FUNCTION: SfEval - realize the action of symmetric functions on polynomials

CALLING SEQUENCE:

SfEval(sf, expr)

SYMF[SfEval](sf, expr)

PARAMETERS:

sf = any symmetric function

expr = any expression depending on some variables

SYNOPSIS:

• Symmetric functions can be considered as operators on polynomials (lambda-ring structure of the ring of polynomials). Let sf be a power- sum pk, then SfEval(pk, c_u u + c_v v + ...) is equal to c_u u^k + c_v v^k + ... where u, v, ... are monomials and c_u, c_v, ... are scalars.

• For a product of power-sums sf=pi pj ..., SfEval(sf, expr) is set to be equal to the product SfEval(pi, expr) * SfEval(pj, expr) * ... The definition is extended by linearity to any symmetric function sf.

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

EXAMPLES:

> with(SYMF):
> SfEval(p2, 3*x*y + 2*z);
 
                                   2  2      2
                                3 x  y  + 2 z
 
> SfEval(p3*p4 + q*p2, 2*x + 3*y);
 
                    3      3      4      4          2      2
                (2 x  + 3 y ) (2 x  + 3 y ) + q (2 x  + 3 y )