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

\begin{Mhelppage}{SfTheta}{V. Prosper}
  
  \begin{MInfo}
    \eng{applies the \Mmath{Theta}{\Theta}-automorphism}
  \end{MInfo}
  
  \begin{MCalls}
    \Mcall{sf,q\Mopt{,t}}
  \end{MCalls}

  \begin{MParameters}
    \param{sf}
    \eng{any symmetric function}
    \param{q, t}
    \eng{any names or expressions}
  \end{MParameters}

  \begin{MIntro}
    \eng{The {\Mthis} function realizes a certain multiplicative
      automorphism of the ring of symmetric functions.  It is defined
      on power-sum functions as follows:
      \begin{itemize}
        \item
        {\Mthis}\Mex{(sf, q)}   gives   the   image  of  sf  under the
        transformation:
        \Mmath{p[i] -->> q x p[i]}{\mathtt{p}[i]\to q\times\mathtt{p}[i]}.
 
        \item
        {\Mthis}\Mex{(sf, q, t)} is the image of \Mex{sf}  under  the  transformation:
        \Mmath{p[i] -->> (1-q\^i)/(1-t\^i) x p[i]}%
        {\mathtt{p}[i]\to (1-q^i)/(1-t^i)\times\mathtt{p}[i]}.
      \end{itemize}
      
      The result is given in the \Mex{p}-basis.
      }
  \end{MIntro}

\begin{MExample}{ex1}
\begin{Mexin}
muEC::SYMF::SfTheta( s[4,1], q );
\end{Mexin}
\begin{Mexout}
  5                            4
(q  p[1, 1, 1, 1, 1]) 1/30 + (q  p[2, 1, 1, 1]) 1/6 +

     3                     2
   (q  p[3, 1, 1]) 1/6 - (q  p[3, 2]) 1/6 - (q p[5]) 1/5
\end{Mexout}
% \begin{Mexout}[hide]
% (q*p[5])*(-1/5) + (q^2*p[3, 2])*(-1/6) + (q^3*p[3, 1, 1])*1/6 +
% (q^4*p[2, 1, 1, 1])*1/6 + (q^5*p[1, 1, 1, 1, 1])*1/30
% \end{Mexout}
\begin{Mexin}
muEC::SYMF::SfTheta( p[3], q, t );
\end{Mexin}
\begin{Mexout}
                           3
                         (q  - 1) p[3]
                         -------------
                              3
                             t  - 1
\end{Mexout}
% \begin{Mexout}[hide]
           % p[3]*(q^3*(-1) + 1)*(t^3*(-1) + 1)^(-1)
% \end{Mexout}
\begin{Mexin}
muEC::SYMF::SfTheta( s[2,1], q, t );
\end{Mexin}
\begin{Mexout}
      /        3            \       /   3           \
      | (q - 1)  p[1, 1, 1] |       | (q  - 1) p[3] |
      | ------------------- | 1/3 - | ------------- | 1/3
      |              3      |       |      3        |
      \       (t - 1)       /       \     t  - 1    /
\end{Mexout}
% \begin{Mexout}[hide]
     % (p[3]*(q^3*(-1) + 1)*(t^3*(-1) + 1)^(-1))*(-1/3) +
     % ((q*(-1) + 1)^3*(t*(-1) + 1)^(-3)*p[1, 1, 1])*1/3
% \end{Mexout}
\end{MExample}

\MRelatedFunctions{SfOmega}
\end{Mhelppage}
