FUNCTION: NcaOnPol - action of an element of the algebra on a polynomial

CALLING SEQUENCE:

NcaOnPol(e_1, exp)

NcaOnPol(e_1, exp, v)

NCA[NcaOnPol](e_1, exp)

NCA[NcaOnPol](e_1, exp, v)

PARAMETERS:

e_1 = any element of the nilCoxeter algebra

exp = any expression

v = any (extra) string

SYNOPSIS:

• The NcaOnPol function realizes the action of an element of the nilCoxeter algebra, say e_1, on an expression exp.

• Simple divided differences Di are defined to be the operators: f(..., x_i, x_{i+1}, ...) - f(..., x_{i+1}, x_i, ...) f --> ----------------------------------------------------- x_i - x_{i+1}

• By default the algebra acts on the variables x1, x2, x3, ...

• When called with a third argument v, being, say `y`, the NcaOnPol function acts on the variables y1, y2, y3, ...

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

EXAMPLES:

> with(NCA):
> NcaOnPol(q^2*A[3,1,2], x1^2*x2*y1^2*y2^3);
 
                                2      2   3
                               q  x1 y1  y2
 
> NcaOnPol(q^2*A[3,1,2], x1^2*x2*y1^2*y2^3, `y`);
 
                           2             2   2
                        - q  (y2 + y3) y1  x1  x2