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\pageno=31
\titrecourant={FOULKES AND LEFSCHETZ CHARACTERS}
\auteurcourant={A. KERBER AND K.-J. TH\"URLINGS}

\eightpoint
\leftline{Publ. I.R.M.A. Strasbourg, $\oldstyle 1984$, 229/S--08}
\leftline{Actes $8^e$ S\'eminaire Lotharingien, p. 31--36}

\tenpoint
\let\it=\sl

\vskip 1.5cm
\centerline{{\bf EULERIAN NUMBERS, FOULKES CHARACTERS}}
\vskip 5pt
\centerline{{\bf AND LEFSCHETZ CHARACTERS OF $S_n$}}
\vskip 2.8mm
\centerline{\sevenrm BY}
\vskip 2.8mm
\centerline{{\petcap Adalbert {KERBER}
{\sevenrm AND} Karl-Joseph {TH\"URLINGS}}}
\vskip 24pt
\abstract{The aim of this talk is to point out a connection
between the characters which \pd FOULKES introduced in order to give a
representation theoretical generalization of Eulerian numbers
and certain Lefschetz characters of $S_n$ which A.~\pd BJ\"ORNER
mentioned at his talk in Feuerstein and which were described in
detail by R. \pd STANLEY in [1]. The missing link is a theorem
on the irreducible constituents of Foulkes' characters.}

\section 1. Eulerian numbers|Let
$\pi =\bigl(\pi (1)\ldots \pi (n)\bigr)$
be an element of the symmetric group $S_n$,
e.g. $(13248765)\in S_8$ (list notation!) with the
{\it up-and-down-sequence} $A(\pi)$ 
(rises indicated by $+,$ falls denoted by $-$),
for example
$$A((13248765))=\ +\,-\,+\,+\,-\,-\,-\ .$$
The number of permutations with a given number of rises, i.e. of entries $+$ 
in its up-down sequence, defines an
{\it Eulerian number:}
$$A(n,k):=\left|\{\pi \in S_n\mid A(\pi )\ {\rm has}\ k\ {\rm rises}
\}\right|.$$
According to
\pd FOULKES, $A(\pi )$ yields a skew diagram via the rule
$$\matrix{
+\leftarrow&\times\cr
           &\downarrow-\cr}
$$
This means, that to an entry $+$ of $A(\pi)$ there corresponds a node
$\times$ that has to be added at the left of the last node added, and in the same row. 
Correspondingly to an entry $-$ there corresponds a node that has to be added 
just below the last node.
For example the sequence $(+-++---)$ mentioned above gives
$$\matrix{
       &&\times&\times\cr
\times&\times&\times\cr
\times\cr
\times\cr
\times\cr}
\ \ {according\ to}\ \ \ \matrix{
                                                    &&+&\times\cr
                                                   +&+&-\cr
                                                   -\cr
                                                   -\cr
                                                   -\cr}
                                       .$$
This resulting skew diagram is the rim part of 
$\lambda (A)$ : $R^{\lambda (A)}_{11}=(431^3)/(2)$.
Recall the definition of
{\it skew representation}
by the Littlewood-Richardson rule:
$$\displaystyle [\lambda /\mu ]
:=\sum _\nu  ([\mu ][\nu ],[\lambda ])[\nu ].$$

\th Theorem {(Foulkes)}|The number of permutations with given up-down 
sequence $A$ can be identified with a dimension of a skew representation:
  $$\left|\{\pi \mid A(\pi )=A\}\right|
=dim[R^{\lambda (A)}_{11}].$$
\finth

For example, if we put
$A:={+\,-\,+\,+\,-\,-\,-}$ then the number of permutations with this sequence isthe dimension of
$[(431^3)/(2)]$ which has the decomposition
$[421^2]+[41^4]+[3^21^2]+[321^3]$ and therefore the dimension
$245.$
More generally we have
the following generalization of Foulkes' result:

\th Theorem {(K/Th)}| The decomposition of $[R^{\lambda (A)}_{11}]$
is obtained from $A$ by successive applications of the rule
$$\matrix{
               \times&\nearrow\!+\cr
               \swarrow\!-\cr}
$$
\finth

