Surjective symmetry classes

### Surjective symmetry classes

In order to derive the number of surjective fixed points of (h,g) we use the preceeding corollaries together with an application of the Principle of Inclusion and Exclusion in order to get rid of the nonsurjective fixed points. We denote by Y n the set of points y ÎY contained in the n-th cyclic factor of bar (h) and put, for each index set I Íc ( bar (h)):
YX(h,g),I:= {f ÎYX(h,g) | " nÎI:f-1[Yn] = Æ}.
Then, by the Principle of Inclusion and Exclusion, we obtain for the desired number of surjective fixed points of (h,g) the following expression:
| YXs,(h,g) | = | YX*(h,g) | = åI Íc( bar (h))(-1) | I | | YX(h,g),I |
= åI Íc( bar (h))(-1)c ( bar (h))- | I | | YX(h,g), c( bar (h)) \I | .
Now we recall that
YX(h,g), c( bar (h)) \I= {f ÎYX(h,g) | " n not ÎI:f-1[Yn]= Æ}.
This set can be identified with [~Y] X( [~h] ,g), where [~h] denotes the product of the cyclic factors of bar (h) the numbers of which lie in I, and where [~Y] is the set of points contained in these cyclic factors. Thus
| YX(h,g), c( bar (h)) \I | = | [~Y] X( [~h] ,g) | = Õj | [~Y] [~h] j | aj( bar (g)).
We can make this more explicit by an application of lemma which yields:
| [~Y] [~h] j | =a1( [~h] j)= åd | jd ·ad( [~h] ).
Putting these things together we conclude
Corollary: The number of surjective fixed points of (h,g) is
| YXs,(h,g) | = åc( bar (h))k=1(-1)c( bar ( h))-k åa Õ | Y | i=1[ai( bar (h)) choose ai] Õj=1 | X | ( åd | jd ·ad )aj( bar (g)),
where the middle sum is taken over all the sequences a=(a1, ...,a | Y | ) of natural numbers aj such that åaj =k (they correspond to all possible choices of [~h] out of h, where ai of the chosen cyclic factors of [~h] are i-cycles). Hence the numbers of surjective fixed points of g and of h amount to:
| YXs,g | = åk=1 | Y | (-1) | Y | -k [ | Y | choose k]kc( bar (g)),
and
| YXs,h | = åk=1c( bar (h))(-1)c( bar (h))-k åa( Õi [ai( bar (h)) choose ai])a1 | X | ,
where the sum is taken over all the sequences (a1, ...,a | Y | ), ai Î N and åai=k.
An application of the Cauchy-Frobenius Lemma finally yields the desired numbers of surjective symmetry classes:
Theorem: The number | (H ´G) \\YXs | of surjective H ´G-classes is
(1)/( | H | | G | ) å(h,g) åk=1c( bar (h))(-1)c( bar (h))-k åa Õi=1 | Y | [ai( bar (h)) choose ai] Õj=1 | X | ( åd | jd ·ad )aj( bar (g)),
where the inner sum is taken over the sequences a=(a1, ...a | Y | ) described in the corollary above. This implies, by restriction, the equations
| G \\YXs | =(1)/( | G | ) åg åk=1 | Y | (-1) | Y | -k[ | Y | choose k]kc( bar (g)),
and
| H \\YXs | =(1)/( | H | ) åh åk=1c( bar (h))(-1)c( bar (h))-k åa( Õi[ai( bar (h)) choose ai])a1 | X | ,
where the last sum is to be taken over all the sequences a=(a1, ...,a | Y | ) such that ai Î N and åai=k.
Try to compute the number of surjective symmetry classes for various group actions.
harald.fripertinger@kfunigraz.ac.at,
last changed: August 28, 2001

 Surjective symmetry classes