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

\begin{Mhelppage}{SfInternal}%
  {V. Prosper}%
  \begin{MInfo}
    \eng{computes the inner tensor product of symmetric functions}
  \end{MInfo}
  

\begin{MCalls}
  \Mcall{sf1, sf2}
\end{MCalls}

\begin{MParameters}
  sf1, sf2 = any symmetric functions
\end{MParameters}

\begin{MIntro}
  \eng{
The {\Mthis} function computes  the  inner tensor product of symmetric
functions \Mex{sf1} and \Mex{sf2}.
This operation is also called the internal product.
Since \Mex{sf1} and \Mex{sf2} may be seen as virtual characters,
{\Mthis} computes the symmetric function  associated with the  product
(Kronecker product) of these two virtual characters.

For the basis of products of power-sums,
\Mmath{\Mthis(\Mex{p[part], p[part']})}%
      {\Mthis(\mathtt{p[part], p[part']})} =  0 if
\Mmath{\Mex{part}<>\Mex{part'}}{\mathtt{part\neq part'}}, 
=  \Mmath{\Mex{SfZee(part) x p[part]}}{\mathtt{SfZee(part)\times p[part]}}
otherwise.
 
The result is expressed in the product of power-sums basis.
}
\end{MIntro}

\begin{MExample}{ex1}
\begin{Mexin}
muEC::SYMF::SfInternal( p[3,3,2,1,1], p[3,3,2,1,1] );
\end{Mexin}
\begin{Mexout}
                      72 p[3, 3, 2, 1, 1]
\end{Mexout}
% \begin{Mexout}[hide]
                     % 72*p[3, 3, 2, 1, 1]
% \end{Mexout}
\begin{Mexin}
muEC::SYMF::SfZee( [3,3,2,1,1] );
\end{Mexin}
\begin{Mexout}
                              72
\end{Mexout}
\begin{Mexin}
muEC::SYMF::SfInternal( s[3,2,1], p[3]*h[3] );
\end{Mexin}
\begin{Mexout}
                    2 p[3, 3]   p[3, 1, 1, 1]
                  - --------- - -------------
                        3             3
\end{Mexout}
% \begin{Mexout}[hide]
            % p[3, 3]*(-2/3) + p[3, 1, 1, 1]*(-1/3)
% \end{Mexout}
\end{MExample}

\MRelatedFunctions{Char2Sf,SfPlethysm,Sf2Char,SfZee}
\end{Mhelppage}
