FUNCTION: HekaOnX - action of an element of the Hecke algebra of the symmetric group on a linear combination of X[perm]

CALLING SEQUENCE:

HekaOnX(e_1, exp)

HEKA[HekaOnX](e_1, exp)

PARAMETERS:

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

exp = any expression

SYNOPSIS:

• The HekaOnX function realizes the action of an element of the Hecke algebra of the symmetric group, 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 HekaOnX(e_1, exp, 'collect'). Moreover, one can use both noexpand and collect options: for instance, HekaOnX(e_1, exp, 'noexpand', 'collect').

• The action of the element SgTranspo(i,n) on the Schubert polynomial X[perm] is : let nu = MultPerm(perm, SgTranspo(i,n)), then X[perm] --> q1 X[perm] if length(nu) > length(perm) X[perm] --> q1 X[perm] + (q2 x_i + q1 x_{i+1}) X[nu] otherwise.

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

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

EXAMPLES:

> with(HEKA):
> HekaOnX(a^4*A[1,3,2] - a^3*A[2,1], z*X[3,2,1] - X[2,1,3]);
 
                      3                       3                    4
   - q2 X[3, 2, 1] z a  - q1 X[2, 4, 1, 3] z a  + q2 X[3, 2, 1] z a
 
                                    4               3
              + q1 X[3, 1, 4, 2] z a  + q2 X[2, 1] a
 
                                 3    4
                + q1 X[1, 3, 2] a  - a  q1 X[2, 1]