FUNCTION: ToXfix - convert any expression to the X Schubert basis of the ring of polynomials as a free module over Sym

CALLING SEQUENCE:

ToXfix(expr)

ToXfix(expr, b)

FM[ToXfix](expr)

FM[ToXfix](expr, b)

PARAMETERS:

expr = any expression

b = any name of a known basis

SYNOPSIS:

• The ToXfix function converts any expression expr to the X Schubert basis of the ring of polynomials as a free module over symmetric polynomials. expr may involve some xi's, simple Schubert polynomials (X[perm], Y[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).

• The call ToXfix(expr, 'X') 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 ToXfix(expr, b, 'collect'). For instance, the following instruction ToXfix(expr, 'X', 'collect') may be used to collect the argument expr.

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

EXAMPLES:

> with(FM):
> FM_n(4);
 
                                      4
 
> ToXfix((1+q)^2*x3*x4):                         # expands the input
> ToXfix((1+q)^2*x3*x4,noexpand):                # does not expand (1+q)^2
> ToXfix((1+q)^2*x3*x4,collect):                 # collects the result
> ToXfix((1+q)^2*x3*x4,noexpand,'x'):            # the most efficient
> ToXfix((1+q)^2*x3*x4,'x',collect):             # specifies a basis
> ToXfix((1+q)^2*x3*x4,noexpand,'x',collect):
> ToXfix(q^2*x3*X[3,1,4,2]*Y[0,1], collect);
 
    2              2                  2
 - q  X[1, 3, 2, 4]  X[1, 2, 3, 4] + q  s[1] X[3, 4, 1, 2]
 
    2                       2                       2
 + q  s[1] X[3, 2, 4, 1] + q  s[1] X[4, 1, 3, 2] + q  X[3, 4, 2, 1]
 
    2                  2
 + q  X[4, 2, 3, 1] - q  s[1, 1, 1] X[2, 3, 1, 4]
 
    2                             2
 - q  s[1, 1, 1] X[3, 1, 2, 4] + q  s[1, 1, 1, 1] X[1, 3, 2, 4]