Burnside ring

Given a finite group G, the generators of its Burnside ring Ω(G) are the formal sums of isomorphism classes of finite G-sets. For the ring structure, addition is given by disjoint union of G-sets and multiplication by their Cartesian product. The Burnside ring is a free Z-module, whose generators are the (isomorphism classes of) orbit types of G. If G acts on a finite set X, then one can write X = \bigcup_i X_i (disjoint union), where each Xi is a single G-orbit. Choosing any element xi in Xi creates an isomorphism G/Gi → Xi, where Gi is the stabilizer (isotropy) subgroup of G at xi. A different choice of representative yi in Xi gives a conjugate subgroup to Gi as stabilizer. This shows that the generators of Ω(G) as a Z-module are the orbits G/H as H ranges over conjugacy classes of subgroups of G. In other words, a typical element of Ω(G) is \sum_{i=1}^N a_i [G/G_i], where ai in Z and G1, G2, ..., GN are representatives of the conjugacy classes of subgroups of G. ==Marks==

Much as character theory simplifies working with group representations, marks simplify working with permutation representations and the Burnside ring. If G acts on X, and H ≤ G (H is a subgroup of G), then the mark of H on X is the number of elements of X that are fixed by every element of H: m_X(H) = \left|X^H\right|, where :X^H = \{ x\in X \mid h\cdot x = x, \forall h\in H\}. If H and K are conjugate subgroups, then mX(H) = mX(K) for any finite G-set X; indeed, if K = gHg−1 then XK = g · XH. It is also easy to see that for each H ≤ G, the map Ω(G) → Z : X ↦ mX(H) is a homomorphism. This means that to know the marks of G, it is sufficient to evaluate them on the generators of Ω(G), viz. the orbits G/H. For each pair of subgroups H,K ≤ G define :m(K, H) = \left|[G/K]^H\right| = \# \left\{ gK \in G/K \mid HgK=gK \right\}. This is mX(H) for X = G/K. The condition HgK = gK is equivalent to g−1Hg ≤ K, so if H is not conjugate to a subgroup of K then m(K, H) = 0. To record all possible marks, one forms a table, Burnside's Table of Marks, as follows: Let G1 (= trivial subgroup), G2, ..., GN = G be representatives of the N conjugacy classes of subgroups of G, ordered in such a way that whenever Gi is conjugate to a subgroup of Gj, then i ≤ j. Now define the N × N table (square matrix) whose (i, j)th entry is m(Gi, Gj). This matrix is lower triangular, and the elements on the diagonal are non-zero so it is invertible. It follows that if X is a G-set, and u its row vector of marks, so ui = mX(Gi), then X decomposes as a disjoint union of ai copies of the orbit of type Gi, where the vector a satisfies, :aM = u, where M is the matrix of the table of marks. This theorem is due to . ==Examples==

The table of marks for the cyclic group of order 6: The table of marks for the symmetric group S3: The dots in the two tables are all zeros, merely emphasizing the fact that the tables are lower-triangular. (Some authors use the transpose of the table, but this is how Burnside defined it originally.) The fact that the last row is all 1s is because [G/G] is a single point. The diagonal terms are m(H, H) = | NG(H)/H |. The numbers in the first column show the degree of the representation. The ring structure of Ω(G) can be deduced from these tables: the generators of the ring (as a Z-module) are the rows of the table, and the product of two generators has mark given by the product of the marks (so component-wise multiplication of row vectors), which can then be decomposed as a linear combination of all the rows. For example, with S3, :[G/\mathbf{Z}_2]\cdot[G/\mathbf{Z}_3] = [G/1], as (3, 1, 0, 0).(2, 0, 2, 0) = (6, 0, 0, 0). ==Permutation representations==

Associated to any finite set X is a vector space V = VX, which is the vector space with the elements of X as the basis (using any specified field). An action of a finite group G on X induces a linear action on V, called a permutation representation. The set of all finite-dimensional representations of G has the structure of a ring, the representation ring, denoted R(G). For a given G-set X, the character of the associated representation is :\chi(g) = m_X(\langle g\rangle) where \langle g\rangle is the cyclic group generated by g. The resulting map :\beta : \Omega(G) \longrightarrow R(G) taking a G-set to the corresponding representation is in general neither injective nor surjective. The simplest example showing that β is not in general injective is for G = S3 (see table above), and is given by :\beta(2[S_3/\mathbf{Z}_2] + [S_3/\mathbf{Z}_3]) = \beta([S_3] + 2[S_3/S_3]). ==Extensions==

The Burnside ring for compact groups is described in . The Segal conjecture relates the Burnside ring to homotopy. == See also ==