Enumeration of symmetry classes



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Enumeration of symmetry classes

  Our paradigmatic examples are the actions of , , and on , obtained from given actions and . The orbits of these groups are called symmetry classes of mappings. If we want to be more explicit, we call them -classes, -classes, -classes and -classes, respectively. Their total number can be obtained by an application of the Cauchy-Frobenius Lemma as soon as we know the number of fixed points for each element of the respective group. In order to derive these numbers we characterize the fixed points of each on and then we use the natural embedding of , and in as described above. Thus the following lemma will turn out to be crucial:

. Lemma   Consider an , an element of and assume that

is the disjoint cycle decomposition of , the permutation of which corresponds to . Then is a fixed point of if and only if the following two conditions hold:

Proof: gif says that is fixed under if and only if its values satisfy the equations

where denotes the length of the cyclic factor of containing the point . Hence in particular the following must be true:

which means that is a fixed point of , as claimed. Thus any fixed clearly has the stated properties, and vice versa.

This, together with the Cauchy-Frobenius Lemma, yields the number of -classes on , and the restrictions to the subgroups , and give the numbers of -, - and -classes on :

. Theorem   If both and are finite actions, then we obtain the following expression for the total number of orbits of the corresponding action of on :

The restriction to , and according to gif yields:

and

In order to apply these results to a specific case it remains to evaluate and or which still can be quite cumbersome as the following example shows.

Try to compute the number of symmetry classes of mappings for various group actions. Furthermore you can compute a transversal of G-classes on .



next up previous contents
Next: Graphs Up: Actions Previous: Examples



Herr Fripertinger
Sun Feb 05 18:28:26 MET 1995