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

CALLING SEQUENCE:

IdcaOnPol(e_1, exp)

IdcaOnPol(e_1, exp, v)

IDCA[IdcaOnPol](e_1, exp)

IDCA[IdcaOnPol](e_1, exp, v)

PARAMETERS:

e_1 = any element of the idCoxeter algebra

exp = any expression

v = any (extra) string

SYNOPSIS:

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

• The action of the element SgTranspo(i,n) is: (exp - exp') exp -> --------------- x_{i+1} x_i - x_{i+1} where exp' is the image of exp under the exchange of x_i and 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 IdcaOnPol function acts on the variables y1, y2, y3, ...

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

EXAMPLES:

> with(IDCA):
> IdcaOnPol(q^2*A[3,1,2], x1^2*x2*y1^2*y2^3);
 
                       2             3   2
                      q  (x2 + x3) y2  y1  x1 x3
 
> IdcaOnPol(q^2*A[3,1,2], x1^2*x2*y1^2*y2^3, `y`);
 
                   2    2             2    2      2
                - q  (y2  + y3 y2 + y3 ) y1  x2 x1  y3