits a cell decomposition, involving only cells of even real dimensions.
More precisely, this cell decomposition is a partition in locally closed
subvarieties, each being algebraically isomorphic to an affine space.
Thus, the odd-dimensional homology groups are zero, and if
$$
\Pi_{\nu\mu}(t^2)=\sum_i t^{2i} \dim H_{2i}(\F_\nu^u,\Z)
$$
is the Poincar\'e polynomial of $\F_\nu^u$, one has 
$|\F_\nu^u |=\Pi_{\nu\mu}(q)$. But this is also equal to
$\tG_{\nu\mu}(q)$, and as this is true for an infinite set of values
of $q$, one has $\Pi_{\nu\mu}(z)=\tG_{\nu\mu}(z)$ as polynomials.
That is, the coefficient of $\tQ_\mu$ on the monomial function
$m_\nu$ is the Poincar\'e polynomial of $\F_\nu^u$, for a unipotent $u$ 
of type $\mu$.

Writing
%
\begin{equation}
\tQ_{\mu} = \sum_{\lambda,\nu}\tK_{\lambda\mu(q)}K_{\lambda\nu}\, m_\nu \ ,
\end{equation}
%
one sees that
%
\begin{equation}
\tG_{\nu\mu}(q) =
\sum_{(\t_1,\t_2)\in\tab(\lambda,\mu)\times\tab(\nu,\mu)} q^{\cc(\t_1)} \ .
\end{equation}
%
Knuth's extension of the Robinson-Schensted correspondence \cite{Kn}
is a bijection between the set
$$
\coprod_\lambda \tab(\lambda,\mu)\times\tab(\lambda,\nu)
$$
of pairs of tableaux with the same shape, and the double coset space
$\S_\mu\backslash \S_n/\S_\nu$ of the symmetric group $\S_n$ modulo
two parabolic subgroups. Double cosets can be encoded by two-line arrays,
integer matrices with prescribed row and column sums, or by {\it tabloids}.

Let $\nu$ and $\mu$ be arbitrary compositions of the same integer $n$.
A $\mu$-tabloid of shape $\nu$ is a filling of the diagram of boxes
with row lengths $\nu_1,\nu_2,\ldots,\nu_r$,  the lowest row being
numbered $1$ (French convention for tableaux), such that the number $i$
occurs $\mu_i$ times, and such that each row is nondecreasing. For example,
$$
\young{ 3 \cr 1 & 1 & 1 \cr 1 & 1& 3 \cr 2 & 3 \cr}
$$
is a $(5,1,3)$-tabloid of shape $(2,3,3,1)$.



We denote by $L(\nu,\mu)$ the set of tabloids of shape $\nu$ and
weight $\mu$.  A tabloid will be identified with the word obtained by reading
it from left to right and top to bottom.
Then,
%
\begin{equation}
\tG_{\nu\mu}(q) =\sum_{T\in L(\nu,\mu)}q^{\cc(T)} \ .
\end{equation}
%
%
\begin{example}{\rm To compute $\tG_{42,321}(q)$ one lists the elements
of $L((4,2),(3,2,1))$, which are
$$
\young{2&3\cr 1&1&1&2\cr}\qquad
\young{2&2\cr 1&1&1&3\cr}\qquad
\young{1&3\cr 1&1&2&2\cr}\qquad
\young{1&2\cr 1&1&2&3\cr}\qquad
\young{1&1\cr 1&2&2&3\cr}
$$
Reading them as prescribed,
we obtain the words
$$
231112\qquad 221113\qquad 131122\qquad 121123\qquad 111223
$$
whose respective charges are $2,1,3,2,4$. The cocharge polynomial is
thus $\tG_{42,321}(q) = 1+q+2q^2+q^3$.
}
\end{example}

In Shimomura's decomposition of the fixed point variety $\F_\mu^u$ of a unipotent
of type $\nu$, the cells are indexed by tabloids of shape $\nu$ and weight
$\mu$. The dimension $d(T)$ of the cell $c_T$ indexed by $T\in L(\nu,\mu)$ is
computed by an algorithm described below, and gives another combinatorial interpretation
of the polynomial $\tG_{\mu\nu}(q)$, exchanging the r\^oles of shape and weight:

%
\begin{equation}
\tG_{\mu\nu}=\sum_{T\in L(\mu,\nu)}q^{\cc(T)}
=\sum_{T\in L(\nu,\mu)}q^{d(T)} \ .
\end{equation}
%
The dimensions $d(T)$ are given by the following algorithm.


\begin{enumerate}

\item If $T\in L(\nu,(n))$ then $d(T)=0$;

\item If $\mu=(\mu_1,\mu_2)$ has exactly two parts, and $T\in L(\nu,\mu)$,
then $d(T)$ is computed as follows. A box $\alpha$ of $T$ is said to
be {\it special} if it contains the rightmost $1$ of its row. For a box
$\beta$ of $T$, put $d(\beta)$=0 if $\beta$ does not contain a $2$,
and if $\beta$ contains a $2$, set $d(\beta)$ equal to the number of
nonspecial $1$'s lying in the column of $\beta$, plus the number of
special $1$'s lying in the same column, but in a lower position. Then
$$d(T)=\sum_\beta d(\beta)\ .$$

\item Let $\mu=(\mu_1,\ldots,\mu_k)$ and $\mu^*=(\mu_1,\ldots,\mu_{k-1})$.
For $T\in L(\nu,\mu)$, let $T_1$ be the tabloid obtained by changing
the entries $k$ into $2$ and all the other ones into $1$. Let $T_2$
be the tabloid obtained by erasing all the entries $k$, {\it and rearranging
the rows in the appropriate order}. Then,
\begin{equation}
d(T)=d(T_1)+d(T_2) \ .
\end{equation}
%
\end{enumerate}

\begin{example}{\rm
With $T=\young{1&4\cr1&2&3\cr1&1&2\cr} \in L(332,4211)$, one has
$$
T_1:=\young{1&2\cr1&1&{\bf 1}\cr1&1&{\bf 1}\cr}
\qquad
T_2=\young{1\cr 1&2&3 \cr 1&1&2\cr}
\qquad
T_{21}=\young{1\cr 1&{\bf 1}&2\cr 1&1&{\bf 1}\cr}
\qquad
T_{22}=\young{1\cr {\bf 1}&2\cr 1&{\bf 1}&2\cr}
$$
where the special entries are printed in boldtype.
Thus, $d(T)=t(T_1)+d(T_2)=2+d(T_{21})+d(T_{22})=4$.
}
\end{example}

We shall need a variant of this construction, in which the shape
$\nu$ is allowed  to be an arbitrary composition, and where in step 3,
the rearranging of the rows is supressed. Such a variant has already
been used by Terada \cite{Te} in the case of complete flags.

That is, we associate to a tabloid $T\in L(\nu,\mu)$ an integer $e(T)$,
defined by

\begin{enumerate}

\item For $T\in L(\nu,(n))$, $e(T)=d(T)=0$;

\item For $T\in L(\nu, (\mu_1,\mu_2))$, $e(T)=d(T)$;

\item Otherwise $e(T)=e(T_1)+e(T_2)$ where $T_1$ is defined as above,
but this time $T_2$ is obtained from $T$ by erasing the entries $k$,
without reordering.
%
\end{enumerate}

\begin{lemma}
Let $\lambda=(\lambda_1,\ldots,\lambda_r)$ be a partition, and let
$\nu=\lambda\cdot\sigma=(\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(r)})$,
$\sigma\in \S_r$. Then, 
the distribution of $e$ on $L(\nu,\mu)$ is the same as the distribution of
$d$ on $L(\lambda,\mu)$.
That is,
$$D_{\lambda\mu}(q)=\sum_{T\in L(\lambda,\mu)}q^{d(T)}
=E_{\nu\mu}(q)=\sum_{T\in L(\nu,\mu)}q^{e(T)} \ . 
$$
In particular, $D_{\lambda\mu}(q)=E_{\lambda\mu}(q)$.
\end{lemma}

