Character of a group

\(\DeclareMathOperator\Hom{Hom}\) A homomorphism of the given group into some standard Abelian group $A$. Usually, $A$ is taken to be either the multiplicative group $k^*$ of a field $k$ or the subgroup

\[ T = \{ z ∈ ℂ: |z| = 1 \} \]

of $ℂ^*$. The concept of a character of a group was originally introduced for finite groups $G$ with $A = T$ (in this case every character $G → ℂ^*$ takes values in $T$).

The study of characters of groups reduces to the case of Abelian groups, since there is a natural isomorphism between the groups $\Hom(G, A)$ and $\Hom(G / (G, G), A)$, where $(G, G)$ is the commutator subgroup of $G$. The characters $G → k^*$ form a linearly independent system in the space of all $k$-valued functions on $G$. A character $G → k^*$ extends uniquely to a character of the group algebra $k[G]$. The characters $G → k^*$ are one-dimensional linear representations of $G$ over $k$; the concept of a character of a representation of a group coincides in the one-dimensional case with the concept of a character of a group. Sometimes characters of a group are understood to mean characters of any of its finite-dimensional representations (and even to mean the representations themselves).

A character of a topological group $G$ is a continuous homomorphism $G → T$. If $G$ is a locally compact Abelian group, then its characters separate points, that is, for any $a,b ∈ G$, $a ≠ b$, there exists a character $α: G → T$ such that $α(a) ≠ α(b)$. For Hausdorff Abelian groups $G$ this assertion is not true, in general (see [3]). A character of an algebraic group $F$ over an algebraically closed field $K$ is a rational homomorphism $G → K^*$.

In number theory an important role is played by the characters of the multiplicative group $ℤ_k^*$ of the residue ring $ℤ_k$ modulo $k$, which correspond one-to-one to Dirichlet characters modulo $k$: To a character $α: ℤ_k^* → T$ there corresponds the Dirichlet character $χ: ℤ → ℂ$ given by the formula

\[ χ(n) = \begin{cases} α(n + kℤ) & \text{if}\quad (n, k) = 1;\\ 0 & \text{if}\quad (n, k) ≠ 1. \end{cases} \]

See also Character group.

References

[1]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2]	S.A. Morris, "Pontryagin duality and the structure of locally compact Abelian groups" , London Math. Soc. Lecture Notes , 29 , Cambridge Univ. Press (1977) MR0442141 Zbl 0446.22006
[3]	E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1 , Springer (1963) MR0156915 Zbl 0115.10603