% $Date: 2003/09/15 16:37:46 $ $Author: nthiery $ $Revision: 1.1.1.1 $

\begin{Mhelppage}{SfCCProd}%
  {V. Prosper}%
  \begin{MInfo}
    \eng{computes the conjugacy class product of symmetric functions}
  \end{MInfo}
  

\begin{MCalls}
%  \Mcall{sf1,sf2,b1,b2}
  \Mcall{sf1,sf2}
\end{MCalls}

\begin{MParameters}
  \param{sf1, sf2}
  \eng{any symmetric functions}
                                %\param{b1, b2}
  %\eng{any strings denoting names of known bases}
\end{MParameters}

\begin{MIntro}
  \eng{

The {\Mthis} function computes the conjugacy class  product of symmetric
functions \Mex{sf1} and \Mex{sf2}.
This product is defined, on the Schur basis, to be
\Mdmath{%
\Mex{s[lambda]} x \Mex{s[mu]} = 0 (\Mex{lambda} <> \Mex{mu}),
\Mex{s[lambda]} x \Mex{s[lambda]} = \Mex{s[lambda]}/\Mex{f(lambda)},%
}{%
\mathtt{s}_\lambda\times\mathtt{s}_\mu = 0,
\mathrm{~when~}\lambda\neq\mu,\mathrm{~and~}
\mathtt{s}_\lambda\times\mathtt{s}_\lambda=\mathtt{s}_\lambda/f(\lambda)
}
in which \Mmath{f\Mex{(lambda)}}{f(\lambda)} stands for the number of
standard tableaux of shape \Mmath{\Mex{lambda}}{\lambda}.

The result is expressed in the basis of Schur functions.
%
%In the case that \Mex{sf1}  and  \Mex{sf2} are solely expressed in the basis
%\Mex{b1} and \Mex{b2}, one may specify these bases.
}
\end{MIntro}

\begin{MExample}{ex1}
\begin{Mexin}
muEC::SYMF::SfCCProd(s^2*s[2,1,1] + s[3,1], q*s[2,1,1]);
\end{Mexin}
\begin{Mexout}
                           2
                        q s  s[2, 1, 1]
                        ---------------
                               3
\end{Mexout}
% \begin{Mexout}[hide]
                    % q*s^2*s[2, 1, 1]*1/3
% \end{Mexout}
\end{MExample}

\MRelatedFunctions{TYP::Sf}
\end{Mhelppage}