\Proof This could be proved by repeating word for word the geometric argument
of \cite{Sh1}. We give here a short combinatorial argument. As the two statistics
coincide on tabloids whose shape is a partition and whose weight has at most
two parts, the only thing to prove, thanks to the recurrence formula, is that
$e$ has the same distribution 
on $L(\beta,(\mu_1,\mu_2))$ as on $L(\alpha,(\mu_1,\mu_2))$ when $\beta$ is
a permutation of $\alpha$. The symmetric group being generated by the elementary
transpositions $\sigma_i=(i,i+1)$, one may assume that $\beta=\alpha\sigma_i$.
We define the image $T\sigma_i$ of a tabloid $T\in L(\alpha,(\mu_1,\mu_2))$ by
distinguishing among the following configurations for rows $i$ and $i+1$:

\begin{enumerate}

\item
$$
\left.\matrix{ x_1 &\ldots & x_k&2&2^r\cr
		1  &\ldots & 1 &{\bf 1} & 2^s\cr}\right.
	\qquad
{\sigma_i \atop \makebox[1cm]{\rightarrowfill} }
\qquad
\left.\matrix{ x_1 &\ldots & x_k&2&2^s\cr
               1  &\ldots & 1 &{\bf 1} & 2^r\cr}\right.
$$
\item
$$
\left.\matrix{1  &\ldots & 1 &{\bf 1} & 2^r\cr
              x_1 &\ldots & x_k&2&2^s\cr}\right.
	      \qquad
	      {\sigma_i\atop \makebox[1cm]{\rightarrowfill}}
	      \qquad
\left.\matrix{1  &\ldots & 1 &{\bf 1} & 2^s\cr
	      x_1 &\ldots & x_k&2&2^r\cr}\right.
$$
\item In all other cases, the two rows are exchanged:
$$
\left.\matrix { x_1&\ldots & x_r\cr y_1&\ldots &&y_s\cr}\right.
\qquad
{\sigma_i\atop \makebox[1cm]{\rightarrowfill}}
\qquad
\left.\matrix {y_1&\ldots &&y_s\cr  x_1&\ldots & x_r\cr}\right.
$$
%
\end{enumerate}
From this definition, it is clear that $e(T\sigma_i)=e(T)$. Moreover, it
is not difficult to check that this defines an $e$-preserving action
of the symmetric group $\S_m$ on the set of $\mu$-tabloids with $m$
rows, such that $L(\alpha,\mu)\sigma=L(\alpha\sigma,\mu)$ (the only point
needing a verification is the braid relation
$\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$.

Thus, for a partition $\lambda$ and a two-part weight $\mu=(\mu_1,\mu_2)$,
$d$ and $e$ coincide on $L(\lambda,\mu)$, and for $\sigma\in\S_m$,
$E_{\lambda\sigma,\mu}(q)=D_{\lambda\mu}(q)$. Now, by induction, for
$\mu=(\mu_1,\ldots,\mu_k)$,
$$
D_{\lambda\mu}(q)=\sum_{T\in L(\lambda,\mu)}q^{d(T_1)}q^{d(T_2)}
$$
$$
=\sum_{\bar\lambda={\rm shape\,}(T_1)}q^{d(T_1)}D_{\bar\lambda,\mu^*}(q)
=\sum_{\bar\lambda={\rm shape\,}(T_1)}e^{e(T_1)}E_{\bar\lambda,\mu^*}(q)
=E_{\lambda\mu}(q) \ .
$$
\cqfd



\begin{example}{\rm
Take $\lambda=(3,2,1)$, $\mu=(4,2)$ and $\nu=\lambda\sigma_1\sigma_2=(3,1,2)$.
The $\mu$-tabloids of shape $\lambda$ are
$$
\matrix{
T
&
\young{2\cr 1&2\cr 1&1&1\cr}
&
\young{2\cr 1&1\cr 1&1&2\cr}
&
\young{1\cr 1&1\cr 1&2&2\cr}
&
\young{1\cr 2&2\cr 1&1&1\cr}
&
\young{1\cr 1&2\cr 1&1&2\cr}
\cr
d(T) & 3 & 2 & 0 & 2 & 1 \cr}
$$
The $\nu$-tabloids of shape $\lambda$ are
$$
\matrix{
T
&
\young{1&1&1\cr 1\cr 2&2\cr}
&
\young{1&1&1\cr 2\cr 1&2\cr}
&
\young{1&1&2\cr 1\cr 1&2\cr}
&
\young{1&1&2\cr 2\cr 1&1\cr}
&
\young{1&2&2\cr 1\cr 1&1\cr}
\cr
e(T) & 2 & 3 & 0 & 2 & 1\cr}
$$
Thus, $D_{\lambda\mu}(q)=E_{\nu\mu}(q)=1+q+2q^2+q^3=\tG_{\mu\lambda}(q)$.
The tabloids contributing a term $q^2$ are apparied in the following way:
$$
\young{2\cr 1&1\cr 1&1&2\cr}\quad\longrightarrow\quad
\young{1&1&2\cr 2\cr 1&1\cr}\qquad
\young{1\cr 2&2\cr 1&1&1\cr}\quad\longrightarrow\quad
\young{1&1&1\cr 1\cr 2&2\cr}
$$
}
\end{example}

{\bf Remark}  The only property that we shall need in the sequel is the
equality $D_{\lambda\mu}(q)=E_{\lambda\mu}(q)$. However, it is
possible to be more explicit by constructing a bijection exchanging
$d$ and $e$. The above action of $\S_m$ can be extended to tabloids
with arbitrary weight, still preserving $e$. Suppose for example
that we want to apply $\sigma_i$ to a tabloid $T$ whose restriction
to rows $i,i+1$ is
$$
\young{1&1&2&3&7&7&9\cr
       1&1&1&2&6&6&6&8&8&9\cr}
$$
One first determines the positions of the greatest entries, which
are the $9$'s, in $T\sigma_i$. Starting with an empty diagram
of the permuted shape $(10,7)$, one constructs $T_1$ as above
by converting all the entries $9$ of $T$ into $2$ and the remaining
ones into $1$. Then we apply $\sigma_i$ to $T_1$, and the positions
of the $2$ in $T_1\sigma_i$ give the positions of the $9$ in  $T\sigma_i$.
Then, the entries $9$ are removed from $T$ ad the procedure is iterated
until one reaches a tabloid whose rows $i$ and $i+1$ are of equal lenghts.
This tabloid is then copied (without permutation) in the remaining part
of the result. On the example, this gives

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\normalsize


\section{Specializations at roots of unity}

As recalled in the preceding section, the Hall-Littlewood functions with
parameter specialized to the cardinality $q$ of a finite field $\GF_q$
provide information about the characters of the linear group $GL(n,\GF_q)$
over this field. It turns out that when the parameter is specialized
to a complex root of unity, one obtains information about representations of $GL(n,\C)$,
that is, a combinatorial decomposition of certain plethysms
\cite{LLT1,LLT2}. We give now a brief review of the main results
of these papers. 

