Character group

Let G be an abelian group. A function ${\displaystyle f:G\to \mathbb {C} \setminus \{0\}}$ mapping the group to the non-zero complex numbers is called a character of G if it is a group homomorphism from ${\displaystyle G}$ to ${\displaystyle \mathbb {C} ^{\times }}$—that is, if ${\displaystyle f(g_{1}g_{2})=f(g_{1})f(g_{2})}$ for all ${\displaystyle g_{1},g_{2}\in G}$.

If f is a character of a finite group G, then each function value f(g) is a root of unity, since for each ${\displaystyle g\in G}$ there exists ${\displaystyle k\in \mathbb {N} }$ such that ${\displaystyle g^{k}=e}$, and hence ${\displaystyle f(g)^{k}=f(g^{k})=f(e)=1}$.

Each character f is a constant on conjugacy classes of G, that is, f(hgh⁻¹) = f(g). For this reason, a character is sometimes called a class function.

A finite abelian group of order n has exactly n distinct characters. These are denoted by f₁, ..., f_n. The function f₁ is the trivial representation, which is given by ${\displaystyle f_{1}(g)=1}$ for all ${\displaystyle g\in G}$. It is called the principal character of G; the others are called the non-principal characters.

If G is a abelian group, then the set of characters f_k forms an abelian group under pointwise multiplication. That is, the product of characters ${\displaystyle f_{j}}$ and ${\displaystyle f_{k}}$ is defined by ${\displaystyle (f_{j}f_{k})(g)=f_{j}(g)f_{k}(g)}$ for all ${\displaystyle g\in G}$. This group is the character group of G and is sometimes denoted as ${\displaystyle {\hat {G}}}$. The identity element of ${\displaystyle {\hat {G}}}$ is the principal character f₁, and the inverse of a character f_k is its reciprocal 1/f_k. If ${\displaystyle G}$ is finite of order n, then ${\displaystyle {\hat {G}}}$ is also of order n. In this case, since ${\displaystyle |f_{k}(g)|=1}$ for all ${\displaystyle g\in G}$, the inverse of a character is equal to the complex conjugate.

Consider the ${\displaystyle n\times n}$ matrix A = A(G) whose matrix elements are ${\displaystyle A_{jk}=f_{j}(g_{k})}$ where ${\displaystyle g_{k}}$ is the kth element of G.

The sum of the entries in the jth row of A is given by

${\displaystyle \sum _{k=1}^{n}A_{jk}=\sum _{k=1}^{n}f_{j}(g_{k})=0}$ if ${\displaystyle j\neq 1}$, and

${\displaystyle \sum _{k=1}^{n}A_{1k}=n}$.

The sum of the entries in the kth column of A is given by

${\displaystyle \sum _{j=1}^{n}A_{jk}=\sum _{j=1}^{n}f_{j}(g_{k})=0}$ if ${\displaystyle k\neq 1}$, and

${\displaystyle \sum _{j=1}^{n}A_{j1}=\sum _{j=1}^{n}f_{j}(e)=n}$.

Let ${\displaystyle A^{\ast }}$ denote the conjugate transpose of A. Then

${\displaystyle AA^{\ast }=A^{\ast }A=nI}$.

This implies the desired orthogonality relationship for the characters: i.e.,

${\displaystyle \sum _{k=1}^{n}{f_{k}}^{*}(g_{i})f_{k}(g_{j})=n\delta _{ij}}$ ,

where ${\displaystyle \delta _{ij}}$ is the Kronecker delta and ${\displaystyle f_{k}^{*}(g_{i})}$ is the complex conjugate of ${\displaystyle f_{k}(g_{i})}$.

• Pontryagin duality

• See chapter 6 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001