%%%%%%%%%   GREENtout.tex   %%%%%%%
% version ancienne sauvee des eaux 
%  inlclut a4new  et les macros  macr2.tex

\magnification=1200
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% Le deuxieme concerne  la \magnification=1200
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%QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ
% enonces de theoremes avec numerotation apres 
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%QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ
% enonces de theoremes avec numerotation d'abord
% #1 = numero (par exemple 3, 3.1, etc.)
% #2 = THEOREME, COROLLAIRE, etc.
% #3 = l'enonce du th proprememnt dit.

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%QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ

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% QQQQQQQQQQQQQQ ce que Knuth n'a pas fait QQQQQQQQQQQQQQQQQQQQQQQQ
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\def\litp{
\vskip 8mm
\setbox3=\hbox{\qquad \qquad \it Alain Lascoux}
\setbox4=\hbox{\qquad L.I.T.P., \it Universit\'e Paris 7}
\setbox5=\hbox{\it 2 Place Jussieu, 75251 Paris Cedex 05}
\hskip 4cm \vbox{\box 3 \box4 \box5} }

\def\tab#1|#2|{ {}_{#2}^{#1} }   % petits tableaux Young 2 lignes

\def\ttab#1|#2|#3|#4|{%                        % petits tableaux Young 4 lignes
\vbox{\hbox{$\tab#1|#2|$} \hbox{$\tab#3|#4|$} } }

\def\preuve{\noindent {\it Preuve} \pointir }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
fichier macr2.tex


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\kern-\thickness}                                                            
                               
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\def\tab#1|#2|{ {}_{#2}^{#1} }   % petits tableaux Young 2 lignes

\def\ttab#1|#2|#3|#4|{%                     % petits tableaux Young 4 lignes
\vbox{\hbox{$\tab#1|#2|$} \hbox{$\tab#3|#4|$} } }




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\font\bigbf=cmbx12
\font\sc=cmcsc10
\def\hl#1#2{\langle #1 ,\ #2 \rangle_q}
\def\mod{{\rm mod\ }}
\def\conj#1{#1\ \tilde{}}
\bigbf
\centerline{GREEN POLYNOMIALS AND HALL-LITTLEWOOD }\bigskip
\centerline{FUNCTIONS AT ROOTS OF UNITY}\vskip 8mm






\centerline{\rm Alain \sc Lascoux\footnote{\dag}{\sevenrm L.I.T.P., Institut
Blaise Pascal,
Universit\'e Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France.},
\rm Bernard \sc Leclerc\ddag \rm\   and Jean-Yves \sc
Thibon\footnote{\ddag}{\sevenrm Institut Gaspard Monge, Universit\'e de
Marne-la-Vall\'ee,  2 All\'ee Jean Renoir,  93160 Noisy-Le-Grand,
France.}}
\vskip 8mm
\sevenbf Abstract. --- \sevenrm We give some applications of our recent
work [LLT] about Hall-Littlewood functions at roots of unity. In particular,
we prove the two conjectures of N. Sultana [Su] on  specializations of
Green polynomials, and we generalize results of Kraskiewicz-Weyman
and Molev-Tsalenko concerning characters of the symmetric group induced
by maximal cyclic subgroups. We also give some new
congruences for Kostka-Foulkes polynomials modulo
cyclotomic polynomials.\rm\vskip 14mm

\noindent\bigbf 1. Introduction\bigskip\rm
Hall-Littlewood functions are symmetric functions of a set of variables $A$
depending on a parameter $q$ (see [Mcd]). When $q=-1$, these functions
coincide with those introduced by Schur in his theory of projective
representations
of the symmetric group.
The investigation of Hall-Littlewood functions at  other roots of unity has been
initiated by A.O. Morris [Mo], in connection with problems in the modular
representation theory of the symmetric group. Recently, new results on this
question have been obtained [Su], [MS], [LLT]. In [Su] and [MS], one finds
some conjectures about  Green polynomials (which are $q$-analogs of the
characters of symmetric groups). These conjectures are proved in section 3. 


