nLab group character

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.

Related concepts

• Pontryagin duality, Pontryagin dual

• conjugacy class of subgroups

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

• Michael Hopkins, Nicholas Kuhn, Douglas Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000), 553-594 (publisher, pdf)