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 C¹,C², ... 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^-1j _n) ...y( p^{-l _n+1}j _n) = yy _p ...y _{p^{l 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 Cⁱ 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 | ⁿn!/ Õ_i,ka_ik!(k | H | / | Cⁱ | )^a_ik.

• 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,kk ·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 ·a₁,l_n ·a₂, ...), 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 _p^ln-1( p^zj_n),

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" 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).

Conjugacy classes in complete monomial groups