Section 4 is concerned with some combinatorial properties of certain
symmetric
functions discovered by Foulkes [Fo]. These functions correspond to
characters of the symmetric group induced by irreducible representations
of a maximal cylic subgroup. A combinatorial interpretation of their
coefficients in the basis of Schur functions has been given by Kraskiewicz
and Weyman [KW] in terms of congruence
properties of the major index of permutations, or, which amounts
to the same, charge of standard tableaux. We give here a generalization
of this result, 
in which the coefficients of certain plethysms with the preceding
functions are interpreted in terms of the charge of more general
tableaux. This result can also be viewed as an isomorphism theorem
for certain submodules of the cohomology of a generalized flag manifold,
or also in terms of harmonic polynomials relative to the symmetric group.

In section 5, we derive some congruences for  Kostka-Foulkes polynomials
modulo cyclotomic polynomials. Such congruences were derived in [LLT] for
certain columns of the $q$-Kostka matrix. Those presented here involve 
 rows of the same matrix.

All the results of this paper are consequences of two theorems 
from [LLT], which
we recall in section 2.
\bigskip\goodbreak


\noindent\bigbf 2. Notations and background\bigskip\rm


Our notations for symmetric functions are as in [LLT].
Partitions are for us weakly increasing vectors of nonnegative integers
$I=(i_1\le i_2\le\ldots\le i_m)$,  Schur functions of an alphabet $A$
are denoted by $S_I(A)$, products of power-sums
 $\psi_{i_1}\psi_{i_2}\cdots\psi_{i_m}$  by $\psi^I(A)$, and
 Hall-Littlewood functions by $P_I(A;q),\ Q_I(A;q)$, as in [Mcd].
The \it modified Hall-Littlewood functions \rm are the
$Q_I'(A;q)=Q_I(A/(1-q);q)$ (in the sense of $\lambda$-rings), that is, the
$Q_I'$ form the adjoint basis of the $P_I$ for the standard scalar product
$(\ ,\ )$
(for which the Schur functions form an orthonormal basis). The
Hall-Littlewood
scalar product is denoted by $\hl{.}{.}$.
\medskip
The \it Green polynomials \rm (see [Mcd]) $X^I_J(q)$ are defined by
$X^I_J(q)=\hl{Q_I}{\psi^J} = (Q_I',\ \psi^J)$.
Given two partitions $I$ and $J$, we denote by $I\vee J$ the partition
obtained by reordering the concatenation of $I$ and $J$, {\it e.g.}
$(122)\vee (1125)=(1112225)$. We write $I^k$ for $I\vee I\vee\cdots\vee I$
($k$ factors). If $I=(i_1,\ldots,i_r)$ we set $kI=(ki_1,\ldots,ki_r)$.
Given a symmetric function $F$, we denote by $D_F$ the adjoint of the
linear operator $G\mapsto FG$ (multiplication by $F$). We shall make
use of the following two results, which were established in [LLT]:\bigskip

\bf Theorem 2.1. --- \it Let $I=(1^{m_1}2^{m_2}\ldots n^{m_n})$ be a
partition, write $m_i=kp_i+r_i$ with $0\le r_i<k$, and
$H=(1^{r_1}2^{r_2}\ldots n^{r_n})$. Then, $\zeta$ being a primitive
$k$-th root of unity,
$$Q'_I(A;\zeta)=Q'_H(A;\zeta)
     \prod_{i\ge 1}\bigl[ Q'_{(i^k)}(A;\zeta)\bigr]^{p_i} \ . $$
\bigskip
\bf Theorem 2.2. --- \it Let $\psi_k(S_n)$ denote the plethysm of the
complete function $S_n$ by the power-sum $\psi_k$, which is defined
by the generating series
$\displaystyle \sum_n z^n\psi_k(S_n(A))=\prod_{a\in A}(1-za^k)^{-1}$.
Then, if $\zeta$ is as above a primitive $k$-th root of unity, one has
$$Q'_{(n^k)}(A;\zeta)=(-1)^{(k-1)n}\psi_k(S_n(A)). $$

\bigskip


\noindent\bigbf 3. Green polynomials at roots of unity\bigskip\rm

Let $\zeta\in {\bb C}$ be a primitive $k$-th root of unity. The
following two results have been conjectured in [Su] (see also [MS]):
\bigskip
\bf Theorem 3.1. --- \it Let $I=(1^{m_1}2^{m_2}\ldots n^{m_n})$ be
a partition, with at least one part $i$ with multiplicity
$m_i\ge k$, and let $J=(j_1,\ldots,j_r)$ be another partition,

with at least one part $j_h\equiv 0\  (\mod k)$. Then,
$$X^I_J(\zeta) \equiv 0\  (\mod k) \ .$$\medskip

