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

\begin{Mhelppage}{SfOmega}{V. Prosper}%
  \begin{MInfo}
    \eng{applies the \Mmath{Omega}{\Omega}-automorphism}
  \end{MInfo}
  

\begin{MCalls}
  \Mcall{sf}
\end{MCalls}

\begin{MParameters}
  \param{sf}
  \eng{any symmetric function} 
\end{MParameters}

\begin{MIntro}
  \eng{

The {\Mthis} function applies the omega-automorphism to the symmetric
function \Mex{sf}.

This involution is defined as:
\Mdmath{%
\begin{itemize}
\item
\Mex{h[i] ->> e[i]},
\item
\Mex{e[i] ->> h[i]},
\item
\Mex{p[i] ->> (-1)\^(i-1) p[i]}.%
\end{itemize}
}%
{%
\begin{array}{ccl}
\mathtt{e}[i]&\to&\mathtt{h}[i],\\
\mathtt{h}[i]&\to&\mathtt{e}[i],\\
\mathtt{p}[i]&\to&(-1)^{i-1}\mathtt{p}[i]%
\end{array}
}

In terms of Schur functions, the involution conjugates the indexing partition.
}
\end{MIntro}

\begin{MExample}{ex1}
\begin{Mexin}
muEC::SYMF::SfOmega( h[3] );
\end{Mexin}
\begin{Mexout}
                             e[3]
\end{Mexout}
\begin{Mexin}
muEC::SYMF::SfOmega( p[4] );
\end{Mexin}
\begin{Mexout}
                             -p[4]
\end{Mexout}
\end{MExample}

\MRelatedFunctions{PART::Part2Conjugate,SfTheta}
\end{Mhelppage}
