FUNCTION: SgaOnXfix - action of an element of the symmetric group algebra on the ring of polynomials as a free module over Sym

CALLING SEQUENCE:

SgaOnXfix(e_1, exp)

SGA[SgaOnXfix](e_1, exp)

PARAMETERS:

e_1 = any element of the symmetric group algebra

exp = any expression

SYNOPSIS:

• The SgaOnXfix 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 with coefficients that are symmetric polynomials in the basis of Schur functions.

• 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 SgaOnXfix(e_1, exp, 'collect'). Moreover, one can use both noexpand and collect options: SgaOnXfix(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 corresponding to a given symmetric group and is not collected.

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

EXAMPLES:

> with(SGA):
> _FMn;
 
                                      4
 
> SgaOnXfix(q*A[3,2,1], s[1,1]*X[2,4,1,3], collect);
 
                           3                  3
   (s[2, 1] + s[1, 1, 1]) q  X[1, 3, 4, 2] - q  X[2, 3, 4, 1] s[1, 1]
 
                                              3
    + (s[2, 2] + s[2, 1, 1] + s[1, 1, 1, 1]) q  X[2, 1, 3, 4]
 
                                                       3
    - (s[2, 2, 1] + s[2, 1, 1, 1] + s[1, 1, 1, 1, 1]) q  X[1, 2, 3, 4]
 
                              3                  3
    - (s[2, 1] + s[1, 1, 1]) q  X[2, 1, 4, 3] + q  X[4, 1, 2, 3] s[1, 1]