\bf Theorem 3.2. --- \it Let $I=(1^{m_1}\ldots i^{m_i}\ldots n^{m_n})$, where
for some $i$,
$m_i\ge 1$, $|I|=n$, $I'=(1^{m_1}\ldots i^{m_i-1}\ldots n^{m_n})$ and suppose
$n=kr$. Then,
$$X^{(r^k)}_I(\zeta) = (-1)^{(k-1)j}k\ X^{(r-j)^k}_{I'}(\zeta)   $$
if $i=jk$, and $X^{(r^k)}_I(\zeta)=0$ otherwise.
\bigskip\goodbreak
Proof of Theorem 3.1: \rm Put $I=(j^k)\vee I'$ and $J=(k\cdot H)\vee J'$,
where no part of $J'$ is multiple of $k$. Then, all HL-functions being
understood
as specialized at $q=\zeta$,
\centerline{
$X^I_J(q)=(\psi^J,\ Q_I')=(\psi^{kH}\psi^{J'},\ Q_{(j^k)}'Q_{I'}')$\qquad
 (by Theorem 2.1)}
\bigskip
\centerline{$=\pm (\psi^{kH}\psi^{J'},\ \psi_k(S_j)Q_{I'}') $ \qquad
 (by Theorem 2.2 )}\bigskip
$$ = \pm (D_{\psi_k(S_j)}(\psi^{kH}\psi^{J'})\ , \ Q_{I'}') $$
so that we just have to check that the coefficients of the expansion
of $D_{\psi_k(S_j)}(\psi^{kH}\psi^{J'})$ in the basis $(\psi^K)$ are
in $k{\bb Z}$. But these coefficients can be explicitely computed: one
has
$$D_{\psi_k(S_j)}(\psi^{kH}\psi^{J'}) = \psi^{J'} D_{\psi_k(S_j)}\psi^{kH}$$
and
$$D_{\psi_k(S_j)}\psi^{kH}
=\sum_{|M|=j}
  {{1}\over{(\psi^M,\ \psi^M)}}
  D_{\psi^{kM}}\psi^{kH} $$
with $(D_{\psi^{kM}}\psi^{kH}, \ \psi^{kN})
=(\psi^{kH}, \ \psi^{k\cdot(M\vee N)} ) \not = 0 $
only when $H=M\vee N$.

Now, let $H=(1^{a_1}2^{a_2}\ldots)$, $M=(1^{b_1}2^{b_2}\ldots)$ and
$N=(1^{c_1}2^{c_2}\ldots)$. We have therefore
$$D_{\psi^{kM}}\psi^{kH} =
(\psi^{kH},\ \psi^{kH})
{{\psi^{kN}}\over{(\psi^{kN},\ \psi^{kN})}} $$
$$=\prod_i
 {{(ki)^{a_i}a_i!}\over {(ki)^{c_i}c_i!}}\psi^{kN}  $$
whence
$${{1}\over{(\psi^M, \ \psi^M)}}D_{\psi^{kM}}\psi^{kH}
=k^{\Sigma a_i - \Sigma c_i}\prod_i{{a_i!}\over{b_i!c_i!}}\psi^{kN} $$
$$= k^{\ell(M)}\prod_i {{a_i}\choose{b_i}} \psi^{kN} $$
and this number is congruent to $0$ since $\ell(M)>0$.\qquad$\qed  $\bigskip

\bf Example. --- \rm With $k=4$ $(\zeta=i=\sqrt{-1})$, we have
$$X^{11114}_{224}(i)=(Q_{1111}'Q_4',\ \psi^4\psi^{22})
=(-1)^3(\psi_4Q_4',\ \psi^4\psi^{22}) $$
$$(Q'_4,\ D_{\psi_4}
(\psi^4\psi^{22}))=-(Q_4',\ 4\psi^{22}) = -4(S_4, \ \psi^{22})=-4 \ . $$\medskip

To prove Theorem 3.2, we need the following lemma, which is easily
established by induction [CT]:
\bigskip
\bf Lemma 3.3. --- \it Let $F\circ G$ denote (as in {\rm [Mcd]}) the
plethysm of the two symmetric functions $F$ and $G$. Then,

$$D_{\psi_m}(F\circ G)=
\sum_{d\mid m}
\Bigl[
 \bigl(D_{\psi_d}F\bigr)\circ G \Bigr]\cdot
 \Bigl[\psi_d\circ\bigl(D_{\psi_{m/d}}G\bigr)\Bigr] \ . \qquad\qed $$

