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,
... .
harald.fripertinger@kfunigraz.ac.at,
last changed: August 28, 2001
