FUNCTION: IdcaOnX - action of an element of the idCoxeter algebra on a linear combination of X[perm]

CALLING SEQUENCE:

IdcaOnX(e_1, exp)

IDCA[IdcaOnX](e_1, exp)

PARAMETERS:

e_1 = any element of the symmetric group algebra

exp = any expression

SYNOPSIS:

• The IdcaOnX function realizes the action of an element of the idCoxeter algebra, 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 IdcaOnX(e_1, exp, 'collect'). Moreover, one can use both noexpand and collect options: for instance, IdcaOnX(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] --> 0 if length(nu) > length(perm) X[perm] --> X[nu] x_{i+1} otherwise.

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

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

EXAMPLES:

> with(IDCA):
> IdcaOnX(q^4*A[1,3,2] - q^3*A[2,1], z*X[3,2,1] - X[2,1,3]);
 
                       4                    4                 3
       - X[3, 2, 1] z q  + X[3, 1, 4, 2] z q  + X[3, 2, 1] z q
 
                             3            3               3
          - X[2, 4, 1, 3] z q  - X[2, 1] q  + X[1, 3, 2] q