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

CALLING SEQUENCE:

HekaOnXfix(e_1, exp)

HEKA[HekaOnXfix](e_1, exp)

PARAMETERS:

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

exp = any expression

SYNOPSIS:

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

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

EXAMPLES:

> with(HEKA):
> _FMn;
 
                                      4
 
> HekaOnXfix(q*A[1,3,2], s[1,1]*X[2,4,1,3], collect);
 
  q2 X[2, 4, 1, 3] s[1, 1] q + (s[2, 1] + s[1, 1, 1]) q1 q X[2, 1, 4, 3]
 
     + q2 X[2, 3, 4, 1] s[1, 1] q - q1 X[4, 1, 2, 3] s[1, 1] q
 
     - (s[2, 2] + s[2, 1, 1] + s[1, 1, 1, 1]) q1 q X[2, 1, 3, 4]