FUNCTION: X2Xfix - from X Schubert basis to X Schubert basis of the ring of polynomials as a free module over Sym

CALLING SEQUENCE:

X2Xfix(schub)

FM[X2Xfix](schub)

PARAMETERS:

schub = any expression in X Schubert basis

SYNOPSIS:

• The X2Xfix function converts an expression from X Schubert basis to the X Schubert basis of the ring of polynomials as a free module over symmetric polynomials.

• All Schubert polynomials are indexed by permutations in the symmetric group of degree _FMn. Coefficients are symmetric polynomials expressed in the Schur basis.

• The result is not expanded.

• One may add 'noexpand' just after the argument schub to choose not to expand the expression schub before treating it.

• One may add 'collect' just after the argument schub or just after the argument 'noexpand' to collect the result.

• Whenever there is a conflict between the function name X2Xfix and another name used in the same session, use the long form FM['X2Xfix'].

EXAMPLES:

> with(FM):
> FM_n(4);
 
                                      4
 
> X2Xfix((1+q)^5*X[3,5,2,4,1]):                  # expands the input
> X2Xfix((1+q)^5*X[3,5,2,4,1],noexpand):         # does not expand (1+q)^5
> X2Xfix((1+q)^5*X[3,5,2,4,1],collect):          # collects the result
> X2Xfix((1+q)^5*X[3,5,2,4,1],noexpand,collect):
> X2Xfix(X[3,2,1] + q*X[3,2,1] + z*X[4,5,1,3,2], collect);
 
            (1 + q) X[3, 2, 1, 4] - z s[1, 1, 1, 1] X[2, 4, 1, 3]
 
               + z s[1, 1, 1] X[3, 4, 1, 2]