\documentstyle[12pt,epic,eepic]{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Marne \textheight 230mm \textwidth 150mm \hoffset -16mm \voffset -20mm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Jussieu %\textheight 235mm %\textwidth 150mm %\hoffset -8mm %\voffset -35mm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \font\goth=eufb10 scaled 1200 \font\sevenbf=cmbx7 \newtheorem{example}{Example}[section] \newtheorem{note}[example]{Note} \newtheorem{theorem}[example]{Theorem} \newtheorem{corollary}[example]{Corollary} \newtheorem{definition}[example]{Definition} \newtheorem{proposition}[example]{Proposition} \newtheorem{algorithm}[example]{Algorithm} \newtheorem{lemma}[example]{Lemma} \newtheorem{conjecture}[example]{Conjecture} \def\Proof{\medskip\noindent {\it Proof --- \ }} \def\cqfd{\hfill $\Box$ \bigskip} \def\adots{\mathinner{\mkern2mu\raise1pt\hbox{.} \mkern3mu\raise4pt\hbox{.}\mkern1mu\raise7pt\hbox{.}}} \def\<{\langle} \def\>{\rangle} \def\fsf{k\!\not< \! \bb{A}\!\not>} \def\cf{{\it cf. }} \def\ie{{\it i.e. }} \def\mat{{\rm Mat}} \def\limproj{\mathop{\oalign{lim\cr \hidewidth$\longleftarrow$\hidewidth\cr}}} \def\col{{\rm col\,}} \def\yam{{\rm Yam}} \def\S{{\bf S}} \def\sym{{\sl Sym}} %\newfont{\bb}{msbm10} \newfont{\bb}{cmbx10} \def\N{{\bf N}} \def\Z{{\bf Z}} \def\E{{\bf E}} \def\Q{{\bf Q}} \def\sgn{{\rm sgn\, }} \def\D{{\rm Des \, }} \def\Q{{\bf Q}} \def\maj{{\rm maj\, }} \def\mod{{\rm \ mod\ }} \def\C{{\bf C}} \def\K{{\bf K}} \def\O{{\cal O}} \def\P{{\cal P}} \def\Sl{\hbox{\goth sl}\hskip 2.5pt} \def\g{\hbox{\goth g}} \def\gl{\hbox{\goth gl}\hskip 2.5pt} \def\tab{{\rm Tab\, }} \def\ch{{\rm ch\,}} \def\I{{\cal I}} \def\h{\hbox{\goth h}} \def\m{\hbox{\goth m}} \def\F{{\cal F}} \def\k{{\bf F}_q} \def\H{{\cal H}} \def\dim{{\rm dim\,}} \def\Hom{{\rm Hom\,}} \def\Ad{{\rm Ad\,}} \def\diag{{\rm diag\,}} \def\Chi{X} \def\F{{\cal F}} \def\tr{{\rm tr\,}} \def\eg{{\it e.g. }} \def\tab{{\rm Tab\,}} \def\t{{\bf t}} \def\tQ{{\tilde Q'}} \def\tK{{\tilde K}} \def\tG{{\tilde G}} \def\tH{{\tilde H}} \def\GF{{\bf F}} \def\cc{{\tilde c}} \def\cs{{\tilde s}} \def\G{{\cal G}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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{} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \newdimen\Squaresize \Squaresize=14pt \newdimen\Thickness \Thickness=0.5pt \def\Square#1{\hbox{\vrule width \Thickness \vbox to \Squaresize{\hrule height \Thickness\vss \hbox to \Squaresize{\hss#1\hss} \vss\hrule height\Thickness} \unskip\vrule width \Thickness} \kern-\Thickness} \def\Vsquare#1{\vbox{\Square{$#1$}}\kern-\Thickness} \def\blk{\omit\hskip\Squaresize} \def\young#1{ \vbox{\smallskip\offinterlineskip \halign{&\Vsquare{##}\cr #1}}} \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) = \ \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. 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\put(365,25){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 2}}}}} \put(305,5){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 1}}}}} \put(325,5){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 1}}}}} \put(345,5){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 1}}}}} \put(365,5){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 1}}}}} \put(565,25){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 1}}}}} \put(585,25){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 1}}}}} \put(605,25){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 2}}}}} \put(625,25){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 3}}}}} \put(645,25){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 6}}}}} \put(665,25){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 6}}}}} \put(685,25){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 6}}}}} \put(705,25){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 8}}}}} \put(725,25){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 