FUNCTION: ToYY - convert any expression to the YY Schubert basis

CALLING SEQUENCE:

ToYY(expr)

ToYY(expr, b)

SP[ToYY](expr)

SP[ToYY](expr, b)

PARAMETERS:

expr = any expression

b = any name of a known basis

SYNOPSIS:

• The ToYY function converts any expression expr to the YY 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, YY, YY and even y that is seen as a basis in the package).

• The call ToYY(expr, 'YY') 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 ToYY(expr, b, 'collect'). For instance, ToYY(expr, 'YY', 'collect') may be used to collect the argument expr.

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

EXAMPLES:

> with(SP):
> Flag(-1):
> ToYY((1+q)^5*x3*x4):                           # expands the input
> ToYY((1+q)^5*x3*x4,noexpand):                  # does not expand (1+q)^5
> ToYY((1+q)^5*x3*x4,collect):                   # collects the result
> ToYY((1+q)^5*x3*x4,noexpand,'x'):              # the most efficient
> ToYY((1+q)^5*x3*x4,'x',collect):               # specifies a basis
> ToYY((1+q)^5*x3*x4,noexpand,'x',collect):
> ToYY(q^2*x3*Y[2,0], collect);
 
     2                   2                    2       2
  - q  y1 YY[2, 0, 0] + q  YY[2, 0, 1, 0] + (q  y1 + q  y2) YY[1, 0, 1, 0]
 
         2       2                    2   2
   + (- q  y2 - q  y1) YY[1, 1, 0] + q  y1  YY[0, 0, 1, 0]
 
       2          2                    2   2                2      2
   + (q  y3 y2 + q  y3 y1) YY[1, 0] - q  y1  YY[0, 1, 0] + q  y3 y1  YY[0]
 
      2
   - q  YY[2, 1, 0]