# Ring of representations

representation ring

A commutative ring $R$ defined as follows. The additive group of $R$ is generated by the equivalence classes of linear representations of a group $G$ over a field $K$, and the defining relations have the form $\pi=\pi_1+\pi_2$, where $\pi$ is the equivalence class of a representation, $\pi_1$ is the equivalence class of a subrepresentation of it and $\pi_2$ is the equivalence class of the corresponding quotient representation of $\pi$; the multiplication of $R$ assigns to two equivalence classes $\pi_1$ and $\pi_2$ the equivalence class of their tensor product. This ring of representations is sometimes called the Grothendieck ring of the group. For locally compact groups $G$ the representation ring is usually meant to be the commutative ring $R$ defined by the operations of direct sum and tensor product in the set of equivalence classes of continuous unitary representations of $G$. The structures of $R$ are very useful if $G$ is compact. Then it leads to the duality theory in terms of block-algebras. In the more general case of groups $G$ of type I, the study of $R$ may be reduced to the study of tensor products of irreducible unitary representations.