FUNCTION: NcaOnXfix - action of an element of the nilCoxeter algebra on the ring of polynomials as a free module over Sym

CALLING SEQUENCE:

NcaOnXfix(e_1, exp)

NCA[NcaOnXfix](e_1, exp)

PARAMETERS:

e_1 = any element of the nilCoxeter algebra

exp = any expression

SYNOPSIS:

• The NcaOnXfix function realizes the action of an element of the nilCoxeter, say e_1, on an expression exp expressed on the X Schubert basis with coefficients that are symmetric polynomials in the basis of Schur functions.

• The expression exp is expanded and the result is not collected.

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

• One may collect the result by adding a third argument: this is done by NcaOnXfix(e_1, exp, 'collect'). Moreover, one can use both noexpand and collect options: NcaOnXfix(e_1, exp, 'noexpand', 'collect').

• A simple divided difference Di acts on a Schubert polynomial X[perm] by sending it to 0 if perm[i] < perm[i+1], or to X[new_perm] where new_perm is obtained by transposing perm[i] and perm[i+1], if perm[i] > perm[i+1].

• The result is expressed on the X Schubert basis corresponding to a given symmetric group and is not collected.

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

EXAMPLES:

> with(NCA):
> _FMn;
 
                                      4
 
> NcaOnXfix(q*A[1,3,2], s[1,1]*X[2,4,1,3]);
 
                           s[1, 1] q X[2, 1, 4, 3]