% $Date: 2003/09/15 16:37:45 $ $Author: nthiery $ $Revision: 1.1.1.1 $

\begin{Mhelppage}{SfAOmega}{V. Prosper}%
  \begin{MInfo}
    \eng{applies the \Mmath{Omega}{\Omega}-automorphism}
  \end{MInfo}
  
  \begin{MCalls}
    \Mcall{sfa\Mopt{,alist}}
  \end{MCalls}

  \begin{MParameters}
    \param{sfa}
    \eng{any symmetric function} 
    \param{alist}
    \eng{a list of alphabets} 
  \end{MParameters}

  \begin{MIntro}
    \eng{The {\Mthis} function applies the
      \Mmath{Omega}{\Omega}-automorphism to the symmetric function
      \Mex{sfa}.

      This involution is defined as:
      \Mdmath{%
        \begin{itemize}
          \item
          {\Mex{h[i](A) ->> e[i](A)},}
          \item
          {\Mex{e[i](A) ->> h[i](A)},}
          \item
          {\Mex{p[i](A) ->> (-1)\^(i-1) p[i](A)}.}%
        \end{itemize}
        }%
      {%
        \begin{array}{ccl}
          \mathtt{e}[i](A)&\to&\mathtt{h}[i](A),\\
          \mathtt{h}[i](A)&\to&\mathtt{e}[i](A),\\
          \mathtt{p}[i](A)&\to&(-1)^{i-1}\mathtt{p}[i](A).%
        \end{array}
        }
      
      In terms of Schur functions, the involution conjugates the
      indexing partition.
      
      One can apply {\Mthis} solely on symmetric functions over the
      alphabets given in the second parameter \Mex{alist}.  }
  \end{MIntro}

\begin{MExample}{ex1}
\begin{Mexin}
muEC::SFA::SfAOmega( p[3,1](A1) - q*s[3](A2) );
\end{Mexin}
\begin{Mexout}
                p[3, 1](A1) - q s[1, 1, 1](A2)
\end{Mexout}
% \begin{Mexout}[hide]
             % p[3, 1](A1) + q*s[1, 1, 1](A2)*(-1)
% \end{Mexout}
\begin{Mexin}
muEC::SFA::SfAOmega( p[3,1](A1) - q*s[3](A2), [ A1 ]);
\end{Mexin}
\begin{Mexout}
                   p[3, 1](A1) - q s[3](A2)
\end{Mexout}
% \begin{Mexout}[hide]
                % q*s[3](A2)*(-1) + p[3, 1](A1)
% \end{Mexout}
\end{MExample}

\MRelatedFunctions{SYMF::SfOmega,PART::Part2Conjugate}
\end{Mhelppage}
