FUNCTION: ToXX - convert any expression to the XX Schubert basis

CALLING SEQUENCE:

ToXX(expr)

ToXX(expr, b)

SP[ToXX](expr)

SP[ToXX](expr, b)

PARAMETERS:

expr = any expression

b = any name of a known basis

SYNOPSIS:

• The ToXX function converts any expression expr to the XX Schubert basis. expr may involve some xi's, simple Schubert polynomials (X[perm], Y[code]), double Schubert polynomials (XX[perm], YY[code], the second alphabet being the yi's), other terms being considered as coefficients.

• The expression expr is expanded and the result is not collected.

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

• The call ToXX(expr, 'XX') does not affect the argument expr, but it simplifies Schubert polynomials indices.

• One may add 'noexpand' just after the argument 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 ToXX(expr, b, 'collect'). For instance, ToXX(expr, 'XX', 'collect') may be used to collect the argument expr.

• Whenever there is a conflict between the function name ToXX and another name used in the same session, use the long form SP['ToXX'].

EXAMPLES:

> with(SP):
> Flag(-1):
> ToXX((1+q)^5*x3*x4):                           # expands the input
> ToXX((1+q)^5*x3*x4,noexpand):                  # does not expand (1+q)^5
> ToXX((1+q)^5*x3*x4,collect):                   # collects the result
> ToXX((1+q)^5*x3*x4,noexpand,'x'):              # the most efficient
> ToXX((1+q)^5*x3*x4,'x',collect):               # specifies a basis
> ToXX((1+q)^5*x3*x4,noexpand,'x',collect):
> ToXX(q^2*x3*X[3,1,2], collect);
 
  2                   2                             2   2
 q  XX[3, 1, 4, 2] + q  (y2 + y1) XX[2, 1, 4, 3] + q  y1  XX[1, 2, 4, 3]
 
      2                2                          2   2
   - q  XX[3, 2, 1] - q  (y2 + y1) XX[2, 3, 1] - q  y1  XX[1, 3, 2]
 
      2                   2                          2   2
   - q  y1 XX[3, 1, 2] + q  y3 (y2 + y1) XX[2, 1] + q  y1  y3 XX[1]