ExamplesComplete monomial groupsCentralizers of elements in finite symmetric groupsConjugacy classes in complete monomial groups

Conjugacy classes in complete monomial groups

Consider an element ( y, p) in H wr S n and assume that C1,C2, ... are the conjugacy classes of H. If

p= Õ nÎ c( p) (j n ...pl n-1j n),

in standard cycle notation, then we associate with its n-th cyclic factor (j n ...pl n-1j n) the element

h n( y, p) := y(j n) y( p-1j n) ...y( p-l n+1j 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 ...pl n-1j n) with respect to ( y, p). In this way we obtain a total of c( p) cycleproducts, ak( p) of them arising from the cyclic factors of p which are of length k. Now let aik( y, p) be the number of these cycleproducts which are associated to a k-cycle of p and which belong to the conjugacy class Ci of H (note that we did not say ''let aik ( y, p) be the number of different cycleproducts ``). We put these natural numbers together into the matrix

a( y, p):=(aik( 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:

aik( y, p) Î N, åi aik( y, p)=ak ( p), åi,k k ·aik( 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:

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 hn( y, p) is conjugate to

yyp ...y pln-1( pzjn),

for each integer z.

The second remark is, that for each p' ÎS n and every y' ÎHn,

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" ÎSn which satisfies p= p" p' p"-1, and for which the cycleproducts hn( y, p) and hn( y' p", p) are conjugate.

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


harald.fripertinger "at" uni-graz.at http://www-ang.kfunigraz.ac.at/~fripert/
UNI-Graz Institut für Mathematik
UNI-Bayreuth Lehrstuhl II für Mathematik
last changed: January 19, 2005

ExamplesComplete monomial groupsCentralizers of elements in finite symmetric groupsConjugacy classes in complete monomial groups