Representation ring

Definition

Given a group $G$, a field $k$, and a subring $R$ of $k$, the representation ring of $G$ over $k$ is a $k$-algebra defined as follows:

• Additively, it is the quotient of the $R$-module generated by all finite-dimensional representations of $G$ over $k$, by the equivalence relation of the sum of two representations being the same as the representation which is their direct sum. Thus, it is, as a $R$-module, freely generated by the indecomposable representations
• The product of two representations is the representation corresponding to their tensor product. Since the representations span the module, this defines the multiplication throughout.

Relation with other functors

The representation ring is closely related to the character ring. In fact, there is a canonical homomorphism from the representation ring to the character ring that sends each representation to the corresponding character function. In the situation of a character-determining field (viz a situation where the character determines the representation) this is actually an isomorphism.

However, in the general (for instance, the modular) case, there may be different representations with the same character, and thus, the homomorphism from the representation ring to the character ring may have a kernel.