FUNCTION: NcaOnY - action of an element of the nilCoxeter algebra on a linear combination of Y[code]

CALLING SEQUENCE:

NcaOnY(e_1, exp)

NCA[NcaOnY](e_1, exp)

PARAMETERS:

e_1 = any element of the nilCoxeter algebra

exp = any expression

SYNOPSIS:

• The NcaOnY function realizes the action of an element of the nilCoxeter algebra, say e_1, on an expression exp expressed on the Y 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 NcaOnY(e_1, exp, 'collect'). Moreover, one can use both noexpand and collect options: for instance, NcaOnY(e_1, exp, 'noexpand', 'collect').

• A simple divided difference Di acts on a Schubert polynomial Y[code] by sending it to 0 if code[i] <= code[i+1], or to Y[new_code] where new_code is defined by new_code[i] = code[i+1] and new_code[i+1] = code[i]-1, if code[i] > code[i+1].

• The result is expressed on the Y Schubert basis and is not collected.

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

EXAMPLES:

> with(NCA):
> NcaOnY(q^4*A[1,3,2] - q^3*A[2,1],  z*Y[2,1] - Y[1,0,0]);
 
                    4                 3            3
                 z q  Y[2, 0, 0] - z q  Y[1, 1] + q  Y[0, 0]