FUNCTION: ToX - convert any expression to the X Schubert basis

CALLING SEQUENCE:

ToX(expr)

ToX(expr, b)

SP[ToX](expr)

SP[ToX](expr, b)

PARAMETERS:

expr = any expression

b = any name of a known basis

SYNOPSIS:

• The ToX function converts any expression expr to the X Schubert basis. expr may involve some xi's, simple Schubert polynomials (X[perm], Y[code]), double Schubert polynomials (XX[perm], YY[code]), 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 ToX(expr, 'X') 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 ToX(expr, b, 'collect'). For instance, ToX(expr, 'X', 'collect') may be used to collect the argument expr.

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

EXAMPLES:

> with(SP):
> Flag(-1):
> ToX((1+q)^5*x3*x4):                            # expands the input
> ToX((1+q)^5*x3*x4,noexpand):                   # does not expand (1+q)^5
> ToX((1+q)^5*x3*x4,collect):                    # collects the result
> ToX((1+q)^5*x3*x4,noexpand,'x'):               # the most efficient
> ToX((1+q)^5*x3*x4,'x',collect):                # specifies a basis
> ToX((1+q)^5*x3*x4,noexpand,'x',collect):
> ToX(q^2*x3*XX[3,1,2]*Y[0,0,1], collect);
 
     2                        2
  - q  y2 y1 X[1, 4, 2, 3] + q  y2 y1 X[1, 2, 5, 3, 4]
 
     2                            2
  + q  (y1 + y2) X[4, 1, 2, 3] + q  (y1 + y2) X[2, 4, 1, 3]
 
     2                  2                  2
  - q  X[4, 2, 1, 3] - q  X[3, 4, 1, 2] - q  (y1 + y2) X[2, 1, 5, 3, 4]
 
     2
  + q  X[3, 1, 5, 2, 4]