# nLab representation ring

### Context

#### Representation theory

representation theory

geometric representation theory

cohomology

# Contents

## Definition

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 $R\left(G\right)$. Elements of the representation ring are hence formal differences (with respect to direct sum) of ordinary representations: virtual representations.

### In components

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.

### In terms of K-theory

Equivalently the representation ring of $G$ is the $G$-equivariant K-theory of the point, or equivalently by the Green-Julg theorem, if $G$ is a compact Lie group, the operator K-theory of the group algebra (the groupoid convolution algebra of the delooping groupoid of $G$):

$R\left(G\right)\simeq {K}_{G}\left(*\right)\simeq \mathrm{KK}\left(ℂ,C\left(BG\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$R(G) \simeq K_G(\ast) \simeq KK(\mathbb{C}, C(\mathbf{B}G)) \,.

## Properties

### Relation to the character ring

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.

### Relation to twisted K-theory

The representation ring of a compact Lie group is equivalent to the $G$-equivariant K-theory of the point.

$\mathrm{Rep}\left(G\right)\simeq {K}_{G}\left(*\right)\phantom{\rule{thinmathspace}{0ex}}.$Rep(G) \simeq K_G(\ast) \,.

The construction of representations by index-constructions of $G$-equivariant Dirac operators (push-forward in $G$-equivariant K-theory to the point) is called Dirac induction.

## References

In the generality of super Lie groups:

Revised on August 10, 2013 18:14:27 by Urs Schreiber (89.204.130.134)