# nLab representation ring

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

More concretely, we get $R\left(G\right)$ as follows. It has a basis $\left({e}_{i}{\right)}_{i}$ 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 $R\left(G\right)$ is commutative thanks to the symmetry of the tensor product.

If $G$ is a finite group and we tensor $R\left(G\right)$ 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.

Revised on July 30, 2009 18:36:46 by Toby Bartels (71.104.230.172)