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

\begin{Mhelppage}{SfEval}%
  {V. Prosper}%
  \begin{MInfo}
    \eng{realizes the action of symmetric functions on polynomials}
  \end{MInfo}
  
\begin{MCalls}
  \Mcall{sf,expr}
\end{MCalls}

\begin{MParameters}
  \param{sf}
  \eng{any symmetric function}
  \param{expr}
  \eng{any expression depending on some variables} 
\end{MParameters}

\begin{MIntro}
  \eng{
Symmetric functions  can  be  considered  as  operators  on  polynomials
(\Mmath{lambda}{\lambda}-ring structure of the ring of polynomials).
Let \Mex{sf} be a power sum \Mex{p}[k],  then
\Mdmath{\Mthis(\Mex{p}[k],  c\_u u  +  c\_v v  +  ...) =
         c\_u u\^k + c\_v v\^k + ...,}%
       {\Mthis(\mathtt{p}[k],  c_u u  +  c_v v  +  ...) =
         c_u u^k + c_v v^k + ...,}
where \Mmath{u, v, ...}{u, v, ...} are monomials  and
\Mmath{c\_u, c\_v, ...}{c_u, c_v, ...} are scalars.

For a product of power-sums \Mex{sf=p[i,j,...]}, \Mthis\Mex{(sf, expr)} is set
to  be equal  to   the   product
\Mmath{\Mthis\Mex{(p[i], expr)} x \Mthis\Mex{(p[j], expr)} x...}%
      {\Mthis\mathtt{(p[i], expr)}\times \Mthis\mathtt{(p[j], expr)}\times...}

The definition is extended by linearity to any symmetric function \Mex{sf}.
}
\end{MIntro}

\begin{MExample}{ex1}
\begin{Mexin}
muEC::SYMF::SfEval( p[2], 3*x*y + 2*z );
\end{Mexin}
\begin{Mexout}
                           2  2      2
                        3 x  y  + 2 z
\end{Mexout}
% \begin{Mexout}[hide]
                       % 2*z^2+3*x^2*y^2
% \end{Mexout}
\begin{Mexin}
muEC::SYMF::SfEval( p[4,3] + q*p[2], 2*x + 3*y );
\end{Mexin}
\begin{Mexout}
               2      2        3      3      4      4
         q (2 x  + 3 y ) + (2 x  + 3 y ) (2 x  + 3 y )
\end{Mexout}
% \begin{Mexout}[hide]
     % q*(x^2*2 + y^2*3) + (x^3*2 + y^3*3)*(x^4*2 + y^4*3)
% \end{Mexout}
\end{MExample}

\MRelatedFunctions{SfAExpand,SfPlethysm}
\end{Mhelppage}
