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.


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