8}}}}} \put(745,25){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 9}}}}} \put(565,5){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 1}}}}} \put(585,5){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 1}}}}} \put(605,5){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 1}}}}} \put(625,5){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 2}}}}} \put(645,5){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 7}}}}} \put(665,5){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{\shortstack[l]{{\twlrm 7}}}}} \end{picture} \end{center} \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 $0\le r_i$ 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 $\$ 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 $$ \ = \left\{ \matrix{ \ & \mbox{if $|\lambda|$ is even}\cr \ & \mbox{if $|\lambda|$ is odd}\cr }\right. $$ 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 $\=2$, 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 $\=5$ 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 $\mu = (1^9)$. \begin{figure}[t] \setlength{\unitlength}{0.25pt} \centerline{ \begin{picture}(1400,500)(0,0) % \put(0,10){$T_1 =$} \put(120,0){\begin{picture}(400,450) \put(0,0){\framebox(50,50){}} \put(50,0){\framebox(50,50){}} \put(0,50){\framebox(50,50){}} \put(0,100){\framebox(50,50){}} \put(100,0){\framebox(150,50){}} \put(0,150){\framebox(50,150){}} \put(0,300){\framebox(50,150){}} \put(50,150){\framebox(150,50){}} \put(200,100){\framebox(150,50){}} \put(150,50){\line(0,1){50}} \put(150,100){\line(-1,0){50}} \put(100,100){\line(0,1){50}} \put(200,50){\line(0,1){50}} \put(300,0){\line(0,1){100}} \put(350,50){\line(0,1){50}} \put(350,50){\line(1,0){50}} \put(400,0){\line(0,1){50}} \put(250,0){\line(1,0){150}} \put(165,10){1} \put(65,60){2} \put(115,110){3} \put(15,160){4} \put(215,60){5} \put(165,160){6} \put(315,10){7} \put(15,310){8} \put(265,110){9} \end{picture}} % \put(880,10){$T_2 = $} \put(1000,0){\begin{picture}(400,450) \put(0,0){\framebox(50,50){}} \put(50,0){\framebox(50,50){}} \put(0,50){\framebox(50,50){}} \put(0,100){\framebox(50,50){}} \put(250,0){\framebox(150,50){}} \put(0,150){\framebox(50,150){}} \put(0,300){\framebox(50,150){}} \put(50,150){\framebox(150,50){}} \put(200,100){\framebox(150,50){}} \put(50,100){\framebox(150,50){}} \put(200,50){\framebox(150,50){}} \put(150,0){\line(0,1){100}} \put(250,0){\line(0,1){50}} \put(100,0){\line(1,0){150}} \put(165,10){4} \put(65,60){2} \put(115,110){6} \put(15,160){1} \put(215,60){7} \put(165,160){8} \put(315,10){5} \put(15,310){3} \put(265,110){9} \end{picture}} % \end{picture}} \caption{\label{TRIBB}} \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 \setlength{\unitlength}{0.25pt} \centerline{ \begin{picture}(300,150) \put(100,0){\begin{picture}(100,100) \put(0,0){\framebox(50,50){1}} \put(50,0){\framebox(50,50){7}} \put(0,50){\framebox(50,50){2}} \put(50,50){\framebox(50,50){9}} \end{picture}} \put(0,20){$t_1 =$} \end{picture} } \bigskip \noindent The complete triple $(t_0,\,t_1,\,t_2)$ of Young tableaux associated to $T_1$ is \setlength{\unitlength}{0.25pt} \centerline{ \begin{picture}(700,150) \put(150,0){\begin{picture}(100,100) \put(0,0){\framebox(50,50){4}} \put(50,0){\framebox(50,50){6}} \put(0,50){\framebox(50,50){8}} \end{picture}} % \put(320,0){\begin{picture}(100,100) \put(0,0){\framebox(50,50){1}} \put(50,0){\framebox(50,50){7}} \put(0,50){\framebox(50,50){2}} \put(50,50){\framebox(50,50){9}} \end{picture}} % \put(490,0){\begin{picture}(100,100) \put(0,0){\framebox(50,50){3}} \put(50,0){\framebox(50,50){5}} \end{picture}} \put(0,40){$\tau^1 =$} \put(110,40){$\Big( $} \put(270,40){,} \put(440,40){,} \put(610,40){$\Big) $} \end{picture}} \bigskip \noindent whereas that corresponding to $T_2$ is \setlength{\unitlength}{0.25pt} \centerline{ \begin{picture}(700,150) \put(150,0){\begin{picture}(100,100) \put(0,0){\framebox(50,50){1}} \put(50,0){\framebox(50,50){8}} \put(0,50){\framebox(50,50){3}} \end{picture}} % \put(320,0){\begin{picture}(100,100) \put(0,0){\framebox(50,50){4}} \put(50,0){\framebox(50,50){5}} \put(0,50){\framebox(50,50){6}} \put(50,50){\framebox(50,50){9}} \end{picture}} % \put(490,0){\begin{picture}(100,100) \put(0,0){\framebox(50,50){2}} \put(50,0){\framebox(50,50){7}} \end{picture}} \put(0,40){$\tau^2 =$} \put(110,40){$\Big( $} \put(270,40){,} \put(440,40){,} \put(610,40){$\Big) $} \end{picture}} \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 \centerline{ \begin{picture}(700,150) \setlength{\unitlength}{0.25pt} \put(150,0){\begin{picture}(100,100) \put(0,0){\framebox(50,50){3}} \put(50,0){\framebox(50,50){3}} \put(0,50){\framebox(50,50){4}} \end{picture}} % \put(320,0){\begin{picture}(100,100) \put(0,0){\framebox(50,50){1}} \put(50,0){\framebox(50,50){3}} \put(0,50){\framebox(50,50){2}} \put(50,50){\framebox(50,50){4}} \end{picture}} % \put(490,0){\begin{picture}(100,100) \put(0,0){\framebox(50,50){2}} \put(50,0){\framebox(50,50){3}} \end{picture}} \put(0,40){$\tau =$} \put(110,40){$\Big( $} \put(270,40){,} \put(440,40){,} \put(610,40){$\Big) $} \end{picture}} \bigskip \noindent with weights $\left( (0,\,0,\,2,\,1),\,(1,\,1,\,1,\,1),\,(0,\,1,\,1,\,0)\right)$ corresponds to the 3-ribbon tableau \centerline{ \begin{picture}(550,500)(0,0) \setlength{\unitlength}{0.25pt} % \put(0,10){$T =$} \put(120,0){\begin{picture}(400,450) \put(0,0){\framebox(50,50){}} \put(50,0){\framebox(50,50){}} \put(0,50){\framebox(50,50){}} \put(0,100){\framebox(50,50){}} \put(100,0){\framebox(150,50){}} \put(0,150){\framebox(50,150){}} \put(0,300){\framebox(50,150){}} \put(50,150){\framebox(150,50){}} \put(200,100){\framebox(150,50){}} \put(150,50){\line(0,1){50}} \put(150,100){\line(-1,0){50}} \put(100,100){\line(0,1){50}} \put(200,50){\line(0,1){50}} \put(300,0){\line(0,1){100}} \put(350,50){\line(0,1){50}} \put(350,50){\line(1,0){50}} \put(400,0){\line(0,1){50}} \put(250,0){\line(1,0){150}} \put(165,10){1} \put(65,60){2} \put(115,110){2} \put(15,160){3} \put(215,60){3} \put(165,160){3} \put(315,10){3} \put(15,310){4} \put(265,110){4} \end{picture}} \end{picture}} \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: \centerline{ \begin{picture}(800,250) \setlength{\unitlength}{0.25pt} \put(0,0){\begin{picture}(200,150) \put(0,0){\framebox(50,50){}} \put(50,0){\framebox(50,50){}} \put(100,0){\framebox(50,50){}} \put(150,0){\framebox(50,50){}} \put(0,50){\framebox(50,50){}} \put(0,0){\line(1,0){200}} \put(0,100){\line(1,0){200}} \put(0,0){\line(0,1){100}} \put(200,0){\line(0,1){100}} \put(50,50){\framebox(150,50){}} \put(0,100){\framebox(150,50){}} \put(65,60){1} \put(115,110){2} \end{picture}} % \put(300,0){\begin{picture}(200,150) \put(0,0){\framebox(50,50){}} \put(50,0){\framebox(50,50){}} \put(100,0){\framebox(50,50){}} \put(150,0){\framebox(50,50){}} \put(0,50){\framebox(50,50){}} \put(0,0){\line(1,0){200}} \put(0,150){\line(1,0){150}} \put(0,100){\line(1,0){50}} \put(50,50){\line(0,1){50}} \put(50,50){\line(1,0){150}} \put(100,50){\line(0,1){100}} \put(0,0){\line(0,1){150}} \put(200,0){\line(0,1){100}} \put(150,150){\line(0,-1){50}} \put(150,100){\line(1,0){50}} \put(65,60){1} \put(115,110){2} \end{picture}} % \put(600,0){\begin{picture}(200,150) \put(0,0){\framebox(50,50){}} \put(50,0){\framebox(50,50){}} \put(100,0){\framebox(50,50){}} \put(50,50){\framebox(50,50){}} \put(0,50){\framebox(50,50){}} \put(0,0){\line(1,0){200}} \put(0,100){\line(1,0){200}} \put(0,0){\line(0,1){100}} \put(200,0){\line(0,1){100}} \put(150,0){\line(0,1){50}} \put(150,50){\line(-1,0){50}} \put(100,50){\line(0,1){50}} \put(0,100){\framebox(150,50){}} \put(165,10){1} \put(115,110){2} \end{picture}} % \end{picture}} \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 $n=kq+r$, $0\le r j$ and there s on the right of $x$ in $w_j$ a letter $u