The first one is a factorization property of the functions $Q'_\lambda(X,q)$
when $q$ is specialized to a primitive root of unity. This is to be seen as
a generalization of the fact that when $q$ is specialized to $1$ the function
$Q'_{\lambda}(X;q)$ reduces to $h_{\lambda}(X) = \prod_i h_{\lambda_i}(X)$. 


\begin{theorem}\label{THLLT1} 
Let $\lambda = (1^{m_1}2^{m_2} \ldots n^{m_n})$ be a
partition written multiplicatively. 
Set $m_i=kq_i+r_i$ with $0\le r_i<k$, and
$\mu=(1^{r_1}2^{r_2} \ldots n^{r_n})$. Then, $\zeta$ being a primitive
$k$-th root of unity,
\begin{equation}\label{FACT}
Q'_{\lambda}(X;\zeta)=Q'_\mu(X;\zeta)
     \prod_{i\ge 1}\bigl[ Q'_{(i^k)}(X;\zeta)\bigr]^{q_i} \ . 
\end{equation}
\end{theorem}

The functions $Q'_{(i^k)}(X;\zeta)$ appearing in the right-hand side of (\ref{FACT}) 
can be expressed as plethysms.

\begin{theorem}\label{THLLT2} 
Let $p_k\circ h_n$ denote the plethysm of the
complete function $h_n$ by the power-sum $p_k$, which is defined
by the generating series
$\displaystyle \sum_n  p_k\circ h_n(X)\,z^n=\prod_{x\in X}(1-zx^k)^{-1}$.
Then, if $\zeta$ is as above a primitive $k$-th root of unity, one has
$$Q'_{(n^k)}(X;\zeta)=(-1)^{(k-1)n}p_k\circ h_n(X). $$
\end{theorem}

\begin{example}
{\rm With $k=3$ $(\zeta = e^{2i\pi /3})$, we have
$$
Q'_{444433311}(X;\zeta)=Q'_{411}(X;\zeta)\,Q'_{333}(X;\zeta)\,Q'_{444}(X;\zeta)
=Q'_{411}(X;\zeta)\,p_4\circ h_{43} \ .
$$
}
\end{example}




Given two partitions $\lambda$ and $\mu$, we denote by $\lambda \vee \mu$
the partition obtained by reordering the concatenation of $\lambda$
and $\nu$, {\it e.g.}  $(2,\,2,\,1)\vee (5,\,2,\,1,\,1) = (5,\,2^3,\,1^3)$.
We write $\mu^k=\mu\vee \mu\vee \cdots\vee \mu$
($k$ factors). If $\mu = (\mu_1,\,\ldots ,\,\mu_r)$, we set
$k\mu = (k\mu_1,\,\ldots ,\,k\mu_r)$.



For $k,n\in {\bf N}$, the \it Ramanujan \rm or \it Von Sterneck\rm
\it sum \rm $c(k,n)$ (also denoted $\Phi(k,n)$) is the sum of the
$k$-th powers of the \it primitive \rm $n$-th roots of unity.  Its value
is given by \it H\"older's formula\rm: if $(k,n)=d$ and $n=md$, then
$c(k,n)=\mu(m)\phi(n)/\phi(m)$, where $\mu$ is the Moebius function and
$\phi$ is the Euler totient function (see {\it e.g.}  \cite{NV}).

Let $P(q)= \sum_{k=0}^{n-1}a_k\,q^k \ \in \Z[q]$ be
a polynomial of degree $\le n-1$. $P$ is said to be \it even modulo $n$ \rm
if $(i,n)=(j,n)\ \Rightarrow a_i=a_j$. 
The following property
\cite{Co} can be regarded as a
generalization of the Moebius inversion formula:
\begin{lemma}
The polynomial $P$ is even modulo $n$ iff for
every divisor $d$ of $n$, the residue of $P(q)$ modulo the cyclotomic
polynomial $\Phi_d(q)$ is a constant $r_d\in{\bb Z}$. In this case, one has
$$
a_k={{1}\over{n}}\sum_{d\mid n}
                  c(k,d)r_d \ , \ \ \ \ 
              r_d=\sum_{t\mid n}c(n/d,t)a_{n/t}\ .
$$
\end{lemma}

With the aid of Ramanujan sums, we define the symmetric functions
\begin{equation}
\ell^{(k)}_n={{1}\over{n}}\sum_{d\mid n}c(k,d)p_d^{n/d} \ . 
\end{equation}
These functions were first encountered by Foulkes as Frobenius characteristics
of the representations of the symmetric group induced by irreducible
representations of transitive  cyclic subgroup \cite{Fo}.
A combinatorial interpretation of the multiplicity $\<s_\lambda\, ,\, \ell^{(k)}_n\>$
has been given by Kraskiewicz and Weyman \cite{KW}. This result is equivalent
to the congruence
$$
Q'_{1^n}(X;q) \equiv \sum_{0\le k \le n-1} q^k \ell_n^{(k)} \ (\mod 1-q^n) \ .
$$
A proof using Cohen's formula can be found in
\cite{De}. Taking into account Theorems \ref{THLLT1} and \ref{THLLT2}, one obtains:

\begin{theorem}{\rm \cite{LLT2} }
 Let $e_i$ be the $\ i$-th elementary symmetric
function, and for $\lambda=(\lambda_1,\ldots,\lambda_m)$, $e_\lambda=
e_{\lambda_1}\cdots e_{\lambda_r}$.
Then, the multiplicity $\<s_\mu\, ,\,\ell^{(r)}_k\circ e_\lambda\>$
of the Schur function $s_\mu$ in the plethysm $\ell^{(r)}_k\circ e_\lambda$
is equal to the number of Young tableaux of shape $\mu'$ (conjugate
partition) and weight $\lambda^k$ whose charge is congruent to $r$ modulo $k$.

This gives as well the plethysms with product of complete functions, since
$$
\<s_{\mu'}\, ,\, \ell_k^{(r)}\circ e_\lambda \> =
\left\{ \matrix{
\<s_\mu\, ,\, \ell_k^{(r)}\circ h_\lambda\> & \mbox{if $|\lambda|$ is even}\cr
\<s_\mu\, ,\, \tilde\ell_k^{(r)}\circ h_\lambda\> & \mbox{if $|\lambda|$ is odd}\cr
}\right.
$$
where $\tilde\ell_k^{(r)}=\omega(\ell_k^{(r)})=\ell_k^{(s)}$ with
$s=k(k-1)/2-r$.
\end{theorem}

\begin{example}{ \rm With $k=4$, $r=2$ and $\lambda=(2)$,
$$\ell^{(2)}_4\circ e_2 = s_{431} +s_{422} +s_{41111}+2s_{3311}+2s_{3221}$$
$$+2s_{32111} +s_{2222}+s_{22211}+2s_{221111}+s_{2111111} \ .$$
To compute the coefficient $\<s_{32111}\, ,\, \ell^{(2)}_4\circ e_2\>=2$, we have
to find the number of tableaux of shape $(3,\,2,\,1,\,1,\,1)'=(5,\,2,\,1)$, weight $(2,\,2,\,2,\,2)$
and charge $\equiv 2\ (\mod 4)$. The two tableaux satisfying
these constraints are:

$$
\young{3\cr 2&4\cr 1&1&2&3&4\cr}\qquad\qquad
\young{4\cr 2&3 \cr 1&1&2&3&4\cr}
$$

\bigskip\noindent
which both have charge equal to $6$. \par
Similarly, the reader can check that $\<s_{732}\, ,\, \ell^{(2)}_4\circ e_{21}\>=5$
is the number of tableaux with shape $(3,\,3,\,2,\,1,\,1,\,1,\,1)$, weight $(2,\,2,\,2,\,2,\,1,\,1,\,1,\,1)$ and
charge $\equiv 2\ (\mod 4)$.
}
\end{example}


A more combinatorial formulation of Theorems \ref{THLLT1} and
\ref{THLLT2} can be presented by means of the notion of
{\it ribbon tableau}, which will also provide the key for their
generalization.



\section{Ribbon tableaux}

To a partition $\lambda$ is associated a $k$-core $\lambda_{(k)}$
and a $k$-quotient $\lambda^{(k)}$
\cite{JK}. The $k$-core is the unique partition obtained by successively removing 
$k$-ribbons (or skew hooks) from $\lambda$. The different possible ways of 
doing so can be distinguished from one another by labelling $1$ the
last ribbon removed, $2$ the penultimate, and so on. Thus Figure~\ref{TRIBB}
shows two different ways of reaching the $3$-core $\lambda_{(3)}=(2,\,1^2)$
of $\lambda = (8,\, 7^2,\,4,\,1^5)$. These pictures represent two $3$-ribbon 
tableaux $T_1,\,T_2$ of shape $\lambda/\lambda_{(3)}$ and weight $\mu = (1^9)$.

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\end{figure}

To define  $k$-ribbon tableaux of general weight and shape, we need some terminology.
The {\it initial cell} of a $k$-ribbon $R$ is its rightmost and bottommost
cell. Let $\theta = \beta/\alpha$ be a skew shape, and set 
$\alpha_+ = (\beta_1)\vee \alpha$, so that $\alpha_+/\alpha$ is the horizontal
strip made of the bottom cells of the columns of $\theta$. We say that $\theta$
is a {\it horizontal $k$-ribbon strip} of weight $m$, if it can be tiled by
$m$ $k$-ribbons the initial cells of which lie in $\alpha_+/\alpha$. (One can
check that if such a tiling exists, it is unique).