\it Proof of Theorem 3.2: \rm The HL-functions being as above specialized
at $q=\zeta$, we have
$$X^{(r^k)}_I(\zeta)=(\psi^I,\ Q_{(r^k)}')=
(-1)^{(k-1)r}(\psi^I,\ \psi_k(S_r)) $$
 (by Theorem 2.2 )
$$=(-1)^{(k-1)r}(\psi^{I'}\psi_i,\ \psi_k(S_r)) =
(-1)^{(k-1)r}(\psi^{I'},\ D_{\psi_i}[\psi_k\circ S_r]) $$
where by Lemma 3.3,
$$D_{\psi_i}[\psi_k\circ S_r] =
\sum_{d\mid i}\Bigl[\bigl(D_{\psi_d}\psi_k\bigr)\circ S_r \Bigr]\cdot
\psi_d\circ (D_{\psi_{i/d}}S_r) $$
$$= k\cdot \psi_k(D_{\psi_j}S_r) = k\psi_k(S_{r-j}) =
(-1)^{(k-1)(r-j)} kQ_{(r-j)^k}'$$
if $i=kj$ and $=0$ otherwise
(by Theorem 2.2 ),        
whence $X^{(r^k)}_I(\zeta)=(-1)^{(k-1)j}(\psi^I,\ kQ_{(r-j)^k}')=
(-1)^{(k-1)j}kX^{(r-j)^k}_{I'}(\zeta)$, as required.\qquad $\qed$\bigskip

 More generally, when $I=H^k=H\vee H\vee \cdots\vee H$
 ($k$ factors), $X^I_J(\zeta)$ can be expressed by means of the value
 of a permutation character of ${\goth{S}}_n$:
 $$X^{H^k}_J(\zeta) = (-1)^{(k-1)|H|}(\psi^J,\ \psi_k(S^H))
 =(-1)^{(k-1)|H|}(\phi_k(\psi^J),\ S^H) \ , $$
 (where $\phi_k$ is the adjoint of the linear operator
 $F\mapsto \psi_k\circ F$, and $S^H=S_{h_1}S_{h_2}\cdots S_{h_r}$)
so that:
\bigskip
\bf Theorem 3.4. --- \it $X^{H^k}_J(\zeta)=0$ when $J$ is not of the form
$kM$ for
some partition $M$, and
$$X^{H^k}_{kM}(\zeta)=(-1)^{(k-1)|H|}k^{\ell(M)}(\psi^M,\ S^H)\ .$$
In particular, $X^{(r^k)}_{kM}(\zeta)=(-1)^{(k-1)r}k^{\ell(M)}$.
\qquad $\qed$\bigskip

\bf Examples. --- {\it (i)} \rm With $k=4$ ($\zeta=i$), $X^{2222}_{224}(i)
=(\psi_4(S_2),\ \psi^{224})=0$ and $X^{24}_{44}(i)$ $=$
$(S_2,\ \phi_4(\psi^{44}))$ $=$ $(S_2,\ 16\psi^{11}) = 16$.\medskip
{\it (ii)} With $k=3$ ($\zeta=e^{2\pi i/3}$),
$X^{222333}_{339}(\zeta)= +3^3(\psi^{113},\ S^{23})$ $=$
$3^3(\psi^{11}, \ S_2D_{\psi_3}S_3)$ $= 27$.\bigskip\bigskip
\goodbreak
\noindent\bigbf 4. Charge congruences, harmonic polynomials and cohomology
of flag manifolds\bigskip\rm


To state our next result, we recall some definitions from number theory.
For $k,n\in {\bb N}$, 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.} [HW] or [NV]).

