Additive characters of a finite abelian group

Let $G$ be a finite abelian group (Group and abelian group), written additively: its operation is written $+$, its identity is written $0$, and its underlying set is finite. Let ${\mathbb{C}}^{\times }$ be the group of units of the field $\mathbb{C}$ ($\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$, The units of a ring are the invertible elements of its multiplicative monoid, and $R^{\times }$ is a group under multiplication; $0\in R^{\times }$ only in the zero ring), that is, the nonzero complex numbers under multiplication.

An additive character of $G$ is a group homomorphism (Monoid homomorphism and group homomorphism)

$\chi :G⟶{\mathbb{C}}^{\times },\chi (x+y)=\chi (x)\chi (y)(x,y\in G)\mathrm{.}$

Write $\widehat{G}$ for the set of all additive characters of $G$. Since $G$ and ${\mathbb{C}}^{\times }$ are groups, every additive character automatically satisfies $\chi (0)=1$ and $\chi (-x)=\chi (x{)}^{-1}$ (A group homomorphism automatically satisfies $f(e)=e^{\mathrm{\prime }}$ and $f(g^{-1})=f(g{)}^{-1}$, and $f(g^n)=f(g{)}^n$ for every $n\in \mathbb{Z}$; for monoid homomorphisms preservation of the identity must be assumed), so multiplicativity is the whole requirement.

This is not the representation-theoretic character. The word "character" already names the function ${\chi }_V(g)=\mathrm{tr}({\rho }_V(g))$ attached to a finite-dimensional complex representation $V$ of $G$ (The character ${\chi }_V(g)=\mathrm{tr}({\rho }_V(g))$ of a finite-dimensional complex representation, A finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ over a field, and its degree). That function need not be multiplicative — only for a one-dimensional representation is it a homomorphism to ${\mathbb{C}}^{\times }$ — and the next item proves the precise bridge: the additive characters of a finite abelian group are exactly the characters of its one-dimensional complex representations, and exactly its irreducible ones.

Well-definedness. $\widehat{G}$ is defined as a set of functions $G\to {\mathbb{C}}^{\times }$, so the definition itself selects nothing. In particular it chooses no decomposition of $G$, no enumeration of $G$, and no isomorphism $G\cong \widehat{G}$, and it asserts nothing here about a group structure on $\widehat{G}$; the dictionary proved in the next item records what structure there is.

• The definition makes sense for an arbitrary abelian group; finiteness is kept because the page works with finite groups throughout, and it is what makes the orthogonality average of the later item a finite sum.

• For $G$ written multiplicatively the same condition reads $\chi (gh)=\chi (g)\chi (h)$. The additive notation is not a change of mathematics: it records that $G$ is abelian and that its operation is being written as addition, so that $\chi$ is a homomorphism of the additive group $G$ into the multiplicative group ${\mathbb{C}}^{\times }$.