  

Now, a {\it $k$-ribbon tableau} $T$ of shape $\lambda/\nu$ and weight
$\mu=(\mu_1,\,\ldots ,\,\mu_r)$ is defined as a chain of partitions
$$
\nu=\alpha^0\subset \alpha^1 \subset \cdots \subset \alpha^r=\lambda
$$
such that $\alpha^i/\alpha^{i-1}$ is a horizontal $k$-ribbon strip of weight
$\mu_i$. Graphically, $T$ may be described by numbering each $k$-ribbon of 
$\alpha^i/\alpha^{i-1}$ with the number $i$. We denote by 
$\tab_k(\lambda/\nu,\,\mu)$ the
set of $k$-ribbon tableaux of shape $\lambda/\nu$ and weight $\mu$, and we set
$$
K_{\lambda/\nu,\,\mu}^{(k)} = |\tab_k(\lambda/\nu,\,\mu)| \ .
$$
Finally we recall the definition of the $k$-sign $\epsilon_k(\lambda/\nu)$. Define
the sign of a ribbon $R$ as $(-1)^{h-1}$, where $h$ is the height of $R$. The
$k$-sign $\epsilon_k(\lambda/\nu)$ is the product of the signs of all the ribbons
of a $k$-ribbon tableau of shape $\lambda/\nu$ (this does not depend on the
particular tableau chosen, but only on the shape).

The origin of these combinatorial definitions is best understood by analyzing
carefully the operation of multiplying a Schur function $s_\nu$ by a plethysm
of the form $p_k \circ h_\mu$. Equivalently, thanks to the involution $\omega$,
one may rather consider a product of the type $s_\nu \, [p_k\circ e_\mu]$. To this end,
since
$$
p_k\circ e_\mu = (e_{\mu_1}\circ p_k)\, \cdots (e_{\mu_n}\circ p_k)
= m_{k^{\mu_1}}\cdots m_{k^{\mu_n}} 
$$
one needs only to apply repeatedly the following multiplication rule due
to Muir \cite{Mu} (see also \cite{Li3}):
$$
s_\nu \, m_\alpha = \sum_\beta s_{\nu + \beta} \ ,
$$
sum over all distinct permutations $\beta$ of 
$(\alpha_1,\,\alpha_2,\,\ldots ,\, \alpha_n,\,0,\,\ldots \, )$.
Here  the Schur functions 
$s_{\nu + \beta}$ are not necessary indexed by partitions and have therefore
to be standardized, this reduction yielding only a finite number of nonzero
summands. For example,
$$
s_{31}\,m_3 = s_{61} + s_{313} + s_{31003} = s_{61} - s_{322} + s_{314} \ .
$$
Other terms such as $s_{34}$ or $s_{3103}$
reduce to $0$. It is easy to deduce
from this rule that the multiplicity
$$
\< s_\nu \, m_{k^{\mu_i}} \, , \, s_\lambda \>
$$
is nonzero iff $\lambda '/\mu '$ is a horizontal $k$-ribbon strip of weight
$\mu_i$, in which case it is equal to $\epsilon_k(\lambda/\nu)$. Hence, applying
$\omega$ we arrive at the expansion
$$
s_\nu \, [p_k\circ h_\mu] = \sum_\lambda \epsilon_k(\lambda/\mu) \, 
K_{\lambda/\nu \,, \mu}^{(k)} \, s_\lambda
$$
from which we deduce by \ref{THLLT1}, \ref{THLLT2} that
$$
K_{\lambda \, \mu}^{(k)} = (-1)^{(k-1)|\mu|} \, \epsilon_k(\lambda) 
\, K_{\lambda \, \mu^k}(\zeta)
$$
and more generally, defining as in \cite{KR} the skew Kostka-Foulkes polynomial
$K_{\lambda/\nu \,, \alpha}(q)$ by
$$
K_{\lambda/\nu \,, \alpha}(q) = \< s_{\lambda/\nu} \, , \, Q'_\alpha(q) \>
$$
we can write
$$
K_{\lambda/\nu \,, \mu}^{(k)} = (-1)^{(k-1)|\mu|} \, \epsilon_k(\lambda/\nu) 
\, K_{\lambda/\nu \,, \mu^k}(\zeta) \ .
$$

It turns out that enumerating $k$-ribbon tableaux is equivalent to enumerating
$k$-uples of ordinary Young tableaux, as shown by the correspondence to be described
now. This bijection was first studied by Stanton and White \cite{SW} in the case
of ribbon tableaux of right shape $\lambda$ (without $k$-core) and standard
weight $\mu = (1^n)$ (see also \cite{FS}). We need some additional definitions.


Let $R$ be a $k$-ribbon of a $k$-ribbon tableau. $R$ contains a unique cell 
with coordinates $(x,\,y)$ 
such that $y-x\equiv 0 \ (\mod k)$. We decide to write in this cell the number attached to
$R$, and we define the {\it type} $i\in \{0,\,1,\,\ldots ,\,k-1\}$
of $R$ as the distance between this cell and the initial cell of $R$.
For example, the $3$-ribbons of 
$T_1$ are divided up into three classes:
\begin{itemize}
\item 4, 6, 8, of type 0;
\item 1, 2, 7, 9, of type 1;
\item 3, 5, of type 2.
\end{itemize}
Define the {\it diagonals} of a $k$-ribbon tableau as the sequences of integers
read along the straight lines $D_i \, : \, y-x = ki $.
Thus $T_1$ has the sequence of diagonals 
$$((8),\,(4),\, (2,\,3,\,6),\,
(1,\,5,\,9),\,(7))\ .$$
This definition applies in particular to $1$-ribbon tableaux, {\it i.e.} ordinary
Young tableaux. It is obvious that a Young tableau is uniquely determined
by its sequence of diagonals. Hence, we can associate to a given $k$-ribbon tableau $T$ of
shape $\lambda/\nu$ a $k$-uple $(t_0,\,t_1,\,\ldots ,\,t_{k-1})$ of 
Young tableaux defined as follows; the diagonals of $t_i$ are obtained by erasing
in the diagonals of $T$ the labels of all the ribbons of type $\not = i$.
For instance, if $T=T_1$ the first ribbon tableau of Figure~\ref{TRIBB}, the sequence of
diagonals of $t_1$ is $\left((2),\,(1,\,9),\,(7)\right)$, and 

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\bigskip
\noindent
The complete triple $(t_0,\,t_1,\,t_2)$ of Young tableaux associated to $T_1$ is

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\bigskip
\noindent
whereas that corresponding to $T_2$ is

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\bigskip
\noindent
One can show that if $\nu =\lambda_{(k)}$, the $k$-core of $\lambda$,
the $k$-uple of shapes $(\lambda^0,\, \lambda^1,\, \ldots,\,\lambda^{k-1})$
of $(t_0,\,t_1,\,\ldots ,\,t_{k-1})$ depends only on the shape $\lambda$ of
$T$, and is equal to the $k$-quotient $\lambda^{(k)}$ of $\lambda$. 
Moreover the correspondence $T \longrightarrow (t_0,\,t_1,\,\ldots ,\,t_{k-1})$ establishes a
bijection between the set of $k$-ribbon tableaux of shape $\lambda/\lambda_{(k)}$
and weight $\mu$, and the set of $k$-uples of Young tableaux of
shapes $(\lambda^0,\,\ldots ,\,\lambda^{k-1})$ and weights $(\mu^0,\, \ldots ,\,\mu^{k-1})$
with $\mu_i = \sum_j \mu^j_i$. 
(See \cite{SW} or \cite{FS}
for a proof in the case when $\lambda_{(k)}= (0)$ and $\mu = (1^n)$).

For example, keeping $\lambda = (8,\, 7^2,\,4,\,1^5)$, the triple

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\bigskip
\noindent
with weights $\left( (0,\,0,\,2,\,1),\,(1,\,1,\,1,\,1),\,(0,\,1,\,1,\,0)\right)$
corresponds to the 3-ribbon tableau 

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\bigskip
\noindent
of weight $\mu=(1,\,2,\,4,\,2)$.