Let $P(q)=\displaystyle \sum_{k=0}^{n-1}a_kq^k \ \in {\bb Z}[q]$ 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 result, due to
E. Cohen ([Co], see also [De] for a simpler proof), provides a
generalization of the Moebius inversion formula:\bigskip
\bf Lemma 4.1. --- \it 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
\medskip
\hskip 8mm (i) $\displaystyle a_k={{1}\over{n}}\sum_{d\mid n}
                  c(k,d)r_d$\medskip
\hskip 8mm (ii) $\displaystyle
              r_d=\sum_{t\mid n}c(n/d,t)a_{n/t}\ .\qquad\qed $\bigskip\rm

With the aid of Ramanujan sums, we define the symmetric functions
$$\ell^{(k)}_n={{1}\over{n}}\sum_{d\mid n}c(k,d)\psi_d^{n/d} \ . $$
These functions were first obtained by Foulkes as Frobenius characteristics
of the representations of the symmetric group induced by irreducible
representations of a maximal cyclic subgroup generated by an $n$-cycle [Fo].
A combinatorial interpretation of the multiplicity $(S_I,\ \ell^{(k)}_n)$
has been discovered by Kraskiewicz and Weyman [KW]. A proof of this result
by means of Cohen's theorem (Lemma 4.1) was subsequently
found by  D\'esarm\'enien
[De]. Adapting the argument of [De] to our case, we arrive at
the following generalization of the Kraskiewicz-Weyman theorem:\bigskip

\bf Theorem 4.2. --- \it Let $\Lambda_i$ be the $\ i$-th elementary symmetric
function, and for $H=(h_1,\ldots,h_m)$, $\Lambda^H=\Lambda_{h_1}\cdots
\Lambda_{h_m}$. Then, the multiplicity $(S_I,\ \ell^{(r)}_k\circ\Lambda^H)$
of the Schur function $S_I$ in the plethysm $\ell^{(r)}_k\circ\Lambda^H$
is equal to the number of Young tableaux of shape $\conj I$ (conjugate
partition) and evaluation $H^k$ whose charge is congruent to $r$ modulo $k$.
\bigskip
Proof: \rm Let $|I|=n$, $|H|=m$ and $d\mid k$. If $\displaystyle P(q)=
\sum_{i=0}^{k-1}a_iq^i$ is the residue modulo $1-q^k$ of the Kostka
polynomial
$K_{I,H^k}(q)$, one has, for $\zeta$ a primitive $d$-th root of unity,
$$P(\zeta)=K_{I,H^k}(\zeta)
=(-1)^{(d-1)mk/d}(\psi_d^{k/d}(S^H),\ S_I) $$
(by Theorem 2.2)
$$=(-1)^n(\psi_d^{k/d}\circ((-1)^mS^H),\ S_I)
=((-1)^n\psi_d^{k/d}\circ\Lambda^H(-A),\ S_I(A))$$
$$=(\psi_d^{k/d}(\Lambda^H),\ S_{\conj I}) = r_d\in {\bb Z}\ . $$
Hence, by Lemma 4.1 {\it (i)},
$$a_i={{1}\over{k}}\sum_{d\mid k}c(i,d)
    (\psi_d^{k/d}(\Lambda^H), \ S_{\conj I})
    =(\ell^{(i)}_k(\Lambda^H),\ S_{\conj I}) \ .\qquad\qed $$

\bf Example. --- \rm With $k=4$, $r=2$ and $H=(2)$,
$$\ell^{(2)}_4(\Lambda^2)=S_{134}+S_{224}+S_{11114}+2S_{1133}+2S_{1223}$$
$$+2S_{11123}
+S_{2222}+S_{11222}+2S_{111122}+S_{1111112} \ .$$
To compute the coefficient $(S_{11123},\ \ell^{(2)}_4(\Lambda^2))=2$, we have
to find the number of tableaux of shape $(11123\conj )=(125)$,  evaluation
$(2222)$ and charge $\equiv 2\ (\mod 4)$. The two tableaux satisfying
these constraints are:
$$\young{ 3  \cr
          2 &4 \cr
          1&1&2&3&4\cr  } \qquad\qquad\qquad
 \young{ 4 \cr
         2&3 \cr
         1&1&2&3&4\cr}  $$
which both have charge equal to $6$. \par
Similarly, the reader can check that $(S_{237},\
\ell^{(2)}_4(\Lambda^{12}))=5$
is the number of tableaux with shape $(1111233)$, evaluation $(11112222)$ and
charge $\equiv 2\ (\mod 4)$.

