Algebraic Combinatorics -- Centralizers of elements in finite symmetric groups

Conjugacy classes in complete monomial groups

Centralizers of elements in finite symmetric groups

Centralizers of elements in finite symmetric groups

As an application of this permutation representation we obtain a description of the centralizers of elements in finite symmetric groups. To show this we note that d[ C_m wr S_n ], where C_m := á(1 ...m) ñ, is just the centralizer of

s:=(1 ...m)(m+1, ...,2m) ...((n-1)m+1, ...,nm) ÎS_mn .

This follows from d[ C_m wr S_n ] ÍC_S( s), which is clear from the formula together with the formula and the fact that | C_S( s) | =mⁿn!= | C_m wr S_n | (cf. Corollary). The general case is now easy:

Corollary: If sÎS_n is of type a=(a₁, ...,a_n), then C_S( s) is a subgroup of S_n which is similar to the direct sum
Å_i(C_i S_{a_i}).

Similarly we can show (recall the examples)

Corollary: The normalizer of the n-fold direct sum
Åⁿ S_m := S_m Å...ÅS_m , n summands,
is conjugate to the plethysm S_m S_n .

Thus centralizers of elements and normalizers of specific subgroups of symmetric groups turn out to be direct sums of complete monomial groups. Since such groups will also occur as acting groups later on, we also describe their conjugacy classes.

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

Centralizers of elements in finite symmetric groups