## Representations of finite groups; basic examples (3)

Posted: February 8, 2011 in Representations of Finite Groups
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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 $K_4.$