% $Date: 2004/10/04 17:25:12 $ $Author: nthiery $ $Revision: 1.1.1.2 $

\begin{Mhelppage}{ToYY}{S. Veigneau}

  \begin{MInfo}
    \eng{converts any expression to the YY Schubert basis}
  \end{MInfo}
      
\begin{MCalls}
 \Mcall{expr}
\end{MCalls}
%
\begin{MParameters}
  \param{expr}
  \eng{any expression}
  \param{b}
  \eng{any name of a known basis} 
\end{MParameters}
%
\begin{MIntro}
  \eng{
The {\Mthis} function converts any  expression expr to the YY Schubert basis.
expr  may  involve  some  \Mex{x}i's,  simple  Schubert  polynomials
(\Mex{X[perm]}, \Mex{Y[code]}), double Schubert polynomials
(\Mex{XX[perm]}, \Mex{YY[code]}, the second alphabet being the \Mex{y}i's),
other terms being considered as coefficients.

The expression \Mex{expr} is expanded and the result is not collected.

One may specify by a second argument, say \Mpar{b}, that expr is solely
expressed in terms of the known basis \Mpar{b} (\Mex{x}, \Mex{X}, \Mex{Y},
\Mex{XX}, \Mex{YY}, \Mex{Pe} and even \Mex{y} that is seen as a basis in the
package).

The call \Mex{{\Mthis}(expr, YY)} does not affect the argument \Mex{expr}, but
it simplifies Schubert polynomials indices.

One may add \Mex{NoExpand} just after the argument \Mex{expr} to choose not to
expand the expression expr before treating it.

One may collect the result by adding a third argument: this is done by
\Mex{ToYY(expr, b, Collect)}.
For  instance,  \Mex{{\Mthis}(expr, YY, Collect)} may be used to collect the
argument \Mex{expr}.
}
\end{MIntro}
%
\begin{MExample}{ex1}
%
\begin{Mexin}
muEC::SP::ToYY((1+q)^5*x3*x4, NoExpand);
\end{Mexin}
\begin{Mexout}
       5
(q + 1)  (y3 YY[0, 0, 1, 0] - y4 YY[0, 1, 0] +

   y3 (y4 YY[0] - YY[0, 0, 1, 0] + YY[0, 0, 0, 1, 0]) +

   YY[0, 2, 0, 0] + YY[0, 0, 1, 1, 0] - YY[0, 1, 0, 1, 0])
\end{Mexout}
\begin{Mexin}
muEC::SP::ToYY(q^2*x3*X[3,1,2], Collect);
\end{Mexin}
\begin{Mexout}
  2       2                        2       2
(q  y1 + q  y2) YY[1, 0, 1, 0] - (q  y1 + q  y2) YY[1, 1, 0] +

     2          2                    2
   (q  y1 y3 + q  y2 y3) YY[1, 0] - q  YY[2, 1, 0] +

    2                   2   2
   q  YY[2, 0, 1, 0] - q  y1  YY[0, 1, 0] +

    2   2                   2                   2   2
   q  y1  YY[0, 0, 1, 0] - q  y1 YY[2, 0, 0] + q  y1  y3 YY[0]
\end{Mexout}
\end{MExample}
%
\MRelatedFunctions{ToXX}
\end{Mhelppage}
