FUNCTION: x2Xfix - from the basis of monomials to X Schubert basis of the ring of polynomials as a free module over Sym

CALLING SEQUENCE:

x2Xfix(pol)

FM[x2Xfix](pol)

PARAMETERS:

pol = any expression in the basis of monomials

SYNOPSIS:

• The x2Xfix function converts any expression from the basis of monomials 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 pol to choose not to expand the expression pol before treating it.

• One may add 'collect' just after the argument pol 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*x1*x3^3):                       # expands the input
> x2Xfix((1+q)^5*x1*x3^3,noexpand):              # does not expand (1+q)^5
> x2Xfix((1+q)^5*x1*x3^3,collect):               # collects the result
> x2Xfix((1+q)^5*x1*x3^3,noexpand,collect):
> x2Xfix(q*x2^2*x4 - x3^3*x4, collect);
 
   (s[2, 2] - q s[1, 1, 1]) X[1, 2, 3, 4] - X[1, 2, 4, 3] s[2, 1]
 
 + (- q s[1] - s[1, 1]) X[2, 3, 1, 4] + (s[2] + s[1, 1]) X[1, 3, 4, 2]
 
 + s[1, 1] X[1, 4, 2, 3] + (- s[1] + q) X[2, 3, 4, 1]
 
 - q s[1] X[3, 1, 2, 4] + X[3, 4, 1, 2] + (- q - s[1]) X[1, 4, 3, 2]
 
 + q X[3, 1, 4, 2] + X[2, 4, 3, 1] + q s[1, 1] X[1, 3, 2, 4]
 
 + q X[4, 1, 2, 3] + q X[2, 4, 1, 3]