FUNCTION: Tox - express any expression in the basis of monomials

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 expresses any expression expr in the basis of monomials. 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 Tox(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 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*X[3,1,2]):                         # expands the input
> Tox((1+q)^5*X[3,1,2],noexpand):                # does not expand (1+q)^5
> Tox((1+q)^5*X[3,1,2],collect):                 # collects the result
> Tox((1+q)^5*X[3,1,2],noexpand,'X'):            # the most efficient
> Tox((1+q)^5*X[3,1,2],'X',collect):             # specifies a basis
> Tox((1+q)^5*X[3,1,2],noexpand,'X',collect):
> factor( Tox(q^2*x3*XX[3,1,2]*Y[0,0,1]) );
 
                                   2
                   (x3 + x1 + x2) q  x3 (x1 - y2) (x1 - y1)
 
> Tox(X[1,3,2]*XX[2,3,1] - q^2*YY[0,1], collect);
 
   2       2        2                     2        2         2    2
  q  y2 + q  y1 - x2  y1 - 2 x2 y1 x1 + x2  x1 + x1  x2 + (y1  - q ) x2
 
             2    2         2
        + (y1  - q ) x1 - x1  y1