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

CALLING SEQUENCE:

SgaOnX(e_1, exp)

SGA[SgaOnX](e_1, exp)

PARAMETERS:

e_1 = any element of the symmetric group algebra

exp = any expression

SYNOPSIS:

• The SgaOnX function realizes the action of an element of the symmetric group 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 SgaOnX(e_1, exp, 'collect'). Moreover, one can use both noexpand and collect options: for instance, SgaOnX(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] --> X[perm] if length(nu) > length(perm) X[perm] --> X[perm] + (x_{i+1} - x_i) 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 SgaOnX and another name used in the same session, use the long form SGA['SgaOnX'].

EXAMPLES:

> with(SGA):
> SgaOnX(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    4                    3               3
    - X[2, 4, 1, 3] z q  - q  X[2, 1] - X[2, 1] q  + X[1, 3, 2] q