nLab
character

Character on a group

There are many notions of a character for an algebraic structure, often topologized. A character on the group $G$ is a homomorphism into the group of units (invertible elements) of the ground field. Regarding that the codomain is abelian, the set of characters is an abelian group under the pointwise multiplication.

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 $ℝ/\mathbb{Z}$. 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 representation

In representation theory, one defines the character of a representation $\rho :G\to \mathrm{End}(V)$ to be the function on $G$ given by $g\mapsto \mathrm{Tr}\rho (g)$, whenever the trace in $V$ makes sense (e.g. when $V$ is finite-dimensional). Since such a function is invariant under conjugation, we 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 $ARing$; 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) $\chi (bb\prime )=\chi (b(\eta \circ \chi )(b\prime ))$ for all $b,b\prime \in B$

(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) $\chi (bb\prime )=\chi ((\eta \circ \chi )(b)b\prime )$ for all $b,b\prime \in B$

(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 $X$.