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

\begin{Mhelppage}{Sf}%
  {V. Prosper}%
  \begin{MInfo}
    \eng{how symmetric functions are encoded in \Mex{SYMF}}
  \end{MInfo}
  

\begin{MIntro}
  \eng{
This part describes how symmetric functions are represented in \Mex{SYMF}.
 
Symmetric function names are compatible with Macdonald's conventions.
 
First of all, one may consider two types of bases:  multiplicative bases
and non-multiplicative ones.
 
The bases declared by default are the following:
 
\begin{itemize}
\item[\Mex{e}:]
               multiplicative   basis  of  elementary  symmetric functions.
               The generators of such  a  basis  are  \Mex{e[]}, \Mex{e[1]},
               \Mex{e[2]}, ..., and  the   elements  of  the  basis  are
               the  monomials  in these variables, which are denoted by 
               \Mex{e[i1,i2,...,ik]}, the indexing vector being a partition.
 
\item[\Mex{h}:]
               multiplicative   basis of  complete  homogeneous  symmetric
               functions. The generators of such  a  basis are  \Mex{h[]},
               \Mex{h[1]}, \Mex{h[2]}, ..., and  the elements  of the  basis
               are the monomials in these variables, which are denoted by
               \Mex{h[i1,i2,...,ik]}, the indexing vector being a partition.
 
\item[\Mex{m}:]
               non-multiplicative  basis of monomial  symmetric  functions.
               The elements of  such a basis  are  indexed  by  partitions.
               For  instance, \Mex{m[3,3,1]} and \Mex{m[]} are elements of
               the m-basis.
 
\item[\Mex{p}:]
               multiplicative   basis of  power sum   symmetric  functions.
               The generators of  such a  basis  are  \Mex{p[1]}, \Mex{p[2]},
               ..., and the  elements  of  the  basis  are  the  monomials in
               these variables, which are denoted by \Mex{p[i1,i2,...,ik]},
               the indexing vector being a partition.

\item[\Mex{s}:]
               non-multiplicative  basis of Schur functions. The  elements
               of such a basis are  indexed  by  partitions. For  instance,
               \Mex{s[3,3,1]} and \Mex{s[]} are elements of the s-basis.
\end{itemize}

}
\end{MIntro}

\begin{MExample}{ex1}
\begin{Mexin}
muEC::TYP::Ish( h[3,1] );
\end{Mexin}
\begin{Mexout}
                             TRUE
\end{Mexout}
\begin{Mexin}
muEC::TYP::Iss( s[2,2,1] );
\end{Mexin}
\begin{Mexout}
                             TRUE
\end{Mexout}
\end{MExample}

\MRelatedFunctions{Ise,Ish,Ism,Isp,Iss,IsPart,SfA}
\end{Mhelppage}
