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A character of a group representation is obtained by taking trace of each matrix in this representation.

The word character is often used in the sense that it is a homomorphism from a group to $(\mathbb C\setminus\{0\},\cdot)$ (sometimes with some additional properties). This kind of characters arises, e.g., in Pontryagin duality or as Dirichlet characters in number theory.

Is there some relation between these two (frequently used) meanings of the world character, or is it just a coincidence that the same word was used for both of them?

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BTW only now I found out what your gravatar is. (But I have to admit that I don't know anything about the mathematics behind it.) – Martin Sleziak Dec 24 '12 at 10:59
Good work, detective. :-) – anon Dec 24 '12 at 11:01
up vote 3 down vote accepted

A $1$-dimensional representation of a group is a continuous homomorphism into $\Bbb C^\times$: in particular it is equal to its own trace (once we identify $\mathrm{GL}_1(\Bbb C)\cong\Bbb C^\times$). So the group homomorphism definition of characters is the $1$-dimensional case of the trace-of-a-representation definition. To copy myself:

The homomorphisms $G\to \Bbb C^\times$ are actually not the whole story of character theory, but are a very tidy chapter in it. If $V$ is a vector space (over $\Bbb C$) and $G$ finite, the homomorphisms $G\to GL(V)$ from $G$ into the general linear group of invertible linear maps are called representations, which are essentially the ways to equip $V$ with a linear $G$ action. If $\rho$ is a representation, then the map given by $\chi_\rho:G\to \Bbb C:g\mapsto\mathrm{tr}\,\rho(g)$ (the trace of the linear map associated to $g$, which is independent of basis or coordinate choice for $V$) is called a character.

If $V$ is one-dimensional (in which case we call $\rho$ and $\chi_\rho$ one-dimensional as well) then $\rho=\chi_\rho$ and the characters are multiplicative. Note that $\mathrm{tr}\,\rho(e_G)=\dim\,V$ shows the dimension can be directly computed from the character, so there is no ambiguity with respect to what dimension a character may have. With a distinguished basis we have $V\cong \Bbb C^n$ in an obvious way, so we can write $GL(V)$ as $GL_n(\Bbb C)$, in which we are working with matrix representations specifically.

(Over here.)

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Why just finite groups? – Andrea Mori Dec 24 '12 at 11:04
@Andrea - Erm, because I didn't want to bother writing "continuous" I guess. – anon Dec 24 '12 at 11:06
@Andrea So far I have only learned about linear representations of finite groups. But maybe later I'll try to understand at least basics of some of other branches of representation theory. – Martin Sleziak Dec 24 '12 at 11:18
Well, it seems to me that given any group $G$ and any homomorphism $\rho:G\rightarrow\Bbb C^\times$, then $\rho(g)=\text{tr}(\rho(g))$. – Andrea Mori Dec 24 '12 at 11:24
@Andrea, right. When a topology on $G$ is assumed and $G$ is infinite, I am not used to the noncontinuous homomorphisms $G\to GL(V)$ being admitted under the label "representation," though I have not studied representation theory this far anyway. Is it standard to accept noncontinuous homomorphisms as representations? – anon Dec 24 '12 at 11:33

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