# 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 $Rep(G)$. This is a symmetric monoidal abelian category and thus has a Grothendieck ring, which is called the representation ring of $G$ and denoted $R(G)$. 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 $R(G)$ 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

$e_i e_j = \sum_k m_{i j}^k e_k ,$

where $m_{i j}^k$ 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.

### 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 $G$):

$R(G) \simeq K_G(\ast) \simeq KK(\mathbb{C}, C(\mathbf{B}G)) \,.$

## Properties

### Relation to the character ring

If $G$ is a finite group and we tensor $R(G)$ 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?.

Similarly for $G$ a compact Lie group, its complex linear representations $\rho \colon G \to U(n) \to Aut(\mathbb{C}^n)$ (for all $n \in \mathbb{N}$) are uniquely specified by their characters $\chi_\tho \coloneqq tr(\rho(-)) \colon G \to \mathbb{C}$. Therefore also here the representation ring is often called the character ring of the group.

### Relation to equivariant K-theory

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

$Rep(G) \simeq K_G(\ast) \,.$

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.

### Completion

On the other hand, by the Atiyah-Segal completion theorem in Borel-equivriant K-theory only the completion of $Rep(G)$ at the augmentation ideal appears

## Examples

### Spin group

The completion of $Rep(Spin(2k))$ is $\mathbb{Z}[ [ e^{\pm x_j} ] ]$ for $1 \leq j \leq k$ (e.g Brylinski 90, p. 9).

## References

Classical results for compact Lie groups:

• Graeme Segal, The representation ring of a compact Lie group, Publications Mathématiques de l’Institut des Hautes Études Scientifiques January 1968, Volume 34, Issue 1, pp 113-128 (NUMDAM)

• Masaru Tackeuchi, A remark on the character ring of a compact Lie group, J. Math. Soc. Japan Volume 23, Number 4 (1971), 555-705 (Euclid)

In the generality of super Lie groups:

With an eye towards loop group representations:

• Jean-Luc Brylinski, Representations of loop groups, Dirac operators on loop space, and modular forms, Topology, 29(4):461–480, 1990.
Revised on March 29, 2014 08:54:04 by Urs Schreiber (89.204.138.114)