\bigskip
A related result, due to Molev and Tsalenko [MT], gives an interpretation
of $\ell^{(k)}_n$ in terms of ${\goth{S}}_n$-harmonic polynomials: let
$H^k_n$ be the Frobenius characteristic of the homogeneous component
${\cal H}^k_n$ of degree $k$ of the module ${\cal H}_n$ of
${\goth{S}}_n$-harmonic polynomials. Then,
$$\ell^{(k)}_n=\sum_{i\equiv k\ (\mod n)}H^i_n \ . $$


More generally, the plethysm $\ell^{(j)}_k(\Lambda^H)$ can be expressed
as a positive sum of Frobenius characteristics of modules of harmonic polynomials.
To each partition $J$ of $n$, it is possible to associate a certain
submodule ${\cal H}_J$ of ${\cal H}_n$ which has been defined in [La]
and called an \it affine Specht module\rm. This module is isomorphic,
as a graded ${\goth{S}}_n$-module, to the rational cohomology
$H^*({\cal F}_J)$ of the variety of flags fixed by an unipotent
of Jordan canonical form corresponding to the partition $J$.
The action of the symmetric group on $H^*({\cal F}_J)$ is defined
in [HS], and the Frobenius characteristic
$H^j_J={\rm ch}(H^{2j}({\cal F}_J))={\rm ch}({\cal H}^j_J) $ 
is given by the equation
$$\sum_j q^j H^j_J = \sum_I \tilde K_{\conj I,J}(q)S_I $$
where the $\tilde K_{I,J}(q)$ are the cocharge polynomials
({\it i.e.} the reciprocals of the Kostka polynomials,
see also [Mcd], ex. 9 p. 136).

Now, suppose $J=H^k$, and let $\tilde K_{\conj I,J}(q)\equiv
\displaystyle \sum_{j=0}^{k-1}a_j q^j \ (\mod 1-q^k)$.
Since $(S_I,\ H^i_J)$ is equal to the coefficient of $q^j$ in
$\tilde K_{\conj I,J}(q)$, we have
$$(S_I,\ \sum_{i\equiv j \ (\mod k)}H^i_J) = a_j \ . $$
On the other hand, $\tilde K_{\conj I,J}(q)\equiv K_{\conj I,J}(q)\
(\mod 1-q^k) $ whence $a_j = (\ell^{(j)}_k(\Lambda^H), \ S_I)$.
 To summarize: \bigskip
 \bf Theorem 4.3. --- \it Let $J=H^k$. Then,
 $$\ell^{(j)}_k(\Lambda^H) = \sum_{i\equiv j\ (\mod k)}H^i_J \ .\qquad\qed
$$\rm
The theorem of Molev and Tsalenko corresponds to the case $H=(1)$
{\it i.e.} to the case of the full flag manifold.\medskip

\bf Example. --- \rm The character of the
preceding example decomposes into three parts
$$\ell^{(2)}_4(\Lambda^2)=H^2_{2222}+
H^6_{2222}+H^{10}_{2222},$$
the three components being
$$H^2_{2222}=S_{111122}+S_{1111112} \ , $$
$$H^6_{2222}=S_{1133}+2S_{1223}+S_{2222}+S_{11114}+2S_{11123}+S_{11222}
+S_{111122}$$
$${\rm and}\qquad H^{10}_{2222}=S_{134}+S_{224}+S_{1133}.$$

\bigskip

\bigskip
\noindent\bigbf 5. Further congruences for Kostka polynomials\bigskip\rm

We have given in [LLT] some congruences modulo cyclotomic polynomials
for certain columns of the $q$-Kostka matrix. We will now present
other congruences for certain rows of the same matrix.

Let $\rho_n$ be the staircase partition $(12\ldots n)$.
It is well-known that for $k=2$ ($\zeta=-1$), one has $S_{\rho_n}=
P_{\rho_n}$. More generally, it can be shown that if $\zeta$ is a
primitive $k$-th root of unity, $P_{\rho^{k-1}_n}(A;\zeta)=
S_{\rho^{k-1}_n}(A)$. This is a consequence of the following result:\bigskip

\bf Theorem 5.1. --- \it Let $\zeta$ be a primitive $k$-th root of $1$, and
$I$ be a partition such that $\ell(\conj I\ )\le n$. Then,
$$S_{I^k\vee\rho^{k-1}_n}(A)
=\sum_J K_{IJ}\ P_{J^k\vee\rho^{k-1}_n}(A;\zeta) $$
where $K_{IJ}$ are the usual Kostka numbers. In other words, we have
the following congruence modulo the cyclotomic polynomial $\Phi_k(q)$:
$$K_{I^k\vee\rho^{k-1}_n,\ H}(q) \equiv
\left\{\matrix{
               K_{IJ} & {\it if} & H=J^k\vee\rho^{k-1}_n \cr
               0      & {\it otherwise.} & \cr
               }\right. $$

