Representations of finite groups; basic examples (3)

Posted: February 8, 2011 in Representations of Finite Groups
Tags: , ,

In this post we are going to look at degree two representations of the dihedral group of order 2m. We will construct m such representations.

Example 1. Let D_{2m} be the dihedral group of order 2m, as described in Example 2. Then, as we mentioned in that example, every element of D_{2m} is written uniquely as g_1^j g_2^k, where 0 \leq j \leq 1 and 0 \leq k \leq m-1. Let

V = \mathbb{C}^2.

Let \zeta be an m-th root of unity and  define T, S : V \longrightarrow V by

T(x,y)=(y,x) and S(x,y)=(\zeta x, \zeta^{- 1}y),

for all x,y \in \mathbb{C}. Clearly T and S are isomorphisms and so they are in \text{GL}(V). It is easily seen that

T^2=S^m=(TS)^2=\text{id}_V.

That means T and S satisfy the same relations as g_1,g_2 do. So the map \rho : D_{2m} \longrightarrow \text{GL}(V) defined by

\rho(g_1^jg_2^k)=T^jS^k,

for 0 \leq j \leq 1 and 0 \leq k \leq m-1, is a well-defined group homomorphism. Thus \rho is a representation of D_{2m} and \deg \rho = \dim_{\mathbb{C}} V = 2. This gives us m representations of degree two for D_{2m} because the number of m-th roots of unity is m.

Remark. As I mentioned in Remark 1, the elements of \text{GL}(V) may also be viewed as invertible matrices: let \zeta be an m-th root of unity and consider V = \mathbb{C}^2 with the standard basis. Let

A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} and B = \begin{pmatrix} \zeta & 0 \\ 0 & \zeta^{-1} \end{pmatrix}.

The linear transformation corresponding to A (resp. B) is clearly T (resp. S) defined in the above example. It might be easier sometimes to work with corresponding matrices instead of linear transformations. So the 2m matrices corresponding to the representations defined in Example 1 are

\begin{pmatrix} \zeta^{k} & 0 \\ 0 & \zeta^{-k} \end{pmatrix}

and

\begin{pmatrix} 0 & \zeta^{-k} \\ \zeta^{k} & 0 \end{pmatrix},

where \zeta is an m-th root of unity and 0 \leq k \leq m-1.

Example 2. Since S_3 \cong D_6, Example 1 gives us three degree two representations of S_3.

Example 3. Since K_4, the Klein four-group is isomorphic to D_4, Example 1 gives us two degree two representations of K_4.

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