Burnside ring

In mathematics, the Burnside ring of a finite group is an algebraic construction that encodes the different ways the group can act on finite sets. The ideas were introduced by William Burnside at the end of the nineteenth century, but the algebraic ring structure is a more recent development, due to Solomon (1967).

Formal definition

Given a finite group G, the elements of its Burnside ring Ω(G) are the formal differences 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 $X = \cup_i X_i$ (disjoint union), where each X_i is a single G-orbit. Choosing any element x_i in X_i creates an isomorphism G/G_i → X_i, where G_i is the stabilizer (isotropy) subgroup of G at x_i. A different choice of representative y_i in X_i gives a conjugate subgroup to G_i 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 a_i in Z and G₁, G₂, ..., G_N are representatives of the conjugacy classes of subgroups of G.

Marks

Much like 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 m_X(H) = m_X(K) for any finite G-set X; indeed, if K = gHg⁻¹ then X^K = g · X^H.

It is also easy to see that for each H ≤ G, the map Ω(G) → Z : X ↦ m_X(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 m_X(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, ..., G_N = G be representatives of the N conjugacy classes of subgroups of G, ordered in such a way that whenever G_i is conjugate to a subgroup of G_j, then i ≤ j. Now define the N × N table (square matrix) whose (i, j)th entry is m(G_i, G_j). 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 u_i = m_X(G_i), then X decomposes as a disjoint union of a_i copies of the orbit of type G_i, where the vector a satisfies,

aM = u,

where M is the matrix of the table of marks. This theorem is due to (Burnside 1897).

Examples

The table of marks for the cyclic group of order 6:

Z₆	1	Z₂	Z₃	Z₆
Z₆ / 1	6	.	.	.
Z₆ / Z₂	3	3	.	.
Z₆ / Z₃	2	0	2	.
Z₆ / Z₆	1	1	1	1

The table of marks for the symmetric group S₃:

S₃	1	Z₂	Z₃	S₃
S₃ / 1	6	.	.	.
S₃ / Z₂	3	1	.	.
S₃ / Z₃	2	0	2	.
S₃ / S₃	1	1	1	1

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) = | N_G(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 = V_X, 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 <g> 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 (tom Dieck 1987).

The Segal conjecture relates the Burnside ring to homotopy.

References

Burnside, William (1897), Theory of groups of finite order, Cambridge University Press
tom Dieck, Tammo (1987), Transformation groups, de Gruyter Studies in Mathematics, 8, Walter de Gruyter, ISBN 978-3-11-009745-0, MR 889050, OCLC 217014538
Dress, Andreas (1969), "A characterization of solvable groups", Math. Zeitschrift, 110 (3): 213–217, doi:10.1007/BF01110213
Kerber, Adalbert (1999), Applied finite group actions, Algorithms and Combinatorics, 19 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-65941-9, MR 1716962, OCLC 247593131
Solomon, L. (1967), "The Burnside algebra of a finite group", J. Combin. Theory, 1: 603–615

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