nLab
representation ring

Any group $G$ has a category of finite-dimensional complex-linear representations, often denoted $\mathrm{Rep}(G)$. This is a symmetric monoidal abelian category and thus has a Grothendieck ring, which is called the representation ring of $G$ and denoted $R(G)$.

More concretely, we get $R(G)$ as follows. It has a basis $(e_i{)}_i$ given by the irreps of $G$: that is, $i$ is an index for an irreducible finite-dimensional complex representation of $G$. It has a product given by

$$e_i{e}_j=\sum _k{m}_{ij}^k{e}_k,$$e_i e_j = \sum_k m_{i j}^k e_k ,

where $m_{ij}^k$ is the multiplicity of the $k$th irrep in the tensor product of the $i$th and $j$th irreps. Note that $R(G)$ is commutative thanks to the symmetry of the tensor product.

If $G$ is a finite group and we tensor $R(G)$ with the complex numbers, it becomes isomorphic to the character ring? of $G$: that is, the ring of complex-valued functions on $G$ that are constant on each conjugacy class. Such functions are called class functions.