FUNCTION: ToY - convert any expression to the Y Schubert basis

CALLING SEQUENCE:

ToY(expr)

ToY(expr, b)

SP[ToY](expr)

SP[ToY](expr, b)

PARAMETERS:

expr = any expression

b = any name of a known basis

SYNOPSIS:

• The ToY function converts any expression expr to the Y 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 ToY(expr, 'Y') does not affect the argument expr.

• 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 ToY(expr, b, 'collect'). For instance, ToY(expr, 'Y', 'collect') may be used to collect the argument expr.

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

EXAMPLES:

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