As before, the significance of this combinatorial construction becomes clearer
once interpreted in terms of symmetric functions. Recall the definition of
$\phi_k$, the adjoint of the linear operator 
$\psi^k:\ F\mapsto p_k\circ F$ acting on the space of symmetric functions.
In other words, $\phi_k$ is characterized by
$$
\< \phi_k(F) \, , \, G \> = \< F \, , \, p_k \circ G \> \ , \ \ \  F,\,G \in \sym \ .
$$
Littlewood has shown \cite{Li3} that if $\lambda$ is a partition whose
$k$-core $\lambda_{(k)}$ is null, then
\begin{equation}\label{PHIQUOT}
\phi_k(s_\lambda) = \epsilon_k(\lambda) \, s_{\lambda^0} \, s_{\lambda^1} \, \cdots
\, s_{\lambda^{k-1}}
\end{equation}
where $\lambda^{(k)} = (\lambda^0 ,\, \ldots \, ,\, \lambda^{k-1} )$ is the
$k$-quotient. Therefore, 
$$
K_{\lambda \, \mu}^{(k)} = \epsilon_k(\lambda) \, \< p_k \circ h_\mu \, , \, s_\lambda \>
= \epsilon_k(\lambda) \, \< \phi_k(s_\lambda) \, , \, h_\mu \>
= \< s_{\lambda^0} \, s_{\lambda^1} \, \cdots
\, s_{\lambda^{k-1}} \, , \, h_\mu \>
$$
is the multiplicity of the weight $\mu$ in the product of Schur functions
$s_{\lambda^0} \, \cdots \, s_{\lambda^{k-1}}$, that is, is equal to the number of 
$k$-uples of Young tableaux of shapes $(\lambda^0,\,\ldots ,\,\lambda^{k-1})$ and 
weights $(\mu^0,\, \ldots ,\,\mu^{k-1})$ with $\mu_i = \sum_j \mu^j_i$. Thus, the
bijection described above gives a combinatorial proof of (\ref{PHIQUOT}).

More generally, if $\lambda$ is replaced by a skew partition $\lambda/\nu$,
(\ref{PHIQUOT}) becomes \cite{KSW}
$$
\phi_k(s_{\lambda/\nu}) = \epsilon_k(\lambda/\nu) \, s_{\lambda^0/\nu^0} \, 
s_{\lambda^1/\nu^1} \, \cdots \, s_{\lambda^{k-1}/\nu^{k-1}}
$$
if $\lambda_{(k)} = \nu_{(k)}$, and $0$ otherwise. This can also be deduced from the 
previous combinatorial correspondence, but we shall not go into further details.

Returning to Kostka polynomials, we may summarize this discussion by stating 
Theorems~\ref{THLLT1} and \ref{THLLT2} in the following way:

\begin{theorem}
Let $\lambda$ and $\nu$ be partitions and set $\nu = \mu^k \vee \alpha$ with
$m_i(\alpha) < k$. Denoting by $\zeta$ a primitive $k$th root of unity, one
has
\begin{equation}\label{KOSTROOT}
K_{\lambda,\,\nu}(\zeta) = (-1)^{(k-1)|\mu|} 
\sum_\beta \epsilon_k(\lambda/\beta)\, K_{\lambda/\beta,\,\mu}^{(k)}\,
K_{\beta,\,\alpha}(\zeta) \ .
\end{equation}
\end{theorem}

\begin{example}{\rm We take $\lambda = (4^2,\,3)$, $\nu = (2^2,\,1^7)$ and
$k=3$ $(\zeta = e^{2i\pi/3})$. In this case, 
$\nu= \mu^k \vee \alpha$ with $\mu = (1^2)$ and
$\alpha = (2^2,\,1)$. The summands of (\ref{KOSTROOT}) are parametrized
by the $3$-ribbon tableaux of external shape $\lambda$ and weight $\mu$.
Here we have three such tableaux:

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\bigskip\noindent
so that 
$$
K_{443,\,221111111}(\zeta) = 2 K_{41,\,221}(\zeta) - K_{32,\,221}(\zeta)
= 2(\zeta^2 + \zeta^3) - (\zeta + \zeta^2) = 2\zeta^2 + 3\ .
$$
}
\end{example}

When $|\alpha|\le |\lambda_{(k)}|$, (\ref{KOSTROOT}) becomes simpler. For  
if $|\alpha| < |\lambda_{(k)}|$ then $K_{\lambda,\,\nu}(\zeta) = 0$, and
otherwise the sum reduces to one single term
$$
K_{\lambda,\,\nu}(\zeta) = (-1)^{(k-1)|\mu|}
\epsilon_k(\lambda/\lambda_{(k)})\, K_{\lambda/\lambda_{(k)},\,\mu}^{(k)}\,
K_{\lambda_{(k)},\,\alpha}(\zeta) \ .
$$
In particular, if $\nu = (1^n)$, one recovers the following theorem of Morris and
Sultana \cite{MS}.
\begin{theorem}
Let $\lambda$ be a partition of $n$ and $\zeta$ a primitive $k$th root of unity. 
Denote by $H(\lambda^{(k)})$ the product of the hook-lengths of the $k$
partitions $\lambda^0,\,\ldots ,\,\lambda^{k-1}$, and by $|\lambda^{(k)}|$ the
sum of their weights.
Set $n=kq+r$, $0\le r <k$. If $r \not = |\lambda_{(k)}|$, 
then $K_{\lambda,\,(1^n)}(\zeta) =0$, otherwise,
$$
K_{\lambda,\,(1^n)}(\zeta) = (-1)^{(k-1)q}
\epsilon_k(\lambda/\lambda_{(k)})\, {|\lambda^{(k)}|!\over H(\lambda^{(k)})}\,
K_{\lambda_{(k)},\,1^r}(\zeta) \ .
$$
\end{theorem}
Indeed, the correspondence  just described between $k$-ribbon tableaux
and $k$-uples of Young tableaux shows at once, in view of the classical
hook-formula \cite{JK}, that
$$
K_{\lambda/\lambda_{(k)},\,1^q}^{(k)} = {|\lambda^{(k)}|!\over H(\lambda^{(k)})} \ .
$$


\section{$H$-functions}

Let $\lambda$ be a partition without $k$-core, and with $k$-quotient
$(\lambda^0,\ldots,\lambda^{k-1})$. For a ribbon tableau $T$ of
weight $\mu$, let $x^T=x_1^{\mu_1}x_2^{\mu_2}\cdots x_r^{\mu_r}$.
Then, the Stanton-White correspondence shows that the generating function
%
\begin{equation}
\G^{(k)}_\lambda
= \sum_{T\in \tab_k(\lambda,\,\cdot\,)}x^T
= \prod_{i=0}^{k-1}\ \sum_{\t_i\in\tab(\lambda^i,\,\cdot\,)}x^{\t_i}
=\prod_{i=0}^{k-1} s_{\lambda^i}
\end{equation}
%
is a product of Schur functions. Introducing in this equation an appropriate statistic 
on ribbon tableaux, one can therefore obtain $q$-analogues of products of Schur functions.
The statistic called {\it cospin},  described below, leads to $q$-analogues
with interesting properties.

Let $R$ be a $k$-ribbon, $h(R)$ its {\it heigth} and $w(R)$ its {\it width}.

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The {\it spin} of $R$, denoted by $s(R)$, is defined as
%
\begin{equation}
s(R) ={h(R)-1\over 2}
\end{equation}
%
and the spin of a ribbon tableau $T$ is by definition the sum of the spins
of its ribbons. For example, the ribbon tableau

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has a spin equal to $6$.

