# Irreducible representation of the dihedral group of order 6

by ericc   Last Updated October 09, 2019 17:20 PM

I know that up to isomorphism there is only one irreducible representation of $$D_3$$ of dimension 2. Write $$D_3=\{1, x, x^2, xy, x^2y, y\}$$, where $$x^3 = 1$$, $$y^2 = 1$$, $$xy = yx^2$$. Given any $$\theta\in[0, 2\pi)$$, I think the following is one of such irreducible representation:

$$R_y = \left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right],$$

$$R_x = \left[ \begin{array}{cc} -\dfrac{1}{2} & -\dfrac{\sqrt{3}}{2}e^{-i\theta} \\ \dfrac{\sqrt{3}}{2}e^{i\theta} & -\dfrac{1}{2} \end{array} \right].$$

Am I correct? If the answer is yes, then what is the isomorphism $$T:\mathbb{C^2}\rightarrow\mathbb{C^2}$$ between this representation and the standard representation

$$R'_y = \left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right],$$

$$R'_x = \left[ \begin{array}{cc} -\dfrac{1}{2} & -\dfrac{\sqrt{3}}{2} \\ \dfrac{\sqrt{3}}{2} & -\dfrac{1}{2} \end{array} \right].$$

On the other hand, if the answer is no, then what forces $$\theta$$ to be $$0$$?

Thanks!

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You have $$R_xA=AR'_x$$, where $$A=\begin{pmatrix}e^{-i\theta/2}&0\\0&e^{i\theta/2}\end{pmatrix}$$ So the two representations are equivalent.

David Hill
October 09, 2019 17:17 PM

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