Given a group G and a field F, the elements of its representation ringRF 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, 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 RC is isomorphic to Z/, where X corresponds to the complex representation sending a generator of the group to a primitive nth root of unity.
For the rational representations of the cyclic group of order 3, the representation ring RQ is isomorphic to Z/, 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 RF is isomorphic to Z/.
The continuous representation ring R for the circle group is isomorphic to Z. The ring of real representations is the subring of R of elements fixed by the involution on R given by X → X −1.
The ring RC for the symmetric group on three points is isomorphic to Z/, 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. If G is finite, the homomorphism R → C is injective, so that R can be identified with a subring of C. For fields F whose characteristic divides the order of the group G, the homomorphism from RF → C defined by Brauer characters is no longer injective. For a compact connected group R is isomorphic to the subring of Rconsisting of those class functions that are invariant under the action of the Weyl group. For the general compact Lie group, see Segal.
Given a representation of G and a natural numbern, we can form the n-th exterior power of the representation, which is again a representation of G. This induces an operation λn : R → R. With these operations, R becomes a λ-ring. The Adams operations on the representation ring R are maps Ψk characterised by their effect on characters χ: The operations Ψk are ring homomorphisms of R to itself, and on representations ρ of dimension d where the Λiρ are the exterior powers of ρ and Nk is the k-th power sum expressed as a function of the d elementary symmetric functions of d variables.