For a partition $\lambda$ without $k$-core, let
%
\begin{equation}
s^*_k(\lambda) = \max\{s(T)\ |\ T\in\tab_k(\lambda,\,\cdot\,)\} \ .
\end{equation}
%
The {\it cospin} $\cs(T)$ of a $k$-ribbon tableau $T$ of shape $\lambda$
is then
%
\begin{equation}
\cs(T)=s^*_k(\lambda)-s(T) \ .
\end{equation}
%
Although $s(T)$ can be a half-integer, it is easily seen that $\cs(T)$
is always an integer. Also, there is one important case where $s(T)$
is an integer. This is when the shape $\lambda$ of $T$ is of the
form $k\mu=(k\mu_1,k\mu_2,\ldots,k\mu_r)$. In this case, the partitions
constituting the $k$-quotient of $\lambda$ are formed by parts of $\mu$,
grouped according to the class modulo $k$ of their indices. More
precisely, $\lambda^i=\{\mu_r\ |\ r\equiv -i \mod k\}$

We can now define three families of polynomials

\begin{equation}
\G^{(k)}_\lambda(X;q)=
\sum_{T\in\tab_k(\lambda,\, \cdot\,)}q^{\cs(T)}x^T
\end{equation}
\begin{equation}
\tH^{(k)}_\mu(X;q)=\sum_{T\in\tab_k(k\mu,\, \cdot\,)}q^{\cs(T)}x^T
=\G^{(k)}_{k\mu}(X;q)
\end{equation}
\begin{equation}
H^{(k)}_\mu(X;q)=\sum_{T\in\tab_k(k\mu,\, \cdot\,)}q^{s(T)}x^T
=q^{s^*_k(k\mu)}\tH^{(k)}_\mu(X;1/q) \ .
\end{equation}

The parameter $k$ will be called the {\it level} of the corresponding
symmetric functions.
There is strong experimental evidence for the following conjectures.


\begin{conjecture}\label{Csym}{\rm (symmetry) }
The polynomials $\tG^{(k)}_\lambda$, $\tH^{(k)}_\mu$ and $H^{(k)}_\mu$ are
symmetric.
\end{conjecture}

\begin{conjecture}\label{Cpos}{\rm (positivity) }
Their coefficients on the basis of Schur functions are polynomials
with nonnegative integer coefficients.
\end{conjecture}

\begin{conjecture}\label{Cmono}{\rm (monotonicity) }
$H^{(k+1)}_\mu-H^{(k)}_\mu$ is positive on the
Schur basis.
\end{conjecture}

\begin{conjecture}\label{Cplet}{\rm (plethysm) }
When $\mu=\nu^k$, for $\zeta$ a primitive $k$-th root of unity,
$$ H^{(k)}_{\nu^k}(\zeta)=(-1)^{(k-1)|\nu|}\ p_k\circ s_\nu $$

Equivalently,
$$
H^{(k)}_{\nu^k}(q) \mod 1-q^k =\sum_{i=0}^{k-1}q^k \ell^{(i)}_k \circ s_\nu
$$
\end{conjecture}

The following statements will be proved in the forthcoming sections.

\begin{theorem}\label{IHL}
The difference $Q'_\mu - H^{(2)}_\mu$ is nonnegative on the Schur basis.
\end{theorem}

\begin{theorem}\label{THL}
For $k\ge \ell(\mu)$, $H^{(k)}_\mu$ is equal to the Hall-Littlewood function
$Q'_\mu$.
\end{theorem}

Taking into account the results of \cite{LLT1,LLT2} and \cite{CL}, this
is sufficient to establish all the conjectures for $k=2$ and $k\ge \ell(\mu)$. 






\begin{example}{\rm
{\bf (i) } The $3$-quotient of $\lambda=(3,3,3,2,1)$ is
$((1),(1,1),(1))$ and


\begin{eqnarray*}
\tG_{33321}(q) & = &
  m_{31} + (1+q) m_{22} + (2+2q+q^2) m_{211}\\
&&  + (3 + 5q + 3q^2 + q^3) m_{1111}\\
& = &  s_{31} + q s_{22} + (q+q^2) s_{211} + q^3 s_{1111}\\
\end{eqnarray*}




is a $q$-analogue of the product
$$
s_1 s_{11} s_1 = s_{31} + s_{22} + 2 s_{211} + s_{1111} \ .
$$


{\bf (ii) } The $H$-functions associated to the partition $\lambda=(3,2,1,1)$ are
\begin{eqnarray*}
H_{3211}^{(2)} & = & s_{3211} +q\, s_{322} +q\, s_{331} +q\, s_{4111} \\
	       &   & +(q+q^2)\, s_{421} +q^2\, s_{43} +q^2\, s_{511} + q^3\, s_{52}\\
H_{3211}^{(3)} & = & s_{3211} + q\, s_{322}+(q+q^2)\,s_{331}+q\, s_{4111}\\
               &   & +(q+2q^2)\,s_{421}+(q^2+q^3)\,s_{43} + (q^2+q^3)\,s_{511}\\
               &   & + 2q^3\, s_{52} + q^4 \, s_{61} \\
H_{3211}^{(4)} & = & s_{3211}+q\,s_{322}+(q+q^2)\,s_{331}+q\,s_{4111}\\
	       &   & +(q+2q^2+q^3)\,s_{421}+(q^2+q^3+q^4)\,s_{43}+(q^2+q^3+q^4)\,s_{511}\\
               &   & +(2q^3+q^4+q^5)\,s_{52} + (q^4+q^5+q^6)\,s_{61} + q^7\, s_7 \\
	       & = & Q'_{3211}\\
\end{eqnarray*}
and we see that
$s_{3211} < H_{3211}^{(2)} < H_{3211}^{(3)} < H_{3211}^{(4)} = Q'_{3211}$ .


{\bf (iii)} The plethysms of $s_{21}$ with the cyclic characters $\ell_3^{(i)}$
are given by the reduction modulo $1-q^3$ of
%
\begin{eqnarray*}
H_{222111}^{(3)}& = &
q^{9}s_{6 3}+  (q+1  )q^{7}s_{6 2 1}+q^{6}s_{6 1 1 1}+ 
( q+1  )q^{7}s_{5 4}+  (q^{3}+2 q^{2}+2 q+1  )q^{5}s_{5 3 1}\\
&&+  (q^{2}+2 q+1  )q^{5}s_{5 2 2}+  (q^{3}+2 q^{2}+
2 q+1  )q^{4}s_{5 2 1 1}+  (q+1  )q^{4}s_{5 1 1 1 1}\\
&&+ (q^{2}+2 q+1  )q^{5}s_{4 4 1}+  (q^{3}+2 q^{2}+3 q+2)q^{4}s_{4 3 2}
  +  (2 q^{3}+3 q^{2}+3 q+1  )q^{3}s_{ 4 3 1 1}\\
&&+  (q^{3}+3 q^{2}+3 q+2  )q^{3}s_{4 2 2 1}+  (q ^{3}+2 q^{2}+2 q+1  )q^{2}s_{4 2 1 1 1}
+q^{3}s_{4 1 1 1 1 1}+ (q^{3}+1  )q^{3}s_{3 3 3}\\
&&  +  (2 q^{3}+3 q^{2}+2 q+1)q^{2}s_{3 3 2 1}+  (q^{2}+2 q+1  )q^{2}s_{3 3 1 1 1}
+  (q^{2}+2 q+1  )q^{2}s_{3 2 2 2}\\
&&+  (q^{3}+2 q^{2}+2 q+1  )qs_{3 2 2 1 1}+  (q+1  )qs_{3 2 1 1 1 1}
+  (q+ 1  )qs_{2 2 2 2 1}+s_{2 2 2 1 1 1}\\
\end{eqnarray*}
%
Indeed,
%
\begin{eqnarray*}
H_{222111}^{(3)}   \mod 1-q^3 &=&  
(2  s_{5 2 1 1}+s_{2 2 2 2 1}+s_{3 2 1 1 1 1}+3  s_{4 3 1 1}\\
&& +2  s_{3 2 2 1 1}+s_{5 2 2}+3  s_{4 3 2}+3  s_{3 3 2 1}+s_{3 3 1 1 1}
 +s_{3 2 2 2}+s_{5 1 1 1 1}\\
&&+3  s_{4 2 2 1}+2  s_{5 3 1}+2  s_{4 2 1 1 1} +s_{ 5 4}+s_{6 2 1}+s_{4 4 1} )q^{2}\\
&& + (2  s_{5 2 1 1}+s_{2 2 2 2 1}+s_{3 2 1 1 1 1}+3  s_{4 3 1 1} 
 +2  s_{3 2 2 1 1}+s_{5 2 2}\\
&&+3  s_{4 3 2}+3  s_{3 3 2 1} + s_{3 3 1 1 1}+s_{3 2 2 2}+s_{5 1 1 1 1} \\
&&+3  s_{4 2 2 1}+2  s_{5 3 1} +2  s_{4 2 1 1 1}+s_{5 4}+s_{6 2 1}+s_{4 4 1} ) q \\
&& +2  s_{3 3 1 1 1}+s_{6 3}+s_{6 1 1 1}+2  s_{5 3 1}+2  s_{5 2 2} 
 +2  s_ {5 2 1 1}+2  s_{4 4 1}\\
&&+2  s_{4 3 2}+3  s_{4 3 1 1} 
 +3  s_{4 2 2 1}+2  s_{4 2 1 1 1}+s_{4 1 1 1 1 1}+2  s_{3 3 3}\\
&& +2  s_{3 3 2 1}+2  s_{3 2 2 2 }+s_{2 2 2 1 1 1}+2  s_{3 2 2 1 1}\\
& =&q^2 \ell_3^{(2)}\circ s_{21}+q\ell_3^{(1)}\circ s_{21}
+\ell_3^{(0)}\circ s_{21} \ .
\end{eqnarray*}

}
\end{example}







\section{The case of dominoes}

For $k=2$, the conjectures can be established by means of the combinatorial
constructions of \cite{CL} and \cite{KLLT}. In this case, conjectures
\ref{Csym}, \ref{Cpos} and \ref{Cplet} follow directly from the results
of \cite{CL}, and the only point remaining to be proved is Theorem \ref{IHL}.

The important special feature of domino tableaux is that there exits a 
natural notion of {\it Yamanouchi domino tableau}. These tableaux correspond
to highest weight vectors in tensor products of two irreducible $GL_n$-modules,
in the same way as ordinary Yamanouchi tableaux are the natural labels for
highest weight vectors of irreducible representations. 

The {\it column reading} of a domino tableau $T$ is the word obtained by reading the
successive columns of $T$ from top to bottom and left to right. Horizontal dominoes, which
belong to two succesive columns $i$ and $i+1$ are read only once, when reading column $i$.
For example, the column reading of the domino tableau

\begin{center}
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\put(5,85){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 4}}}}}
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\end{picture}
\end{center}
%
is $\col(T)=431212$.

A {\it Yamanouchi word} is a word $w=x_1x_2\cdots x_n$ such that each right factor
$v=x_i\cdots x_n$ of $w$ satisfies $|v|_j\ge |v|_{j+1}$ for each $j$, where $|v|_j$
denotes the number of occurences of the letter $j$ in $v$.

A {\it Yamanouchi domino tableau} is a domino tableau whose column reading is a
Yamanouchi word. We denote by $\yam_2(\lambda,\mu)$ the set of Yamanouchi domino
tableaux of shape $\lambda$ and weight $\mu$.

It follows from the results of \cite{CL}, Section 7, that the Schur expansions of the
$H$-functions of level $2$ are given by
%
\begin{equation}
H^{(2)}_\lambda =
\sum_\mu \sum_{T\in\yam_2(2\lambda,\mu)}q^{s(T)} s_\mu \ .
\end{equation}
%
On the other hand,
%
\begin{equation}
Q'_\lambda =
\sum_\mu \sum_{\t\in\tab(\mu,\lambda)} q^{c(\t)}s_\mu \ .
\end{equation}
%
To prove Theorem \ref{IHL}, it is thus sufficient to exhibit an injection
$$
\eta :\quad \yam_2(2\lambda,\mu) \longrightarrow  \tab(\mu,\lambda)
$$
satisfying 
$$
c(\eta(T)) = s(T) \ .
$$
%
To achieve this, we shall make use of a bijection described in \cite{BV}, and
extended in \cite{KLLT}, which sends a domino tableau $T\in\tab_2(\alpha,\mu)$
over the alphabet $X=\{1,\ldots,n\}$, to an ordinary tableau $\t=\phi(T)\in\tab(\alpha,\bar{\mu}\mu)$
over the alphabet $\bar{X}\cup X=\{\bar n<\ldots < \bar 1<1<\ldots <n\}$. The weight $\bar\mu\mu$
means that $\t$ contains  $\mu_i$ occurences of $i$ and of $\bar i$. The tableau $\phi(T)$
is invariant under Sch\"utzenberger's involution $\Omega$, and the spin of $T$ can be recovered
from $\t$ by the following procedure \cite{KLLT2}. 

Let $\alpha=2\lambda$, $\beta=\alpha'$, $\beta_{\rm odd}=(\beta_1,\beta_3,\ldots\,)$
and $\beta_{\rm even}=(\beta_2,\beta_4,\ldots\,)$. Then, there exists a
unique factorisation $\t=\tau_1\tau_2$ in the plactic monoid ${\rm Pl\,}(X\cup\bar X)$,
such that $\tau_1$ is a contretableau of shape $\alpha^1=(\beta_{\rm even})'$
and $\tau_2$ is a tableau of shape $\alpha^2=(\beta_{\rm odd})'$. The
spin of $T=\phi^{-1}(\t)$ is then equal to the number $|\tau_1|_+$ of positive letters
in $\tau_1$, which is also equal to the number $|\tau_2|_-$ of negative letters in
$\tau_2$. Moreover, $\tau_2=\Omega(\tau_1)$.

\begin{example}{\rm With the following  tableau $T$ of shape $(4,4,2,2)$, one finds

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%
By jeu-de-taquin, we find that in the plactic monoid
$$
\t \ \  = \  \
\young{\bar 1 & 1 \cr \bar 2 & \bar 1\cr \blk &\bar 2\cr \blk & \bar 3\cr}
\ 
\young{3\cr 2\cr 1 & 2 \cr \bar 1 & 1\cr}
\quad =  \ \tau_1\tau_2 \ .
$$
%
The number of positive letters of $\tau_1$ and the number of
negative letters of $\tau_2$ are both equal to $1$, which is
the spin of $T$.
}
\end{example}

This correspondence still works in the general case ($\alpha$ need not
be of the form $2\lambda$) and the invariant tableau associated to a
domino tableau $T$ admits a similar factorisation $\t=\tau_1\tau_2$,
but in general $\tau_2\not =\Omega(\tau_1)$ and the formula for the
spin is $s(T)={1\over 2}(|\tau_1|_+ +|\tau_2|_-)$.

The map $\eta :\ \yam_2(2\lambda,\mu)\longrightarrow \tab(\mu,\lambda)$
is given by the following algorithm: to compute
$\eta (T)$,
%
\begin{enumerate}

\item construct the invariant tableau $\t=\phi(T)$

\item apply the jeu-de-taquin algorithm to $\t$ to 
obtain the plactic factorization $\t=\tau_1\tau_2$,
and keep only $\tau_2$.

\item Apply the evacuation algorithm to the {\it negative} letters
of $\tau_2$, keeping track of the successive stages. 
After all the negative letters have been evacuated, one
is left with a Yamanouchi tableau $\tau$ in positive letters.

