% $Id: preface.tex,v 1.1.1.1 2003/09/15 16:37:46 nthiery Exp $
\eng{
  \section*{The library \Mex{muEC::SP}}
  
  The \Mex{muEC::SP} library provides functions to work on simple and
  double Schubert polynomials.  Schubert polynomials form a linear
  basis of the ring of multivariate polynomials.
  
  All defined bases are stored in the global variable
  \Mex{muEC::SP::SPBases} that is by default \Mex{[x, y, Pe, X, Y, XX,
    YY]}.  Note that the \Mex{y}i's are seen as special variables
  since they are the second alphabet for double Schubert polynomials.
  
  Schubert polynomials may be indexed by permutations or integer
  vectors.  In the first case, they are denoted by the expression
  \Mex{X[}permutation\Mex{]} and in the second one, by
  \Mex{Y[}code\Mex{]} where code is a sequence of nonnegative integers
  (interpreted as the code of a permutation).
 
  In the case of double Schubert polynomials we denote by
  \Mex{XX[}permutation\Mex{]} the double Schubert polynomial indexed
  by a permutation and by \Mex{YY[}code\Mex{]} the corresponding
  double Schubert polynomial indexed by a vector of non-negative
  integers (interpreted as the code of a permutation).
  
  The \Mex{Pe}-basis is the dual basis of the basis of increasing
  monomials (less than the `staircase' monomial), according to the
  scalar product \Mex{SP::SpScalarPol}.  It is the basis of products
  of elementary symmetric functions on a flag of alphabets.
  }
