Given a group G and a field F, the elements of its representation ring RF(G) are the formal differences of isomorphism classes of finite-dimensional 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.
The representation ring of G is the Grothendieck ring of the category of finite-dimensional representations of G. 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).
For fields F whose characteristic divides the order of the group G, the homomorphism from RF(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).
The Adams operations on the representation ring R(G) are maps Ψk characterised by their effect on characters χ: The operations Ψk are ring homomorphisms of R(G) 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.