nLab character




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 GG (a group character) is a group homomorphism from GG to the group of units k ×k^\times of the ground field kk:

χ:Gk × \chi \;\colon\; G \longrightarrow k^\times

Since the codomain k ×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 GG.

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

For more see at group character.

For topological groups one considers continuous characters. Specifically, for a locally compact Hausdorff group GG (often further assumed to be an abelian group), a character of GG is a continuous homomorphism to the circle group /\mathbb{R}/\mathbb{Z}. If GG 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 ρ:GEnd(V)\rho\colon G\to End(V) to be the group character on GG given by gTrρ(g)g \mapsto Tr \rho(g), whenever the trace in VV makes sense (e.g. when VV 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 GG.

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

A left character of an AA-ring (B,μ B,η B)(B,\mu_B,\eta_B) is a map χ:BA\chi:B\to A such that

(i) (left AA-linearity) χ(η(a)b)=aχ(b)\chi(\eta(a)b) = a\chi(b) for all aAa\in A, bBb\in B

(ii) (associativity) χ(bb)=χ(b(ηχ)(b))\chi(b b') = \chi(b (\eta\circ\chi)(b')) for all b,bBb,b'\in B

(iii) (unitality) χ(1 B)=χ(1 A)\chi(1_B) = \chi(1_A)

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

Dually, a right character of an AA-ring (B,μ B,η B)(B,\mu_B,\eta_B) is a map χ:BA\chi:B\to A such that

(i) (right AA-linearity) χ(bη(a))=χ(b)a\chi(b\eta(a)) = \chi(b)a for all aAa\in A, bBb\in B

(ii) (associativity) χ(bb)=χ((ηχ)(b)b)\chi(b b') = \chi((\eta\circ\chi)(b) b') for all b,bBb,b'\in B

(iii) (unitality) χ(1 B)=χ(1 A)\chi(1_B) = \chi(1_A)

This is in turn equivalent to extending the right regular action of AA to the action of BB on AA.

Character of a topological space

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




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

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

Last revised on August 22, 2022 at 14:23:39. See the history of this page for a list of all contributions to it.