\rm This theorem is itself a direct consequence of the following three lemmas.
Details about $k$-cores and rim hook tableaux can be
found in [KSW] and [SW].
\bigskip
\bf Lemma 5.2. --- \it Let $I$ be a partition such that $\ell(\conj I\ )\le n$.
\smallskip
{\rm a)} The $k$-core of the partition $I^k\vee\rho^{k-1}_n$ is
$\rho^{k-1}_n$. \smallskip
{\rm b)} The only $k$-rim hooks which are removable from
$I^k\vee\rho^{k-1}_n$ are those of shape $(1^k)$. After removal of
such a rim hook, one gets a partition $I'^k\vee\rho^{k-1}_n$ with
$\ell(\conj{I'}\ )\le n$.\qquad $\qed$ \bigskip

\bf Lemma 5.3. --- \it If $(Q_H',\ S_{I^k\vee\rho^{k-1}_n}) \not = 0$,
then $H=J^k\vee\rho^{k-1}_n$ for some partition $J$.\medskip

Proof:\ \rm If $(Q_H',\ S_{I^k\vee\rho^{k-1}_n}) \not = 0$ then
$\ell(H)\ge\ell(I^k\vee\rho^{k-1}_n)$. This forces $H$ to have a
part $j$ with multiplicity $\ge k$. Set $H=(j^k)\vee H'$. Then,
$$(Q_H',\ S_{I^k\vee\rho^{k-1}_n})=
(-1)^{(k-1)j}(Q'_{H'},\ D_{\psi_k(S_j)}S_{I^k\vee\rho^{k-1}_n}) $$
(by Theorems 2.1 and 2.2)
$$=(-1)^{(k-1)j}(-1)^{(k-1)j}\sum_{I'}\alpha_{I'}
               (Q'_{H'},\ S_{I'\vee\rho^{k-1}_n})
=\sum_{I'}\alpha_{I'}(Q'_{H'}, \ S_{I'^k\vee\rho^{k-1}_n}) $$
(by Lemma 5.2)
where the summation is over partitions $I'$ of weight $|I|-kj$, and the
$\alpha_{I'}$ are certain nonnegative integers. By induction, we obtain
after a finite number of steps
$$(Q'_H,\ S_{I^k\vee\rho^{k-1}_n})
=\sum_{H''}\alpha_{H''}(Q'_{H''}, \ S_{\rho^{k-1}_n}) $$
where $\alpha_{H''}\ge 0$. For this sum to be non zero, it is necessary
that for some $H''$, $(Q'_{H''},\ S_{\rho^{k-1}_n})\not = 0$. This implies
that $\ell(H'')\ge\ell(\rho^{k-1}_n)$ and that no multiplicity of $H''$
be $\ge k$. The only possibility is $H''=\rho^{k-1}_n$, whence
$H=H''\vee J^k = \rho^{k-1}_n\vee J^k$.\qquad $\qed$
\bigskip
\bf Lemma 5.4. --- \it For $|I|=|J|$ and $\ell(\conj I)\le n$,
$$D_{\psi_k(S^J)}S_{I^k\vee\rho^{k-1}_n}=
(-1)^{(k-1)|J|}\ K_{IJ} S_{\rho^{k-1}_n} \ . $$
Proof:\ \rm Since $|I|=|J|$ and the $k$-core of $I^k\vee\rho^{k-1}_n$ is
$\rho^{k-1}_n$, we see that
$$D_{\psi_k(S^J)}S_{I^k\vee\rho^{k-1}_n} = \beta_{IJ}\cdot S_{\rho^{k-1}_n}$$
where $\beta_{IJ}$ is, up to sign, the number of $k$-rim hook tableaux
of shape $I^k\vee\rho^{k-1}_n/\rho^{k-1}_n$ and evaluation $J$.
From lemma 5.2, it results that the rim hooks of these tableaux are all of
shape $(1^k)$, so that $\beta_{IJ}=(-1)^{(k-1)|J|}K_{IJ}$.\qquad $\qed$
\bigskip
\it Proof of Theorem 5.1:\ \rm
$$S_{I^k\vee\rho^{k-1}_n} =
\sum_H (Q'_H,\ S_{I^k\vee\rho^{k-1}_n})\ P_H $$
$$=\sum_J (Q'_{J^k\vee\rho^{k-1}_n},\ S_{I^k\vee\rho^{k-1}_n})
           P_{J^k\vee\rho^{k-1}_n}$$
(by Lemma 5.3)
         $$  =\sum_J (-1)^{(k-1)|J|}
               (Q'_{\rho^{k-1}_n},\ D_{\psi_k(S^J)}
                       S_{I^k\vee\rho^{k-1}_n})P_{J^k\vee\rho^{k-1}_n} $$
(by Theorems 2. and 2.2)
$$=\sum_J K_{IJ}P_{J^k\vee\rho^{k-1}_n}  $$
(by Lemma 5.4). \qquad $\qed $
\bigskip
\bf Example. --- \rm Take $k=3$, $n=3$, $I=(2,3)$ and $J=(1,2,2)$.
Then,
$$K_{112222233333,111112222222233}(q) =
  q^3 ( 1+ 2q  +6q^2  + 13 q^3  + 24q^4 + 39 q^5 $$
$$ +59 q^6  +   81  q^7 +  105 q^8  + 128  q^9 
+148 q^{10} +163 q^{11}  +171q^{12} + 172 q^{13}  +165 q^{14}
+  153 q^{15}  +   134 q^{16}$$
$$   +114 q^{17}  + 92 q^{18}  + 72 q^{19}
+   53 q^{20} + 38 q^{21} + 25 q^{22}  +16  q^{23}  + 9  q^{24}  
+ 5  q^{25}  + 2 q^{26}    +q^{27}    )$$
$$\equiv K_{23,122} =2 \ (\mod 1+q+q^2) \ . $$



