nLab character

Definition

There are many notions of a character for an algebraic structure, often topologized.

Character of a group

For groups, there are two different, but related, concepts which are both often just referred to as characters of a group:

1. multiplicative group characters,

2. characters of linear representations of a group.

Here the first notion is the special case of the second notion for 1-dimensional linear representations. Since for abelian groups all irreducible representations over the complex numbers are 1-dimensional, in this case both notions agree, but in general they do not.

Multiplicative character of a group

A multiplicative character on a (discrete) group $G$ (a group character) is a group homomorphism from $G$ to the group of units $k^\times$ of the ground field $k$:

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

Since the codomain $k^\times$ is an abelian group we have that

1. such a group homomorphism is invariant under conjugation, so that a group character descends to a function on the set of conjugacy classes of elements in $G$.

2. the collection of characters is itself an abelian group under 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 more see at group character.

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 a 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.)

Character of a linear representation of a group

In representation theory, one defines the character of a linear representation $\rho\colon G\to End(V)$ to be the group character on $G$ given by $g \mapsto Tr \rho(g)$, whenever the trace in $V$ makes sense (e.g. when $V$ is finite-dimensional). Since such a function is invariant under conjugation, one may equivalently consider it a function on the set of conjugacy classes of elements in $G$.

Sometimes we also extend a character linearly to the free vector space on the set of conjugacy classes. This version of the character can be identified with the bicategorical trace of the identity map of the representation, considered as a $k[G]$-$k$-module.

There is a different notion of an infinitesimal character in Harish–Chandra theory and also a notion of the formal character.

There are important formulas concerning characters in representation theory, like Weyl character formula, Kirillov character formula, Demazure character formula and so on. There is also a formula for the induced character of an induced representation.

Left and right characters on a ring over a ring

Let $A$ be a unital, not necessarily commutative, ring (or, more generally, $k$-algebra for $k$-commutative); then a monoid in a category of $A$-bimodules (which are respectively also compatibly $k$-modules), is called an $A$-ring. In other words an $A$-ring $B$ is an object in the coslice category $A\Ring$; it is thus a ring $B$ equipped with multiplication $\mu_B$ and a map $\eta: A\to B$ of rings.

A left character of an $A$-ring $(B,\mu_B,\eta_B)$ is a map $\chi:B\to A$ such that

(i) (left $A$-linearity) $\chi(\eta(a)b) = a\chi(b)$ for all $a\in A$, $b\in B$

(ii) (associativity) $\chi(b b') = \chi(b (\eta\circ\chi)(b'))$ for all $b,b'\in B$

(iii) (unitality) $\chi(1_B) = \chi(1_A)$

where we denoted multiplication in $A$ and in $B$ by concatenation. The conditions on $\chi$ can be restated as the requirement that the map $B\otimes A\to A$ given by $b\otimes a\mapsto \chi(b\eta(a))$ is a $B$-action extending the left regular $A$-action (i.e. the multiplication on $A$ considered as a left action).

Dually, a right character of an $A$-ring $(B,\mu_B,\eta_B)$ is a map $\chi:B\to A$ such that

(i) (right $A$-linearity) $\chi(b\eta(a)) = \chi(b)a$ for all $a\in A$, $b\in B$

(ii) (associativity) $\chi(b b') = \chi((\eta\circ\chi)(b) b')$ for all $b,b'\in B$

(iii) (unitality) $\chi(1_B) = \chi(1_A)$

This is in turn equivalent to extending the right regular action of $A$ to the action of $B$ on $A$.

Character of a topological space

The character $\chi(X,x)$ of a topological space $X$ at a point $x$ is the minimal cardinality of a local basis of neighborhoods of point $x$ (local basis of topology on $X$) if it is infinite and aleph zero otherwise. The character of a topological space is the supremum of $\chi(X,x)$ when $x$ runs through $X$.

Properties

• characters are cyclotomic integers

Examples

• character table of 2D4=Dic2=Q8

• character table of 2T

• character table of 2O

• character table of 2I

• character table of GL(2,3)

Related concepts

• irreducible character

• Weyl character formula

• Kac character formula

• Kac-Weyl character

• Cheeger-Simons differential character

• Chern root, splitting principle

• character ring

• table of marks

References

Character rings of compact Lie groups are discussed in

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

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

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

Discussion 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)

for review see

• Arpon Raksit, Characters in global equivariant homotopy theory, 2015 pdf

Examples of characters of linear representations of finite groups are discussed and listed at

• James Montaldi, representations, 2008