   Special symmetry classes

## Special symmetry classes

We now return to YX and consider its subsets consisting of the injective and the surjective maps f only:
YXi:= {f ÎYX | f injective } and YsX:= {f ÎYX | f surjective }.
It is clear that each of these sets is both a G-set and an H-set and therefore it is also an H ´G-set, but it will not in general be an H wr X G -set. The corresponding orbits of G,H and H ´G on YiX are called injective symmetry classes, while those on YsX will be called surjective symmetry classes. We should like to determine their number. In order to do this we describe the fixed points of (h,g) ÎH ´G on these sets to prepare an application of the Cauchy-Frobenius Lemma. A first remark shows how the fixed points of (h,g) on YX can be constructed with the aid of bar (h) and bar (g), the permutations induced by h on Y and by g on X (use lemma):
Corollary: If bar (g)= Õ n(x n ...gl n-1x n) , then f ÎYX is fixed under (h,g) if and only if the following two conditions are satisfied:
f(x n) ÎYhl n,
and the other values of f arise from the values f(x n) according to
f(x n)=hf(g-1x n)=h2f( g-2x n)= ... .
This together with lemma yields:
Corollary: The fixed points of (h,g) are the f ÎYX which can be obtained in the following way:
• To each cyclic factor of bar (g), let l denote its length, we associate a cyclic factor of bar (h) of length d dividing l.
• If x is a point in this cyclic factor of bar (g) and y a point in the chosen cyclic factor of bar (h), then put
f(x):=y, f(gx):=hy,f(g2x):=h2y, ... .
• Injective symmetry classes
• Surjective symmetry classes
• Various combinatorial numbers
• Exercises

• harald.fripertinger@kfunigraz.ac.at,
last changed: August 28, 2001   Special symmetry classes