Burnside ring

For a finite group G, the Burnside ring Ω(G) of G is defined to be the ring generated by formal differences of isomorphism classes of G-sets. The ring structure is given by disjoint union and Cartesian product of G-sets.

Let \(\mathcal{O}\) be a single orbit of a finite group G, and let \(x\in \mathcal{O}\). Then \(\mathcal{O}\) is isomorphic as a G-set to G/H, where H=Gx (the isotropy subgroup of x). If one chose a different 'starting point' \(y\in\mathcal{O}\), then H would be replaced by a conjugate subgroup. The set of isomorphism classes of G-orbits is in this way in 1-1 correspondence with the set of conjugacy classes of subgroups of G.

The Burnside ring Ω(G) is therefore the Z-module generated by the conjugacy classes of subgroups of G. See the wikipedia page.

Let S be a finite G-set (a set upon which G acts). Then S can be decomposed into a finite disjoint union of G-orbits, and so corresponds to a sum $$ S = \sum_i a_i [G/H_i] $$ where the Hi represent the conjugacy classes of subgroups of G, and the ai are (positive) integers.

Permutation Representation

If G acts on a finite set S, then there is associated a permutation representation (over any ring or field, but let us use Q) as follows. Let V be the vector space whose basis consists of the elements of S. Thus a general element of V is of the form \(\mathbf{v} = \sum_{s\in S}a_s s\), where the as are elements of the field/ring. Then \(g\in G\) acts by \(g(\mathbf{v}) = \sum_{s\in S} a_s g(s)\), which is a permutation of the coefficients as. The matrix representing g then has a 1 at the intersection of the column corresponding to s and the row corresponding to g(s), and 0 at other entries of that row and column.

If we fix a base field, this defines a homomorphism from the Burnside ring Ω(G) to the ring of representations over that field which we denote $$\beta:\Omega(G)\longrightarrow R(G).$$

Properties of β

  • β is only surjective for the trivial group and the group of order 2;
  • β is seldom injective

Table of marks

The element in row G/K and column H represents the number of points in the orbit (type) G/K fixed by H. By the definition of the action of G on G/K, a coset gK is fixed by H whenever HgK = gK. But this is equivalent to, g-1Hg < K so that $$m(G/K, H) = \# \left\{ gK \in G/K \mid g^{-1}Hg < K \right\}.$$

  • In particular, the diagonal elements are m(K, K) = |NG(K)/K|.
  • If H is not conjugate to a subgroup of K then m(G/K, H) = 0.
  • If K is a normal subgroup of G and H < K then \(g^{-1}Hg \subset K\) for all \(g\in G\), so that \(m(G/K,\,H)=|G/K|\).
  • If G is Abelian then every subgroup is normal and so, again if H < K then \(m(G/K,\,H)=|G/K|\), otherwise \(m(G/K,\,H)=0\).

Examples

S3 S4