   Products of cycle indices

#### Products of cycle indices

Let GX and HY be two finite group actions then the direct product G´H induces group actions on the sets XÈY, X´Y or YX and the wreath product  H wr XG:={(y,g) | yÎHX, gÎG} with a multiplication (y,g)(y', g')=(yy'g, gg'), where yy'g(x):=y(x)y'g (x) and y'g(x):=y'(g-1x), acts in a natural way on the sets X´Y and YX.

In his famous article  Pólya demonstrated how to compute some of these cycle indices.

The direct sum of GX and HY acts on the disjoint union X DOTCUP Y by

(G´H)´(X DOTCUP Y) -> X DOTCUP Y,
where (g,h)(z)=gz for zÎX and (g,h)(z)=hz for zÎY.

The direct product of GX and HY acts on the cartesian product X´Y by

(G´H)´(X´Y) -> X´Y
((g,h),(x,y)) -> (gx,hy).
The cycle indices of these actions can be computed from the cycle indices of the actions GX and HY.

The corresponding routines in SYMMETRICA are

```INT zykelind_dir_summ(a,b,c)     OP a,b,c;
INT zykelind_dir_prod(a,b,c)     OP a,b,c;
```
In both cases `a` is Z(G,X), `b` is Z(H,Y), and `c` is the constructed cycle index. `c` must be different from `a` and `b`.

In order to compute the direct sum or the direct product of n copies of the same cycle index one can use

```INT zykelind_hoch_dir_summ(a,b,c)  OP a,b,c;
INT zykelind_hoch_dir_prod(a,b,c)  OP a,b,c;
```
Here in this context `b` is an INTEGER object, which tells, how many copies of `a` shall be combined to compute `c`. `a` `b` and `c` must be different.

Furthermore there are

```INT zykelind_dir_summ_apply(a,b)     OP a,b;
INT zykelind_dir_prod_apply(a,b)     OP a,b;
```
which compute `b` to be the direct sum or the direct product of the two cycle indices `a` and `b`.

The wreath product of G and H acts on the cartesian product X´Y by

H wr XG´(X´Y) -> X´Y
((y,g)(x,y)) -> (gx, y(x)y).
This action is called the composition  G[H] of G and H. In the case that X={1,...,n} and Y={1,...,m} there is a similar action of H wr G on the set {1,...,nm} , which is called the plethysm  H pleth G of G and H.

The following action of the wreath product on YX is called the exponentiation of H by G. It is given by

H wr X G´YX -> YX         ((y,g),f) -> y(·)f(g-1·).
The cycle indices of these actions can be computed by
```INT zykelind_kranz(a,b,c)                 OP a,b,c;
INT zykelind_plethysm(b,a,c)              OP a,b,c;
INT zykelind_exponentiation(a,b,c)        OP a,b,c;
```
In all these cases `a` is Z(G,X), `b` is Z(H,Y), and `c` is the cycle index of the wreath product action. `c` must be different from `a` and `b`. It should be mentioned that `zykelind_kranz(a,b,c)` equals `zykelind_plethysm(b,a,c)`.
harald.fripertinger@kfunigraz.ac.at,
last changed: November 19, 2001   Products of cycle indices