Character group

Definition

If G is an 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.

Alternative definition

There is another definition of character group^{pg 29} which uses ${\displaystyle U(1)=\{z\in {\mathbb{C}}^{\ast }:|z|=1\}}$ as the target instead of just ${\displaystyle {\mathbb{C}}^{\ast }}$. This is useful when studying complex tori because the character group of the lattice in a complex torus ${\displaystyle V/\mathrm{\Lambda }}$ is canonically isomorphic to the dual torus via the Appell–Humbert theorem. That is,

${\displaystyle \text{Hom}(\mathrm{\Lambda },U(1))\cong V^{\vee }/{\mathrm{\Lambda }}^{\vee }=X^{\vee }}$

We can express explicit elements in the character group as follows: recall that elements in ${\displaystyle U(1)}$ can be expressed as

${\displaystyle e^{2\pi ix}}$

for ${\displaystyle x\in \mathbb{R}}$. If we consider the lattice as a subgroup of the underlying real vector space of ${\displaystyle V}$, then a homomorphism

${\displaystyle \varphi :\mathrm{\Lambda }\to U(1)}$

can be factored as a map

${\displaystyle \varphi :\mathrm{\Lambda }\to \mathbb{R}\stackrel{\exp (2\pi i\cdot )}{\to }U(1)}$

This follows from elementary properties of homomorphisms. Note that

${\displaystyle \begin{array}{cc}\varphi (x+y)& =\exp (2\pi if(x+y))\\ & =\varphi (x)+\varphi (y)\\ & =\exp (2\pi if(x))\exp (2\pi if(y))\end{array}}$

giving us the desired factorization. As the group

${\displaystyle \text{Hom}(\mathrm{\Lambda },\mathbb{R})\cong \text{Hom}({\mathbb{Z}}^{2n},\mathbb{R})}$

we have the isomorphism of the character group, as a group, with the group of homomorphisms of ${\displaystyle {\mathbb{Z}}^{2n}}$ to ${\displaystyle \mathbb{R}}$. Since ${\displaystyle \text{Hom}(\mathbb{Z},G)\cong G}$ for any abelian group ${\displaystyle G}$, we have

${\displaystyle \text{Hom}({\mathbb{Z}}^{2n},\mathbb{R})\cong {\mathbb{R}}^{2n}}$

after composing with the complex exponential, we find that

${\displaystyle \text{Hom}({\mathbb{Z}}^{2n},U(1))\cong {\mathbb{R}}^{2n}/{\mathbb{Z}}^{2n}}$

which is the expected result.

Examples

Finitely generated abelian groups

Since every finitely generated abelian group is isomorphic to

${\displaystyle G\cong {\mathbb{Z}}^n\oplus \underset{i=1}{\overset{m}{⨁}}\mathbb{Z}/a_i}$

the character group can be easily computed in all finitely generated cases. From universal properties, and the isomorphism between finite products and coproducts, we have the character groups of ${\displaystyle G}$ is isomorphic to

${\displaystyle \text{Hom}(\mathbb{Z},{\mathbb{C}}^{\ast }{)}^{\oplus n}\oplus \underset{i=1}{\overset{k}{⨁}}\text{Hom}(\mathbb{Z}/n_i,{\mathbb{C}}^{\ast })}$

for the first case, this is isomorphic to ${\displaystyle ({\mathbb{C}}^{\ast }{)}^{\oplus n}}$, the second is computed by looking at the maps which send the generator ${\displaystyle 1\in \mathbb{Z}/n_i}$ to the various powers of the ${\displaystyle n_i}$-th roots of unity ${\displaystyle {\zeta }_{n_i}=\exp (2\pi i/n_i)}$.

Orthogonality of characters

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 ${\displaystyle j\ne 1}$, 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 ${\displaystyle k\ne 1}$, 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

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

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

${\displaystyle \sum _{k=1}^n{f_k}^{\ast }(g_i)f_k(g_j)=n{\delta }_{ij}}$ ,

where ${\displaystyle {\delta }_{ij}}$ is the Kronecker delta and ${\displaystyle f_k^{\ast }(g_i)}$ is the complex conjugate of ${\displaystyle f_k(g_i)}$.