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## `SYMF::SfEval` -- realizes the action of symmetric functions on polynomials

#### Call(s)

`SYMF::SfEval(sf,expr)`

#### Parameters

`sf` | - | any symmetric function
`expr` | - | any expression depending on some variables

#### Introduction

Symmetric functions can be considered as operators on polynomials (lambda-ring structure of the ring of polynomials). Let `sf` be a power sum `p`[k], then

    SYMF::SfEval(p[k],  c_u u  +  c_v v  +  ...) =
             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=p[i,j,...]`, `SYMF::SfEval``(sf, expr)` is set to be equal to the product `SYMF::SfEval``(p[i], expr)` x `SYMF::SfEval``(p[j], expr)` x...

The definition is extended by linearity to any symmetric function `sf`.

#### Example 1

    >> muEC::SYMF::SfEval( p[2], 3*x*y + 2*z );

                                     2  2      2
                                  3 x  y  + 2 z

    >> muEC::SYMF::SfEval( p[4,3] + q*p[2], 2*x + 3*y );

                         2      2        3      3      4      4
                   q (2 x  + 3 y ) + (2 x  + 3 y ) (2 x  + 3 y )

#### Related Functions

`SfAExpand`, `SfPlethysm`

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MuPAD Combinat, an open source algebraic combinatorics package