By this pictorial description we mean 
that to an entry $+$ of $A$ there corresponds a node
$\times$ which has to be added to the right of the last node, maybe 
in a higher  
row,   
while to an entry $-$ there corresponds 
a node added to the left of the last node or in a lower row. Consider 
once more the example $A=(+-++---)$. We start with a node $\times$, 
and the first 
entry of $A$ is a $+$, so the corresponding
node has to be added, according to the rule,
to the right of the starting node, i.e. we obtain the diagram $\times\otimes$, 
where the last node added is encircled. Now the second entry of 
$A$ is a minus sign, hence the corresponding addition of a node is again 
uniquely determined, and we get the diagram
$$\matrix{
               \times&\times\cr 
               \otimes\cr}
 .$$
The next entry of $A$ is a plus sign, so that there are two places open for 
an additional node which are to the right of the 
node which was added last time:
$$\matrix{
        \times&\times&\otimes\cr 
        \times\cr}
  \ \ {and}\ \ \matrix{
                                     \times&\times\cr 
                                     \times&\otimes\cr}
                                 .$$
The next steps yield the following cascade of diagrams:
$$\matrix{
        \times&\times&\times&\otimes\cr
        \times\cr
        &\swarrow&\searrow\cr}
  \ \ \ \ \ \ \ \ \ \ \matrix{
                            \times&\times&\otimes\cr
                            \times&\times\cr
                            &\swarrow&\searrow\cr}
                              $$
$$\matrix{
  \times&\times&\times&\times\cr
  \times\cr
  \otimes\cr
  &&\downarrow\cr}
  \ \ \matrix{
                    \times&\times&\times&\times\cr
                    \times&\otimes\cr
                                 \cr
                     &\downarrow\cr}
              \ \ \matrix{
                               \times&\times&\times\cr
                               \times&\times\cr
                               \otimes\cr
                               &\downarrow\cr}
                         \ \ \matrix{
                                           \times&\times&\times\cr
                                           \times&\times&\otimes\cr
                                                               \cr
                                            &\downarrow\cr}
$$
$$\matrix{
         \times&\times&\times&\times\cr
         \times\cr
         \times\cr
         \times\cr
         \times\cr}
   \ \ \matrix{
                     \times&\times&\times&\times\cr
                     \times&\times\cr
                     \times\cr
                     \times\cr}
               \ \ \matrix{
                                 \times&\times&\times\cr
                                 \times&\times\cr
                                 \times\cr
                                 \times\cr
                                 \times\cr}
                          \ \ \matrix{
                                            \times&\times&\times\cr
                                            \times&\times&\times\cr
                                            \times\cr
                                            \times\cr}
                                      $$
Hence from $A=(+-++---)$ we obtain the diagrams
$$[4,1^4],[4,2,1^2],[3,2,1^3],[3^2,1^2],$$
and each one of them exactly once. 

{\it Proof}\pointir ``By example''
  $$\displaylines{\quad
R^{\lambda (({+}{-}{-}{+}{+}{-}{-}{+}{+}{+}))}_{11}
=\det\pmatrix{[2]&[3]&[6]&[7]&[11]\cr
                1&[1]&[4]&[5]&[9]\cr
                0&1&[3]&[4]&[8]\cr
                0&0&1&[1]&[5]\cr
                0&0&0&1&[4]\cr}\hfill\cr
\hfill\eqalign{&=[4]\det\pmatrix{[2]&[3]&[6]&[7]\cr
                         1&[1]&[4]&[5]\cr
                         0&1&[3]&[4]\cr
                         0&0&1&[1]\cr}
-\det\pmatrix{[2]&[3]&[6]&[11]\cr
                1&[1]&[4]&[9]\cr
                0&1&[3]&[8]\cr
                0&0&1&[5]\cr}\cr
&=[4][(7,6,6,4)/(5,5,3,3)]-[(8,7,7,5)/(6,6,4)]\cr
&=[4][R^{\lambda (({+}{-}{-}{+}{+}{-}))}_{11}]
-[R^{\lambda (({+}{-}{-}{+}{+}{-}{+}{+}{+}{+}))}_{11}].\cr}
\quad\cr}
$$
Hence, the following lemma completes the proof.

\th Lemma|Let $A$ denote an up-and-down sequence. Then, for each
$k\in {\bf N}$ we have
  $$[k+1][R^{\lambda (A)}_{11}]=
[R^{\lambda ((A{+}{+}\cdots{+}))}_{11}]
+[R^{\lambda ((A{-}{+}\cdots{+}))}_{11}].$$
\finth

\section 2. Foulkes characters| Foulkes' result gives the following 
interpretation of Eulerian numbers as sums of dimensions of skew 
representations:
  $$A(n,k)=\sum _{A,k\ ups}
{\rm dim}[R^{\lambda (A)}_{11}],$$
while the above generalization gives a more general result in terms of 
characters:
$$\chi ^{n,k}:=\sum _{A,k\ ups} \chi ^{R^{\lambda (A)}_{11}}.$$
We suggest to call these characters
{\it Foulkes\ characters.} They have the following remarkable properties:

\th Theorem {(Foulkes)}|

\vskip 3pt
\indent\indent (i) {\it no. of cycles of
$\pi  ={}$no. of cycles of $\rho  \Rightarrow
\chi ^{n,k}(\pi )=\chi ^{n,k}(\rho )$} ;

\vskip 3pt
\indent\indent (ii)
$\chi ^{n,0}=\zeta ^{(1^n)}$,\quad
$\chi ^{n,n-1}=\zeta ^{(n)}$,\quad
$\chi ^{n,k}=\zeta ^{(1^n)}\otimes \chi ^{n,n-1-k}$ ;

\vskip 3pt
\indent\indent
(iii) {\it The Foulkes characters satisfy the following recursion:}
$$\chi^{n,k}_\mu=\chi^{n-1,k-1}_{\mu^*}-\chi^{
n-1,k}_{\mu^*},
\mu^*:=(\mu_1,\ldots,\mu_{i-1},\mu_i-1,\mu_{i+1},\ldots).$$\finth
\vskip 5pt
{\petcap Further Properties}.

\vskip 3pt
\indent \indent
(i) $(\chi ^{n,k},\zeta ^\lambda )>0
\Rightarrow \lambda _1\le k+1$,\quad $\lambda '_1\le n-k$ ;

\vskip 3pt
\indent \indent
(ii) $(\chi ^{n,k},\zeta ^{(j+1,1^{n-j-1})})>0\Leftrightarrow j=k$ ;

\vskip 3pt
\indent \indent
(iii) {\it The $\chi ^{n,k}$ are linearly independent;}

\vskip 3pt
\indent \indent
(iv) {\it If $\chi :S_n\rightarrow {\bf C}$ denotes a character,
depending only on the 
\indent\indent number of
cyclic factors, then we have} 
  $$\chi =\sum _i {(\chi ,\zeta ^{(i+1,1^{n-i-1})})
\over f^{(i+1,1^{n-i-1})}}\chi^{n,i}.$$
%\finth

\noindent
%For example, $\chi ^{5,2}=6\zeta ^{(3,1^2)}+3\zeta ^{(2^2,1)}
%+3\zeta ^{(3,2)}$,
%take $(1/3)\chi ^{5,2}$.
Using 5.8.30 in KERBER-TH\"URLINGS we obtain 

\th Theorem| The ``P\'olya-char\-ac\-ter'' $\chi,$ defined by
$$\chi (\pi ):=m^{\hbox{\sevenrm no. of cycles of }\pi }$$
has the following decomposition into irreducibles:
$$\chi =\sum _k {m+k\choose n} \chi ^{n,k}.$$
\finth

\section 3. Foulkes tables|
This section contains the Foulkes tables 
$\displaystyle F_i:=(\chi _j^{n,k})$
of the symmetric groups $S_n$, for
$n\leq7$. We recall that the $j$-th column of the Foulkes table contains 
in its $i$-th row the value of the Foulkes characters $\chi^{n,i}$ on the classes of 
elements which consist of $j$ cyclic factors. The 0-th row indicates the 
column numbers $j,$ while the $0$-th column shows the row numbers $i.$
$$F_1=\matrix{
i\backslash j&1\cr 
0&1\cr}
,\ F_2=
\matrix{
i\backslash j&2&1\cr 
0&1&-1\cr
1&1&1},
\ F_3=
\matrix{
i\backslash j&3&2&1\cr 
0&1&-1&1\cr
1&4&0&-2\cr
2&1&1&1},
$$
$$
F_4=\matrix{
i\backslash j&4&3&2&1\cr 
0&1&-1&1&-1\cr
1&11&-3&-1&3\cr
2&11&3&-1&-3\cr
3&1&1&1&1},
\ F_5=
\matrix{
i\backslash j&5&4&3&2&1\cr 
0&1&-1&1&-1&1\cr
1&26&-10&2&2&-4\cr
2&66&0&-6&0&6\cr
3&26&10&2&-2&-4\cr
4&1&1&1&1&1},
$$
$$F_6=\matrix{
i\backslash j&6&5&4&3&2&1\cr 
0&1&-1&1&-1&1&-1\cr
1&57&-25&9&-1&-3&5\cr
2&302&-40&-10&8&2&-10\cr
3&302&40&-10&-8&2&10\cr
4&57&25&9&1&-3&-5\cr
5&1&1&1&1&1&1},
$$
$$ F_7=
\matrix{
i\backslash j&7&6&5&4&3&2&1\cr 
0&1&-1&1&-1&1&-1&1\cr
1&120&-56&24&-8&0&4&-6\cr
2&1191&-245&15&19&-9&-5&15\cr
3&2416&0&-80&0&16&0&-20\cr
4&1191&245&15&-19&-9&5&15\cr
5&120&56&24&8&0&-4&-6\cr
6&1&1&1&1&1&1&1}.
$$

\section 4. The connection with a result of Stanley|We consider groups
acting on posets $M$ such that
$x\le y\Leftrightarrow gx\le gy$. An important example is the action of
$S_n$ on $2^n,$ the power set of $n,$ with the inclusion as partial order.

