## Representations of finite groups; basic definitions & remarks

Posted: February 7, 2011 in Representations of Finite Groups
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Definition 1. Let $V$ be a finite dimensional vector space over $\mathbb{C}.$ The set of all invertible $\mathbb{C}$-linear transformations $T : V \longrightarrow V$ is a group under composition of functions. This group is denoted by $\text{GL}(V).$

Remark 1.  We know from linear algebra that an element of $T \in \text{GL}(V)$ corresponds to an invertible matrix $[T]_{\mathcal{B}},$ where $\mathcal{B}$ is a $\mathbb{C}$-basis for $V.$ So if $\dim_k V = m,$ then $\text{GL}(V)$ is just the general linear group $\text{GL}(m, \mathbb{C}),$ i.e. $\text{GL}(V)$ can also be viewed as the group of $m \times m$ invertible matrices with entries in $\mathbb{C}.$

Definition 2. Let $G$ be a finite group. A (complex) representation of $G$ is a group homomorphism $\rho : G \longrightarrow \text{GL}(V).$ The degree of this representation is defined to be $\dim_{\mathbb{C}} V.$ We will denote by $\deg \rho$ the degree of $\rho.$

Remark 2. Let $\rho : G \longrightarrow \text{GL}(V)$ be a representation of $G$ and $|G|=n.$ Let $g \in G.$ Then $g^n = 1,$ where $1$ is the identity element of $G.$ Thus, since $\rho$ is a group homomorphism, we have

$(\rho(g))^n = \rho(g^n)=\rho(1) = \text{id}_V,$

where $\text{id}_V$ is the identity map of $V.$

Remark 3. A degree one representation of $G$ is just a group homomorphism $\rho : G \longrightarrow \mathbb{C}^{\times},$ where $\mathbb{C}^{\times}$ is the multiplicative group of $\mathbb{C}.$

Proof. If $\dim_{\mathbb{C}} V =1,$ then, as we mentioned in Remark 1, $\text{GL}(V)$ can be viewed as the group of $1 \times 1$ invertible matrices with entries from $\mathbb{C},$ which is obviously $\mathbb{C}^{\times}.$

Remark 4. Suppose that $\rho$ is a degree one representation of $G$ and let $|G|=n.$ Let $g \in G.$ Then, by Remark 2 and 3, $(\rho(g))^n = 1 \in \mathbb{C}^{\times}.$ So in this case $\rho(g)$ is an $n$-th root of unity in $\mathbb{C},$ for all $g \in G.$