FUNCTION: Flag - fix the degree of the symmetric group

CALLING SEQUENCE:

Flag(n)

SP[Flag](n)

PARAMETERS:

n = any (extra) positive integer

SYNOPSIS:

• To compute in the ring of polynomials modulo the ideal of generated by symmetric polynomials in x1, ..., xn without constant term, use Flag(n). Equivalently, all Schubert polynomials indexed by permutations which cannot be restricted to S(n), i.e. which do not fix n+1, n+2, ..., are put to zero. The ring can be interpreted as the cohomology ring of the flag manifold of rank n over the field of complex numbers.

• In the case of double Schubert polynomials, the ideal is generated by identifying any symmetric polynomial in x to the same polynomial in y. All double Schubert polynomials indexed by permutations which cannot be restricted to S(n) are null.

• Use Flag(-1) to compute in the ring of polynomials, not modulo the ideal. This is the default case.

• When called without any argument, it returns the current value which is
  1 when computing without truncation and a positive integer otherwise.

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

EXAMPLES:

> with(SP):
> Flag(-1):
> x2X(x3);
 
                          - X[1, 3, 2] + X[1, 2, 4, 3]
 
> Flag(3):
> x2X(x3);
 
                                  - X[1, 3, 2]