Injective symmetry classes |

while the second item of corollary says that we have to multiply this number byÕ_{j}[a_{j}( bar (h)) choose a_{j}( bar (g))]a_{j}( bar (g))!,

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

Try to compute the number of injective symmetry classes for various group actions.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 | ].

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001

Injective symmetry classes |