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^{X}_{(h,g),I}:= {f ÎY^{X}_{(h,g)}| " nÎI:f^{-1}[Y_{n}] = 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:

| Y^{X}_{s,(h,g)}| = | Y^{X*}_{(h,g)}| = å_{I Íc( bar (h))}(-1)^{ | I | }| Y^{X}_{(h,g),I}|

= å_{I Íc( bar (h))}(-1)^{c ( bar (h))- | I | }| Y^{X}_{(h,g), c( bar (h)) \I}| .

Now we recall that

Y^{X}_{(h,g), c( bar (h)) \I}= {f ÎY^{X}_{(h,g)}| " n not ÎI:f^{-1}[Y_{n}]= Æ}.

This set can be identified with * [~Y] ^{X}_{( [~h] ,g)}*, where

| Y^{X}_{(h,g), c( bar (h)) \I}| = | [~Y]^{X}_{( [~h] ,g)}| = Õ_{j}| [~Y]_{ [~h] j}|^{aj( bar (g))}.

We can make this more explicit by an application of lemma which yields:

| [~Y]_{ [~h] j}| =a_{1}( [~h]^{j})= å_{d | j}d ·a_{d}( [~h] ).

Putting these things together we conclude

Corollary:The number of surjective fixed points of(h,g)iswhere the middle sum is taken over all the sequences| Y^{X}_{s,(h,g)}| = å^{c( bar (h))}_{k=1}(-1)^{c( bar ( h))-k}å_{a}Õ^{ | Y | }_{i=1}[a_{i}( bar (h)) choose a_{i}] Õ_{j=1}^{ | X | }( å_{d | j}d ·a_{d})^{aj( bar (g))},a=(aof natural numbers_{1}, ...,a_{ | Y | })asuch that_{j}åa(they correspond to all possible choices of_{j}=k[~h]out ofh, whereaof the chosen cyclic factors of_{i}[~h]arei-cycles). Hence the numbers of surjective fixed points ofgand ofhamount to:and| Y^{X}_{s,g}| = å_{k=1}^{ | Y | }(-1)^{ | Y | -k}[ | Y | choose k]k^{c( bar (g))},where the sum is taken over all the sequences| Y^{X}_{s,h}| = å_{k=1}^{c( bar (h))}(-1)^{c( bar (h))-k}å_{a}( Õ_{i}[a_{i}( bar (h)) choose a_{i}])a_{1}^{ | X | },(a,_{1}, ...,a_{ | Y | })aand_{i}ÎNåa._{i}=k

An application of the Cauchy-Frobenius Lemma finally yields the desired numbers of surjective symmetry classes:

Theorem:The number| (H ´G) \\Yof surjective^{X}_{s}|H ´G-classes iswhere the inner sum is taken over the sequences(1)/(| H | | G |)å_{(h,g)}å_{k=1}^{c( bar (h))}(-1)^{c( bar (h))-k}å_{a}Õ_{i=1}^{ | Y | }[a_{i}( bar (h)) choose a_{i}] Õ_{j=1}^{ | X | }( å_{d | j}d ·a_{d})^{aj( bar (g))},a=(adescribed in the corollary above. This implies, by restriction, the equations_{1}, ...a_{ | Y | })and| G \\Y^{X}_{s}| =(1)/(| G |)å_{g}å_{k=1}^{ | Y | }(-1)^{ | Y | -k}[ | Y | choose k]k^{c( bar (g))},where the last sum is to be taken over all the sequences| H \\Y^{X}_{s}| =(1)/(| H |)å_{h}å_{k=1}^{c( bar (h))}(-1)^{c( bar (h))-k}å_{a}( Õ_{i}[a_{i}( bar (h)) choose a_{i}])a_{1}^{ | X | },a=(asuch that_{1}, ...,a_{ | Y | })aand_{i}ÎNåa._{i}=k

Try to compute the number of surjective symmetry classes for various group actions.

harald.fripertinger "at" uni-graz.at | http://www-ang.kfunigraz.ac.at/~fripert/ |

UNI-Graz | Institut für Mathematik |

UNI-Bayreuth | Lehrstuhl II für Mathematik |

Surjective symmetry classes |