Injective symmetry classes |

Such an *f* is injective if and only if the mapping described
in the first item of corollary is
injective and corresponding cyclic factors of * bar (g)* and
* bar (h)* have the same length. The number of such mappings is

Õ_{j}[a_{j}( bar (h)) choose a_{j}( bar (g))]a_{j}( bar (g))!,

while the second item of corollary
says that we have to multiply this number by * P _{j}j^{aj( bar (g))}* in order to get the number

Corollary:The number of fixed points of(h,g)onYis_{i}^{X}and hence, by restriction, the numbers of fixed points of| Y^{X}_{i,(h,g)}| = Õ_{j}[a_{j}( bar (h)) choose a_{j}( bar (g))] j^{aj( bar (g))}a_{j}( bar (g))!,gand ofhare:| Y^{X}_{i,g}| = [ | Y | choose | X | ] | X | ! if bar (g)=1and| Y^{X}_{i,g}| = 0 otherwise,| Y^{X}_{i,h}| = [a_{1}( bar (h)) choose | X | ] | X | !.

An application of the Cauchy-Frobenius Lemma yields the desired number of injective symmetry classes:

Theorem:The number of injectiveH ´G-classes isso that we obtain by restriction the number of injective| (H ´G) \\Y_{i}^{X}| =(1)/(| H | | G |)å_{(h,g)}Õ_{j}[a_{j}( bar (h)) choose a_{j}( bar (g))]j^{aj ( bar (g))}a_{j}( bar (g))!,G-classesand the number of injective| G \\Y_{i}^{X}| =(| X | !)/(| bar (G) |)[ | Y | choose | X | ]= [ | Y | choose | X | ] | S_{X}/ bar (G) | ,H-classes| H \\Y_{i}^{X}| =(| X | !)/(| H |)å_{k= | X | }^{ | Y | }| {h ÎH | a_{1}( bar (h))=k } | [k choose | X | ].

Try to compute the number of injective 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 |

Injective symmetry classes |