FUNCTION: NcaOnX - action of an element of the nilCoxeter algebra on a linear combination of X[perm]

CALLING SEQUENCE:

NcaOnX(e_1, exp)

NCA[NcaOnX](e_1, exp)

PARAMETERS:

e_1 = any element of the nilCoxeter algebra

exp = any expression

SYNOPSIS:

• The NcaOnX function realizes the action of an element of the nilCoxeter algebra, say e_1, on an expression exp expressed on the X Schubert basis.

• 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 NcaOnX(e_1, exp, 'collect'). Moreover, one can use both noexpand and collect options: for instance, NcaOnX(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 and is not collected.

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

EXAMPLES:

> with(NCA):
> NcaOnX(q^4*A[1,3,2] - q^3*A[2,1], z*X[3,2,1] - X[2,1,3]);
 
                  4                 3               3
               z q  X[3, 1, 2] - z q  X[2, 3, 1] + q  X[1, 2]