# nLab group character

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

group theory

# Contents

## Idea

A multiplicative character of a group $G$ is a group homomorphism into the circle group $U(1)$, or more generally into the group of units $k^\times$ of a given ground field (for instance $\mathbb{C}^\times = U(1)$):

$\chi \;\colon\; G \longrightarrow k^\times \,.$

Since $k^\times$ is an abelian group, this means that group characters are in particular class functions.

Dually a co-character is a homomorphism out of $k^\times$ into $G$.

The collection of characters is itself an abelian group under the pointwise multiplication, this is called the character lattice $Hom(G,k^\times)$ of the group. Similarly the cocharacter lattice is $Hom(k^\times, G)$.

For topological groups one considers continuous characters. Specifically, for a locally compact Hausdorff group $G$ (often further assumed to be an abelian group), a character of $G$ is continuous homomorphism to the circle group $\mathbb{R}/\mathbb{Z}$. If $G$ is profinite, then this is the same as an continuous homomorphism to the discrete group $\mathbb{Q}/\mathbb{Z}$. (See MO.)

## Properties

### Restriction of group characters to maximal tori – weights

Let $G$ be a connected compact Lie group.

By the general properties of maximal tori in this case, it follows that every group character $G \to U(1)$ is already fixed by its restriction along a maximal torus inclusion

$T \hookrightarrow G \to U(1) \,.$

Now the group characters of the abelian maximal torus $T\simeq U(1)^n$ are n-tuples of group characters of the circle group $U(1)$, which are integers – the weights.

Explicitly, under a given identification of the circle group as a quotient of the additive group of real numbers

$U(1) \simeq \mathbb{R}/h \mathbb{Z}$

for $h \in (0,\infty)$, then the character $\lambda$ on $U(1)^n\simeq T$ labeled by $(x_1, \cdots, x_n) \in \mathbb{Z}^n$ is

$\lambda(t_1,\cdots t_n) = \exp(\tfrac{i}{\hbar} \sum_{j = 1}^n t_j n_j )$

(where $\hbar \coloneqq h/2\pi$ is “Planck's constant”).

(e.g. Johansen, section 2.10)

### Relation to Chern roots and the splitting principle

A group character, hence a group homomorphism $G \to U(1)$ induces a map of classifying spaces $B G \to B U(1) \simeq K(\mathbb{Z},2)$. Similarly for the restriction to the maximal torus above, which induces

$B U(1)^n \simeq B T \to B G \to B U(1)\simeq K(\mathbb{Z},2) \,.$

Under this identification the weights $x_i$ of the group character, as above, are the “Chern roots” as the appear in the splitting principle. See there for more.

### Characters and fundamental group of tori

Write $S^1$ for the circle group.

Let $T$ be a torus, regarded as an abelian group. Write $[T,S^1]$ for its character group.

There is a bilinear form

$\pi_1(T)\otimes [T, S^1] \longrightarrow \mathbb{Z}$

on the fundamental group of the torus and its character group, given by sending a homotopy class $[\gamma]$ of a continuous map

$\gamma \colon S^1 \longrightarrow T$

to the homotopy class $c(\gamma)$ of the composition with a character $c \colon T \longrightarrow S^1$

$c(\gamma) \;\colon\; S^1 \stackrel{\gamma}{\longrightarrow} T \stackrel{c}{\longrightarrow} S^1$

regarded as an element $[c(\gamma)] \in \pi_1(S^1) \simeq \mathbb{Z}$.

This bilinear form is non-degenerate, and hence constitutes an isomorphism

$\pi_1(T) \simeq [T,S^1] \,.$

### Inner product and orthogonality

The complex class functions on a finite group $G$ have an inner product given by

$\langle \alpha, \beta\rangle \coloneqq \frac{1}{{\vert G \vert}} \underset{g \in G}{\sum} \alpha(g) \overline{\beta(g)} \,.$

The Schur orthogonality relation is the statement that the irreducible group characters $\{\chi_i\}_i$ form an orthonormal basis of the space of class functions under this inner product:

$\langle \chi_i, \chi_j \rangle = \left\{ \array{ 1 & if \; i = j \\ 0 & otherwise } \right.$

Such properties arise from characters occurring as traces of group representations.

### In terms of the classifying space of the group

Consider the classifying space, $B G$, of the group. Then its free loop space, $Map (S^1, B G)$, has as components $G$ modulo conjugation. Then, the characters of $G$ may be expressed as the zeroth cohomology of this loop space, $H^0(\mathcal{L} B G, \mathbb{C})$. This construction is useful in the generalisation to transchromatic characters.

## References

Original articles on character rings/representation rings of compact Lie groups include

• 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:PMIHES_1968__34__113_0)

• 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:jmsj/1259849785)

Lecture notes on characters of finite and compact Lie groups include

• Troels Roussauc Johansen, Character Theory for Finite Groups and Compact Lie Groups pdf

• Andrei Yafaev, Characters of finite groups (pdf)

Discussion for finite groups in the more general context of equivariant complex oriented cohomology theory (transchromatic character) is in

Last revised on April 27, 2021 at 08:49:26. See the history of this page for a list of all contributions to it.