Enumeration by stabilizer type

Enumeration by stabilizer type

Let U≤ G be a subgroup of G, then a partition π∈ Π_n is called U-invariant if gπ=π for all g∈ U. The set of all U-invariant partitions will be denoted by (Π_n)_U. The stabilizer of a partition π is the subgroup U := { g∈ G | gπ=π} of G. The stabilizers of all partitions in the orbit G(π) of π lie in the conjugacy class Ũ of the stabilizer U of π. So the orbit G(π) is called of stabilizer type Ũ . The set of all orbits of type Ũ is also called the U-stratum and it will be indicated as Ũ \\\Π_n. The Lemma of Burnside provides a formula which allows to compute the numbers of G-orbits of type Ũ from the number of V-invariant partitions for U,V≤ G. In [10] it is formulated in the following way: Let Ũ ₁,...,Ũ _d be the conjugacy classes of subgroups of G then

where B(G) := (b_ij)_{1≤ i,j≤ d} is the Burnside matrix of G given by

b_ij=^._. |Ũ _i|

|G/U_i|
^._.
∑
V∈ Ũ _j μ(U_i,V),^._.

where μ is the Moebius function in the incidence algebra over the subgroup lattice L(G) of G. So, for computing the number of U-strata, we have to determine the number of V-invariant partitions. In [18] the following formula is proved: Theorem: Let T be a system of representatives of the G-orbits on X and let H be a system of representatives of the conjugacy classes of G (e.g. H={ U₁,...,U_d}). Then the number of G-invariant partitions of X is given by

^._.
∑
δ∈ Π_T ^._.
∏
A∈ δ ^._.
∑
H∈ H ^._. 1

m_H(H)
^._.
∏
t∈ A m_H(G_t),^._.

where Π_T denotes the set of all partitions δ of T. The blocks of the partition δ are indicated as A. For t∈ X the stabilizer of t in G is denoted by G_t. Finally for subgroups U,V of G

m_U(V)=^._. |N_G(U)|

|U|
^._.
∑
W∈ Ũ ζ(V,W)^._.

is the mark of U at V, where ζ is the zeta-function in the incidence algebra over L(G). In the case that U∈ Ũ _j and V∈ Ũ _i then

m_U(V)=m_ij := ^._. |N_G(U_j)|

|U_j|
^._.
∑
W∈ Ũ _j ζ(U_i, W).^._.

The matrix M(G) := (m_ij)_{1≤ i,j≤
d} is called the table of marks of G and it is the inverse of the Burnside matrix B(G). The conjugacy classes of subgroups of C_n are well known. In [5] it is shown that a system of representatives of the conjugacy classes of subgroups of D_n (for n∈ ℕ) is given as a disjoint union

⨃ _{d |
n}R(d),^._.

where

harald.fripertinger "at" uni-graz.at, May 10, 2016

Enumeration by stabilizer type