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

CALLING SEQUENCE:

HekaOnPol(e_1, exp)

HekaOnPol(e_1, exp, v)

HEKA[HekaOnPol](e_1, exp)

HEKA[HekaOnPol](e_1, exp, v)

PARAMETERS:

e_1 = any element of the Hecke algebra of the symmetric group

exp = any expression

v = any (extra) string

SYNOPSIS:

• The HekaOnPol function realizes the action of an element of the Hecke algebra of the symmetric group, say e_1, on an expression exp.

• The action of the element SgTranspo(i,n) is : (exp - exp') (q2 x_i + q1 x_{i+1}) exp --> ---------------------------------- + q1 exp (x_i - x_{i+1}) where exp' is the image of exp under SgTranspo(i,n).

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

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

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

EXAMPLES:

> with(HEKA):
> HekaOnPol(A[2,1], 1);
 
                                    q1
 
> HekaOnPol(A[2,1], x1);
 
                          x2 q1 + x1 q2 + q1 x1
 
> HekaOnPol(a^2*A[3,1,2], x1^2*x2*y1*y2^3);
 
     2    2   2        2                                       2
    a  (x3  q1  + x3 q1  x1 + x2 q2 x3 q1 + x3 x1 q2 q1 + x3 q1  x2
 
                       2           2        2           2   2          3
+ 2 x2 x1 q2 q1 + x1 q2  x2 + x2 q1  x1 + x2  q1 q2 + x2  q1 ) x1 y1 y2
 
> HekaOnPol(a^2*A[3,1,2], x1^2*x2*y1^2*y2^3, `y`);
 
      2    2           2                           2          2      3
   - a  (y2  q2 + q1 y2  + y2 y3 q2 + y2 y3 q1 + y3  q1) q2 x1  x2 y1