Denote by
$R$ a subset of the set of ranks, and by
$K_R(M)$ a set of rank selected chains.
Put $K_R(M,{\bf C}):={\bf C}^{K_R(M)}$ and denote by
$H_i(M_R,{\bf C})$ the {\it homology group.} Using these notions we can 
introduce 
  $$\kappa _R(g):=\hbox{trace of $g$ on }K_R(M,{\bf C}),
\gamma _{R,i}(g):=\hbox{trace of $g$ on }H_i(M_R,{\bf C}),$$
and
$$\nu _R(g):=\sum _{i=0}^r (-1)^{\left|R\right|-i}
\gamma _{R,i}(g),$$
the {\it 
Lefschetz\ character}. Then, to begin with, we have the following well known
facts:
$$
\kappa _R=\sum _{T\subseteq R}\nu _T,\ \ 
or,\ equivalently,\ \ 
\nu _R
=\sum _{T\subseteq R}(-1)^{\left|R\setminus T\right|}T.$$


\th Theorem {(Stanley)}|If
$R:=(n_1,\ldots,n_k)_<$,
$\rho :=(n_1,n_2-n_1,\ldots,n_k-n_{k-1},n-n_k)$ ;
$\rho ^*:=\hbox{partition obtained by reordering}$, then

{\rm (i)} $\kappa _R=\xi ^{\rho ^*}$, the Young character,
$\displaystyle {}=\sum _{\lambda \vdash n}\left|ST^{\lambda '}
(\rho ^*)\right|\zeta ^\lambda $
(standard tableaux, shape $\lambda '$, content $\rho ^*$).

{\rm (ii)} $\displaystyle \nu _R
=\sum _{\lambda \vdash n}\left|ST_R^{\lambda '}(1^n)\right|
\zeta ^\lambda $
(standard Young tableaux with $R$ as set of ascents).\finth

%\vskip 2cm
%\rightline{then $\mu _1$ is an ascent.\qquad}
%\vskip 2cm

Hence we obtain from the above discussion of Foulkes characters :

\vskip 5pt
{\petcap Theorem.}
  $$\chi ^{n,n-k-1}=\sum _R \nu _R\qquad(\left|R\right|=k).$$

This shows the connection between Foulkes characters and
the Lefschetz characters of $S_n$ on $2^n$.

\goodbreak
\vskip 1cm
\eightpoint
\centerline{REFERENCES}
\nobreak
\vskip.5cm
\article 1|\pd FOULKES (H.O.)|Eulerian numbers, Newcomb's problem
and representations of symmetric groups|Discrete Math.|30|1980|3--49|

\divers 2|\pd KERBER (Adalbert) and \pd TH\"URLINGS
(Karl-Josef)|Symmetrieklassen
von Funktionen und ihre Abz\"ahlungstheorie~II (in print), III (in
preparation), {\sl Bayreuther Math. Schriften}|

\article 3|\pd STANLEY (Richard)|Some aspects of groups acting on
finite posets|J. Combinatorial Theory, Ser.~A|32|1982|132--161|

\nobreak
\vskip 1cm
\line{\quad\vtop{\hbox{Adalbert {\petcap Kerber} and 
Karl-Josef {\petcap Th\"urlings},}
             \hbox{Lehrstuhl II f\"ur Mathematik,}
              \hbox{Universit\"at Bayreuth,}
             \hbox{D-95440 Bayreuth, Deutschland}
            \hbox{email : {\eightrm
kerber@btm2x6.mat.uni-bayreuth.de}}}\
%hfill
%\vtop{\hbox{Dominique {\petcap Foata,}}
                 %\hbox{D\'epartement de math\'ematique,}
                 %\hbox{Universit\'e Louis-Pasteur,}
                 %\hbox{7, rue Ren\'e-Descartes,}
                 %\hbox{F-67084 Strasbourg, France.}
                 %\hbox{email : {\eightrm
%foata@math.u-strasbg.fr}}}\quad}
%\tenpoint
%\adresse
%{{\petcap {Adalbert} Kerber} and Karl-Josef {\petcap Th\"urlings},
%Lehrstuhl II f\"ur Mathematik,
%Universit\"at Bayreuth,
%Postfach 3008,
%D-8580 Bayreuth.
%
%\fin
}

\bye

**************************endofpaper********************************