\item Complete the tableau $\tau$ to obtain the tableau $\t'=\eta(T)$
using the following rule: suppose that at some stage of the evacuation,
the box of $\tau_2$ which disappeared after the elimination of $\bar i$  was
in row $j$ of $\tau_2$. Then add a box numbered $j$ to row $i$ of $\tau$.
\end{enumerate}

\begin{theorem}\label{etainj}
The above algorithm defines an injection
$$\eta :\ \yam_2(2\lambda,\mu)\longrightarrow \tab(\mu,\lambda)
$$
satifying $c\circ\eta=s$.
\end{theorem}

\begin{corollary}
$H^{(2)}_\lambda \le Q'_\lambda$
\end{corollary}


\begin{example}{\rm Let $T$ be the following Yamanouchi domino tableau, which is
of shape $2\lambda=(6,4,4,2,2)$, of weight $\mu=(4,3,2)$ and has spin $s(T)=3$
%
\begin{center}
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%
Then,
$$
\phi(T)\ = \ 
\young{ 3 & 3\cr 1 & 2 \cr \bar 1 & \bar 1 & 2 & 2 \cr
          \bar 2 & \bar 2 & \bar 1 & 1 \cr
           \bar 3 & \bar 3 & \bar 2 & \bar 1 & 1 & 1\cr}
\qquad \equiv \qquad
\young{ \bar 1 & 1 & 3 \cr \blk & \bar 1 & 2 \cr
        \blk & \bar 2 & \bar 1\cr
        \blk & \blk & \bar 2\cr
        \blk & \blk & \bar 3\cr}
\ 
\young{ 3 \cr 2 \cr 1 & 2\cr \bar 2 & 1\cr \bar 3 & \bar 1 & 1\cr}
$$
and the succesive stages of the evacuation process are
$$
\matrix{
\young{3\cr 2\cr 1&2\cr \bar 2&1\cr \bar 3&\bar 1&1\cr} 
&
\longrightarrow
&
\young{\times\cr 3\cr 2&2\cr 1&1\cr \bar 2&\bar 1&1\cr}
&
\longrightarrow
&
\young{3\cr 2&\times\cr 1&2\cr \bar 1&1&1\cr}
&
\longrightarrow
&
\young{\times\cr 3\cr 2&2\cr 1&1&1\cr}
\cr
& \bar i = \bar 3 & & \bar i =\bar 2 & & \bar i=\bar 1 & \cr
&      j = 5      & &      j = 3     & &      j =    4 & \cr}
$$
so that we find
$$
\eta(T) =
\young{ 3&5\cr 2&2&3\cr 1&1&1&4\cr}
$$
a tableau of shape $\mu=(4,3,2)$, weight $\lambda=(3,2,2,1,1)$ and charge $c(\t')=3$.

}
\end{example}







\section{The stable case}

As the $Q'$-functions are known to verify all the conjectured properties 
of  $H$-functions,
the stable case of the conjectures will be a consequence of
theorem \ref{THL}. This result will be proved by means of 
Shimomura's cell decomposition of unipotent varieties.

A tabloid $\t$ of shape $\nu=(\nu_1,\ldots,\nu_k)$ can be identified
with a $k$-tuple $(w_1,\ldots,w_k)$ of words, $w_i$ being a row tableau of
lenght $\nu_i$. The Stanton-White correspondence $\psi$ assciates to such
a $k$-tuple of tableaux a $k$-ribbon tableau $T=\psi(\t)$. Thus, the cells
of a unipotent variety $\F^u_\mu$ (where $u$ is of type $\nu$)
are  labelled by $k$-ribbon tableaux of a special kind.  The following
theorem, which implies the stable case of the conjectures, shows that
this labelling is natural from a geometrical point of view.

\begin{theorem}\label{e2cs}
The Stanton-White correspondence $\psi$ sends a tabloid $\t\in L(\nu,\mu)$
onto a ribbon tableau $T=\psi(\t)$ whose cospin is equal to the dimension
of the cell $c_\t$ of $\F^u_\mu$ labelled by $\t$, when one uses the modified
indexation for which the dimension of $c_\t$ is $e(\t)$ (see Section \ref{HLUV}).
That is,
$$
\cs(\psi(\t))=e(\t) \ .
$$
\end{theorem}

At this point, it is useful to observe, following \cite{Te}, that
the $e$-statistic can be given a nonrecursive definition, as a kind
of inversion number. Let $\t=(w_1,\ldots,w_k)$ be a tabloid, identified
with a $k$-tuple of row tableaux.  Let $y$ be the $r$-th letter of $w_i$
and $x$ be the $r$-th letter of $w_j$, and suppose that $x<y$. Then, 
the pair $(y,x)$ is said to be an $e$-inversion if either 

\medskip
(a) $i<j$

\medskip\noindent or

\medskip
(b) $i>j$ and there s on the right of $x$ in $w_j$ a letter $u<y$

\medskip
Then $e(\t)$ is equal to the number of inversions $(y,x)$ in $\t$.

\begin{example}{\rm Let $\t\in L((2,3,2,1),(2,3,1,1,1))$ be the
following tabloid (the number under a letter $y$ is the number of
$e$-inversions of the form $(y,x)$):
$$
\t \quad = \quad
\left(
\matrix{
\young{2&3\cr} & , &\young{1&1&2\cr}&,&\young{4&5\cr}&,&\young{2\cr} \cr
\matrix{1&1}   &   &\matrix{0&0&0\cr} &&\matrix{3&1\cr}&&\matrix{1\cr} \cr}
\right)
$$
so that $e(\t)=7$. Its image under the SW-correspondence is the $4$-ribbon tableau

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%
whose cospin is equal to $7$.
}
\end{example}
















\footnotesize

\begin{thebibliography}{ABCD}

\bibitem[\bf BV]{BV} {\sc D. Barbasch and D. Vogan}, {\it Primitive ideals and orbital integrals
in complex classical groups}, Math. Ann. 259, (1982) 153-199

\bibitem[\bf CL]{CL} {\sc C. Carr\'e and B. Leclerc}, {\it Splitting the square of a
Schur function into its symmetric and antisymmetric parts}, 
Institut Gaspard Monge, preprint, 1993 ({\it to appear} in J. Alg. Comb.).

\bibitem[\bf Co]{Co} {\sc E. Cohen}, {\it A class of arithmetical functions}\rm,  Proc. Nat.
Acad. Sci. U.S.A. {\bf 41} (1955), 939-944.

\bibitem[\bf De]{De} {\sc J. D\'esarm\'enien}, {\it Etude modulo $n$ des statistiques
mahonniennes}\rm,
Actes du $22^{\rm e}$ s\'eminaire Lotharingien de Combinatoire, IRMA,
Strasbourg
(1990), 27-35.

\bibitem[\bf FS]{FS} {\sc S. Fomin and D. Stanton}, {\em Rim hook lattices}, Mittag-Leffler institute,
preprint No 23, 1991/92.

\bibitem[\bf Fo]{Fo} {\sc H.O. Foulkes}, \it Characters of symmetric groups induced by
characters of cyclic subgroups\rm, in \it Combinatorics \rm (Proc.
Conf. Comb. Math. Inst. Oxford 1972), Inst. Math. Appl.,
Southend-on-Sea, 1972, 141-154.

\bibitem[\bf Ga]{Ga} {\sc  D. Garfinkle}, {\it On the classification of primitive ideals for
complex classical Lie algebras, I}, Compositio Mathematica, 75 (1990)
2, 135-169

\bibitem[\bf Gr]{Gr} {\sc J.A. Green}, {\it The characters of the finite general linear groups},
Trans. Amer. Math. Soc. {\bf 80} (1955), 402-447.

\bibitem[\bf HSh]{HSh}{\sc R. Hotta} and {\sc N. Shimomura}, {\it The fixed point
subvarieties of unipotent transformations on generalized flag varieties
and the Green functions}, Math. Ann. {\bf 241} (1979), 193-208.

\bibitem[\bf HS]{HS} {\sc R. Hotta and T.A. Springer}, \it A specialization theorem for
certain Weyl groups\rm, Invent. Math {\bf 41} (1977), 113-127.

\bibitem[\bf JK]{JK} {\sc G. D. James and A. Kerber}, {\it The representation theory of the symmetric group},
Addison-Wesley, 1981.

\bibitem[\bf KSW]{KSW} {\sc A. Kerber, F. S\"anger and B. Wagner}, {\it Quotienten und 
Kerne von Young-Diagrammen, Brettspiele und Plethysmen gew\"ohnlicher irreduzibler 
Darstellungen symmetrischer Gruppen}, Mitt. Math. Sem. Giessen, {\bf 149} (1981), 131-175.

\bibitem[\bf KKL]{KKL} {\sc A. Kerber, A. Kohnert and A. Lascoux}, {\it SYMMETRICA, an object oriented
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