% $Id: SymmetricFunctions.tex,v 1.1.1.3 2004/10/04 17:25:13 nthiery Exp $
\begin{Mhelppage}{SymmetricFunctions}{N.~Thiéry}
  [ImplState=stable,TestState=stable,DocState=beta]
%\Mproofread{user}{date}{comment}

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

\begin{MIntro}
  \eng{\Mthis\Mex{(R)} creates a domain for the symmetric functions
    over the ground ring $R$. }
\end{MIntro}

\MKeywords{combinatorics, symmetric functions}

\begin{MDetails}
  \eng{}
\end{MDetails}

\begin{MDomEntries}
  \MDomEntry{powersum}
  \MDomEntry{p}
     \eng{The domain of symmetric functions expanded on the powersums basis}
  \MDomEntry{Schur}
  \MDomEntry{s}
     \eng{The domain of symmetric functions expanded on the Schur basis}
  \MDomEntry{elementary}
  \MDomEntry{e}
     \eng{The domain of symmetric functions expanded on the elementary basis}
  \MDomEntry{complete}
  \MDomEntry{h}
     \eng{The domain of symmetric functions expanded on the complete basis}
  \MDomEntry{monomial}
  \MDomEntry{m}
     \eng{The domain of symmetric functions expanded on the monomial basis}
\end{MDomEntries}

\begin{MMethodDetails}{antipode}[DocState=beta]
  \infoEng{antipode of symmetric functions}
  \call{\Mtype{symmetric function} f}
  %
  \eng{Returns the antipode of the symmetric function \Mpar{f}; the
    result need not be expressed in the same basis as \Mpar{f}.}
\end{MMethodDetails}

\begin{MMethodDetails}{omega}[DocState=beta]
  \infoEng{omega operator on symmetric function}
  \call{\Mtype{symmetric function} f}
  %
  \eng{Returns the image of the symmetric function \Mpar{f} by the
    \Mex{omega} operator; the result need not be expressed in the same
    basis as \Mpar{f}.}
  
  \eng{omega is an algebra homomorphism and an involution. For any
    homogeneous symmetric function $f$, and alphabet $X$, one has
    $\omega f(X) = (-1)^{\deg f} f(-X)$.}
\end{MMethodDetails}

\begin{MMethodDetails}{fromPoly}[DocState=beta]
  \infoEng{conversion from symmetric polynomials}
  \call{\Mtype{symmetric polynomial (in \Mex{DOM_POLY})} f}
  %
  \eng{Express the symmetric polynomial $f$ in the monomial basis.}
  \eng{The polynomial $f$ is assumed to be symmetric; no check is done.}
\end{MMethodDetails}

See the \Mup[index,demonstration]{demonstration} for examples.

%\begin{MBackground}
%\end{MBackground}

\begin{MChanges}
\end{MChanges}

\end{Mhelppage}

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