Representation ring

In

Sylow p-subgroups are cyclic

Formal definition

Given a group G and a field F, the elements of its representation ring R_F(G) are the formal differences of isomorphism classes of finite dimensional linear F-representations of G. For the ring structure, addition is given by the direct sum of representations, and multiplication by their tensor product over F. When F is omitted from the notation, as in R(G), then F is implicitly taken to be the field of complex numbers.

Succinctly, the representation ring of G is the

Grothendieck ring

of the category of finite-dimensional representations of G.

Examples

For the complex representations of the cyclic group of order n, the representation ring R_C(C_n) is isomorphic to Z[X]/(Xⁿ − 1), where X corresponds to the complex representation sending a generator of the group to a primitive nth root of unity.
More generally, the complex representation ring of a finite abelian group may be identified with the group ring of the character group.
For the rational representations of the cyclic group of order 3, the representation ring R_Q(C₃) is isomorphic to Z[X]/(X² − X − 2), where X corresponds to the irreducible rational representation of dimension 2.
For the modular representations of the cyclic group of order 3 over a field F of characteristic 3, the representation ring R_F(C₃) is isomorphic to Z[X,Y]/(X² − Y − 1, XY − 2Y,Y² − 3Y).
The continuous representation ring R(S¹) for the circle group is isomorphic to Z[X, X⁻¹]. The ring of real representations is the subring of R(G) of elements fixed by the involution on R(G) given by X ↦ X⁻¹.
The ring R_C(S₃) for the symmetric group on three points is isomorphic to Z[X,Y]/(XY − Y,X² − 1,Y² − X − Y − 1), where X is the 1-dimensional alternating representation and Y the 2-dimensional irreducible representation of S₃.

Characters

Any representation defines a character χ:G → C. Such a function is constant on conjugacy classes of G, a so-called class function; denote the ring of class functions by C(G). If G is finite, the homomorphism R(G) → C(G) is injective, so that R(G) can be identified with a subring of C(G). For fields F whose characteristic divides the order of the group G, the homomorphism from R_F(G) → C(G) defined by Brauer characters is no longer injective.

For a compact connected group R(G) is isomorphic to the subring of R(T) (where T is a maximal torus) consisting of those class functions that are invariant under the action of the Weyl group (Atiyah and Hirzebruch, 1961). For the general compact Lie group, see Segal (1968).

λ-ring and Adams operations

Given a representation of G and a natural number n, we can form the n-th

exterior power of the representation, which is again a representation of G. This induces an operation λⁿ : R(G) → R(G). With these operations, R(G) becomes a λ-ring

.

The

Adams operations

on the representation ring R(G) are maps Ψ^k characterised by their effect on characters χ: