Character Theory

Recall

Let $k$ be a field and vector space $V$ over $k$. The trace operator

$$Tr:En{d}_k(V)\to k$$

does not depend on the basis choice.

Definition of Character

Let $k$ be a field and $V$ be a vector space over $k$. If we have a group representation $\rho :G\to G{L}_k(V)$. The character of the representation $\rho$ is a map

$${\chi }_{\rho }:G\to k,\text{where}{\chi }_{\rho }(g)=Tr(\rho (g)).$$

Special Characters

Trivial Character

Trivial Representation.

Regular Character

Equivalent to Regular Representation.

Let $G=\left\{g_1,\dots ,g_n\right\}$ such that $\left|G\right|=n$. As a terminolog in Serre,

$$\{\begin{array}{ll}1& g_i=g{g}_j\\ 0& \text{else}\end{array}=\{\begin{array}{ll}1& g=g_i{g}_j^{-1}\\ 0& \text{else}\end{array}.$$

Therefore, if $g=e$, ${\chi }_e\equiv \left|G\right|$, and otherwise 0.

Dual Character

Given Dual Representation. So ${\chi }_{{\rho }^{\ast }}(g)={\chi }_{\rho }(g^{-1})$.

Definition of General Characters

1. $f$ is a 1 dimentional Character if $f:G\to k^{\times }$ is a group homomorphism.
2. $f$ is a Simple Character if $f={\chi }_V$ where $V$ is a simple $kG$-module.
3. $f$ is a Virtual Character if $\exists \rho ,{\rho }^{\prime }$ such that $f={\chi }_{\rho }-{\chi }_{{\rho }^{\prime }}$.
4. $f$ is a Class function$if$\forall g, h \in G$,$f(g) = f(hgh^{-1})$.

Remark 1

$$\left\{1d-Character\right\}\subset \left\{SimpleCharacter\right\}\subset \left\{Character\right\}\subset \left\{VirtualCharacter\right\}\subset \left\{ClassFunctions\right\}.$$

Remark 2

$\left\{Characters\right\}$ is as monoid Ring but does not have additive inverse.

$\left\{VirtualCharacters\right\}$ is commutative ring.

$\left\{ClassFunctions\right\}$ is a $k$-algebras.

Property

Proposition