% $Date: 2003/10/28 21:59:22 $ $Author: nthiery $ $Revision: 1.1.1.2 $

\begin{Mhelppage}{SfPlethysm}{V. Prosper}

  \begin{MInfo}
    \eng{plethysm of symmetric functions}
  \end{MInfo}
  

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

  \begin{MParameters}
    \param{sf1, sf2}
    \eng{any symmetric functions}
    \param{b1, b2}
    \eng{any known bases}
  \end{MParameters}

  \begin{MIntro}
    \eng{The {\Mthis} function computes the plethysm \Mex{sf1[sf2]}.
      
      One may specify that \Mex{sf1} and \Mex{sf2} are expressed on
      the bases \Mex{b1} and \Mex{b2} by using either \Mex{\Mthis(sf1,
        sf2, b1)}, \Mex{\Mthis(sf1, sf2, b1, b2)}.
      
      The plethysm operation is defined as follows: let \Mex{p[i]} be
      the \Mex{i}-th power-sum symmetric function and \Mex{sf1},
      \Mex{sf2}, \Mex{sf3} be any symmetric functions, then,
 
      \Mdmath{%
        \begin{itemize}
          \item
          {\Mex{(sf1 + sf2)[sf3] = sf1[sf3] + sf2[sf3]}}
          \item
          {\Mex{(sf1 x sf2)[sf3] = sf1[sf3] x sf2[sf3]}}
          \item
          {\Mex{sf1[p[i]] = p[i][sf1]}}
          \item
          {\Mex{p[i][p[j]] = p[i x j]}}
        \end{itemize}
        }%
      {%
        \begin{array}{ccc}
          \mathtt{(sf1 + sf2)[sf3]}&=&\mathtt{sf1[sf3] + sf2[sf3]}\\
          \mathtt{(sf1\times sf2)[sf3]}&=&\mathtt{sf1[sf3]\times sf2[sf3]}\\
          \mathtt{sf1[p[i]]}&=&\mathtt{p[i][sf1]}\\
          \mathtt{p[i][p[j]]}&=&\mathtt{p[i\times j]}
        \end{array}%
        }
      
      The default is to compute plethysms on the \Mex{p}-basis and
      return the result expressed on the \Mex{p}-basis.
      
      Special algorithms have been included to compute plethysms.  In
      particular, the result is expressed on the basis of Schur
      functions when:
      \begin{itemize}
        \item
        {\Mex{b1=e}, \Mex{b2=e,h,s},}
        \item
        {\Mex{b1=h}, \Mex{b2=e,h,s},}
        \item
        {\Mex{b1=s}, \Mex{b2=e,h,s},}
        \item
        {\Mex{b1=p}, \Mex{b2=e,h,p,s,m},}
        \item
        {\Mex{b1=m}, \Mex{b2=e,h,p,s,m}.}
      \end{itemize}
      }
  \end{MIntro}

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

   s[4, 2, 1, 1, 1] + s[3, 2, 2, 2] + s[3, 3, 2, 1] +

   s[4, 2, 2, 1] + 2 s[4, 3, 1, 1] + s[5, 2, 1, 1] +

   s[6, 1, 1, 1] + s[3, 3, 3] + s[4, 3, 2] + s[5, 2, 2] +

   s[4, 4, 1] + s[5, 3, 1]
\end{Mexout}
\end{MExample}

\MRelatedFunctions{Char2Sf}
\end{Mhelppage}