\vfill\eject

\bigbf References
\rm\bigskip

[CT] C. Carr\'e and J.-Y. Thibon, \it Some formulas for plethysm
coefficients\rm, L.I.T.P. Internal report 91.63 (1991).\medskip

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

[De] 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.\medskip

[Fo] 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.\medskip

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

[HW] G.M. Hardy and E.M. Wright, \it An introduction to the theory of
numbers\rm, (5th ed.), Oxford, Clarendon Press, 1979.\medskip

[KW] W. Kraskiewicz and J. Weyman, \it Algebra of invariants and the
action of a Coxeter element\rm, Preprint Math. Inst. Univ. Copernic,
Toru\'n, Poland.
\medskip

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

[La] A. Lascoux, \it Cyclic permutations on words, tableaux and
harmonic
polynomials\rm, Proc. of the Hyderabad Conference on Algebraic
groups,
Manoj Prakashan, Madras, 1991, 323-347.\medskip

[LLT] A. Lascoux, B. Leclerc and J.-Y. Thibon, \it Fonctions de
Hall-Littlewood et polyn\^omes de Kostka-Foulkes aux racines de
l'unit\'e\rm, C.R. Acad. Sci. Paris, t. {\bf 316}, ser. I (1993),
1-6.
\medskip

[Mcd] I.G. Macdonald, \it Symmetric functions and Hall
polynomials\rm,
Oxford Math. Monographs, Oxford, 1979.\medskip

[MT] A.I. Molev and L.M. Tsalenko, \it Representations of symmetric
groups in a free Lie (super) algebra and in the space of harmonic
polynomials\rm, Funct. Anal. Appl. {\bf 20}:2 (1986),
150-152.\medskip

[Mo] A.O. Morris, \it On an algebra of symmetric functions\rm, Quart.
J.
Math. Oxford Ser. (2) {\bf 16} (1965), 53-64.\medskip

[MS] A.O. Morris and N. Sultana, \it Hall-Littlewood functions at
roots of
1 and modular representations of the symmetric group\rm, Math. Proc.
Cambridge Phil. Soc. {\bf 110} (1991), 443-453.\medskip

[NV] C.A. Nicol and H.S. Vandiver, \it A Von Sterneck arithmetical
function
and restricted partitions with respect to a modulus\rm, Proc. Nat.
Acad.
Sci. U.S.A. {\bf 40} (1954), 825-835.\medskip

[SW] D.W. Stanton and D.E. White, \it A Schensted algorithm for rim
hook tableaux\rm, J. Comb. Theory A {\bf 40} (1985), 211-247.\medskip

[Su] N. Sultana, \it Hall-Littlewood functions and their applications
to representation theory\rm, Thesis, University of Wales,
1990.\medskip

\bye


