| | | **Conjugacy classes in complete monomial groups** |

### Conjugacy classes in complete monomial groups

Consider an element *( y, p)* in *H wr S*_{ n} and assume that
*C*^{1},C^{2}, ... are the conjugacy classes of *H*. If
* p= Õ*_{ nÎ c( p)} (j_{ n} ...p^{l n-1}j_{ n}),

in standard cycle notation, then we associate with its * n*-th cyclic factor
*(j*_{ n} ...p^{l n-1}j_{ n}) the element
*
h*_{ n}( y, p) := y(j_{ n}) y( p^{-1}j_{ n}) ...y( p^{-l n+1}j_{ n}) = yy_{ p} ...y_{ pl n-1}(j_{ n})

of *H* and call it the * n*-th cycleproduct of *( y, p)* or the
cycleproduct *associated* to *(j*_{ n} ...p^{l n-1}j_{ n})
with respect to *( y, p)*. In this way we obtain a total of
*c( p)* cycleproducts,
*a*_{k}( p) of them arising from the cyclic factors of * p*
which are of length *k*. Now let *a*_{ik}( y, p) be the number of these
cycleproducts which are associated to a *k*-cycle of * p* and which belong
to the conjugacy class *C*^{i} of *H* (note that we did *not*
say
''let *a*_{ik}
( y, p) be the number of *different* cycleproducts ``).
We
put these natural numbers together into the matrix
*a( y, p):=(a*_{ik}( y, p)),

This matrix has *n* columns (*k* is the column index)
and as many rows as there are conjugacy classes
in *H* (*i* is the row index). Its entries satisfy the following conditions:
*
a*_{ik}( y, p) Î **N**, å_{i} a_{ik}( y, p)=a_{k}
( p), å_{i,k} k ·a_{ik}( y, p)=n.

We call this matrix *a( y, p)* the *type*
of *( y, p)* and we say that *( y, p)* is *of type* *a( y, p)*.
**Lemma: **
*
The conjugacy classes of
complete monomial groups **H wr S*_{ n} have the following properties:
*C*^{H wr S n }( y', p')=C^{H wr S n }( y,
p) if and only if a( y', p')=a( y, p).
- The order of the conjugacy class of elements of type
*(a*_{ik}) in
*H wr S*_{ n} , *H* finite, is equal to
* | H | *^{n}n!/ Õ_{i,k}a_{ik}!(k | H | / | C^{i} | )^{aik}.

- Each matrix
*(b*_{ik}) with *n* columns and as many rows as *H* has
conjugacy classes, the elements of which satisfy
*b*_{ik} Î **N**, å_{i,k}k ·b_{ik}=n,

occurs as the type of an element *( y, p) ÎH wr S*_{ n} .
- If
*H* is a permutation group and * a:= a(h*_{n}
( y, p)), then the cycle partition * a( d( y, p))*, where
* d* denotes the permutation representation of
the formula, is equal
to
* å*_{n} l_{n} ·a(h_{n}( y, p)),

where *l*_{n} ·a, a:= a(h_{n}( y, p)), is defined to be
*(l*_{n} ·a_{1},l_{n} ·a_{2}, ...), and where * å*_{n} ...
means that the proper partition has to be formed that consists of all the
parts of all the summands *l*_{n} ·a(h_{n}( y, p)).

Proof: A first remark concerns the cycleproducts introduced in
the formula. Since
in each group *G* the products *xy* and *yx* of two elements are conjugate, we
have that
*h*_{n}( y, p) is conjugate to
* yy*_{p} ...y_{ pln-1}( p^{z}j_{n}),

for each integer *z*.
The second remark is, that for each * p' ÎS*_{ n} and every * y' ÎH*^{n},

*a( y, p)=a(( i, p')( y, p)( i, p')*^{-1})=a(( y',1)( y,
p)( y',1)^{-1}).

This follows from the fact that both *( y*_{ p'}, p' pp'^{-1}) and
*( y' y y'*_{p}^{-1}, p) are of type *a( y, p)*.
A third remark is that *a( y, p)=a( y', p')* implies the existence of an
element * p" ÎS*_{n} which satisfies * p= p" p' p"*^{-1}, and for which
the cycleproducts *h*_{n}( y, p) and *h*_{n}( y'_{ p"}, p) are
conjugate.

It is not difficult to check these remarks and then to derive the statement
(exercise).

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001

| | | **Conjugacy classes in complete monomial groups** |