Character (mathematics)

Character of a representation

The character ${\displaystyle \chi :G\to F}$ of a representation ${\displaystyle \varphi :G\to \mathrm{G}\mathrm{L}(V)}$ of a group G on a finite-dimensional vector space V over a field F is the trace of the representation ${\displaystyle \varphi }$ (Serre 1977), i.e.

${\displaystyle {\chi }_{\varphi }(g)=\mathrm{Tr}(\varphi (g))}$ for ${\displaystyle g\in G}$

In general, the trace is not a group homomorphism, nor does the set of traces form a group. The characters of one-dimensional representations are identical to one-dimensional representations, so the above notion of multiplicative character can be seen as a special case of higher-dimensional characters. The study of representations using characters is called "character theory" and one-dimensional characters are also called "linear characters" within this context.

Alternative definition

If restricted to finite abelian group with ${\displaystyle 1\times 1}$ representation in ${\displaystyle \mathbb{C}}$ (i.e. ${\displaystyle \mathrm{G}\mathrm{L}(V)=\mathrm{G}\mathrm{L}(1,\mathbb{C})}$), the following alternative definition would be equivalent to the above (For abelian groups, every matrix representation decomposes into a direct sum of ${\displaystyle 1\times 1}$ representations. For non-abelian groups, the original definition would be more general than this one):

A character ${\displaystyle \chi }$ of group ${\displaystyle (G,\cdot )}$ is a group homomorphism ${\displaystyle \chi :G\to {\mathbb{C}}^{\ast }}$ i.e. ${\displaystyle \chi (x\cdot y)=\chi (x)\chi (y)}$ for all ${\displaystyle x,y\in G.}$

If ${\displaystyle G}$ is a finite abelian group, the characters play the role of harmonics. For infinite abelian groups, the above would be replaced by ${\displaystyle \chi :G\to \mathbb{T}}$ where ${\displaystyle \mathbb{T}}$ is the circle group.