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## `SYMF::SfTheta` -- applies the Theta-automorphism

#### Call(s)

`SYMF::SfTheta(sf,q <,t>)`

#### Parameters

`sf` | - | any symmetric function
`q, t` | - | any names or expressions

#### Introduction

The `SYMF::SfTheta` function realizes a certain multiplicative automorphism of the ring of symmetric functions. It is defined on power-sum functions as follows:

- `SYMF::SfTheta``(sf, q)` gives the image of sf under the transformation: p[i] -->> q x p[i].

- `SYMF::SfTheta``(sf, q, t)` is the image of `sf` under the transformation: p[i] -->> (1-q^i)/(1-t^i) x p[i].

The result is given in the `p`-basis.

#### Example 1

    >> muEC::SYMF::SfTheta( s[4,1], q );

            5                            4
          (q  p[1, 1, 1, 1, 1]) 1/30 + (q  p[2, 1, 1, 1]) 1/6 +

               3                     2
             (q  p[3, 1, 1]) 1/6 - (q  p[3, 2]) 1/6 - (q p[5]) 1/5

    >> muEC::SYMF::SfTheta( p[3], q, t );

                                     3
                                   (q  - 1) p[3]
                                   -------------
                                        3
                                       t  - 1

    >> muEC::SYMF::SfTheta( s[2,1], q, t );

                /        3            \       /   3           \
                | (q - 1)  p[1, 1, 1] |       | (q  - 1) p[3] |
                | ------------------- | 1/3 - | ------------- | 1/3
                |              3      |       |      3        |
                \       (t - 1)       /       \     t  - 1    /

#### Related Functions

`SfOmega`

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MuPAD Combinat, an open source algebraic combinatorics package