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## `SYMF::SfInternal` -- computes the inner tensor product of symmetric functions

#### Call(s)

`SYMF::SfInternal(sf1, sf2)`

#### Parameters

sf1, sf2 = any symmetric functions

#### Introduction

The `SYMF::SfInternal` function computes the inner tensor product of symmetric functions `sf1` and `sf2`. This operation is also called the internal product. Since `sf1` and `sf2` may be seen as virtual characters, `SYMF::SfInternal` computes the symmetric function associated with the product (Kronecker product) of these two virtual characters.

For the basis of products of power-sums, `SYMF::SfInternal`(`p[part], p[part']`) = 0 if `part`<>`part'`, = `SfZee(part) x p[part]` otherwise.

The result is expressed in the product of power-sums basis.

#### Example 1

    >> muEC::SYMF::SfInternal( p[3,3,2,1,1], p[3,3,2,1,1] );

                                72 p[3, 3, 2, 1, 1]

    >> muEC::SYMF::SfZee( [3,3,2,1,1] );

                                        72

    >> muEC::SYMF::SfInternal( s[3,2,1], p[3]*h[3] );

                              2 p[3, 3]   p[3, 1, 1, 1]
                            - --------- - -------------
                                  3             3

#### Related Functions

`Char2Sf`, `SfPlethysm`, `Sf2Char`, `SfZee`

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MuPAD Combinat, an open source algebraic combinatorics package