| | | **Products of cycle indices** |

#### Products of cycle indices

Let _{G}X and _{H}Y be two finite group actions then the direct product
*G´H* induces group actions on the sets *XÈY*, *X´Y* or
*Y*^{X} and the wreath product
*H wr *_{X}G:={(y,g) | yÎH^{X}, 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^{-1}x),
acts in a natural way on the sets *X´Y* and *Y*^{X}.
In his famous article [13] Pólya demonstrated how to
compute some of these cycle indices.

The direct sum
of _{G}X and _{H}Y 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 _{G}X and _{H}Y 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 _{G}X and _{H}Y.
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 *_{X}G´(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 *Y*^{X} is called the
*exponentiation*
of *H* by *G*. It is given by

*H wr *_{X} G´Y^{X} -> Y^{X} ((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** |