Special symmetry classes

Special symmetry classes

We now return to Y^X and consider its subsets consisting of the injective and the surjective maps f only:

Y^X_i:= {f ÎY^X | f injective } and Y_s^X:= {f ÎY^X | 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 Y_i^X are called injective symmetry classes, while those on Y_s^X 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 Y^X 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 ...g^{l _n-1}x _n) , then f ÎY^X is fixed under (h,g) if and only if the following two conditions are satisfied:

f(x _n) ÎY_{h^{l 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 ÎY^X 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

Special symmetry classes