# Certain Isomorphic Representations of the dihedral group $D_{3}$

Using the following presentation of the dihedral group $$D_{3}$$ $$$$D_{3} = \left\langle r,s \mid r^{2} = s^{2} = (rs)^{3} = e \right\rangle$$$$ There is one (up to isomorphism) irreducible 2-dimensional complex representation $$\begin{equation*} \begin{matrix} \rho : D_{3} \to \operatorname{GL}(2, \mathbb{C}) \\ r \mapsto \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \\ s \mapsto \begin{pmatrix} 0 & e^{\frac{-2\pi i}{3}} \\ e^{\frac{2\pi i}{3}} & 0 \end{pmatrix} \end{matrix} \end{equation*}$$ is there an isomorphic complex irreducible representation to this one such that the entries in the matrices are all in $$\mathbb{Z}$$? I think there should be since $$D_{3} \cong S_{3}$$.

• would you like to see one in $SL_2({\Bbb{Z}}_2)$? Mar 20, 2019 at 2:26
• I commented cuz $\Bbb Z$ hasn't finite subgroups. Mar 20, 2019 at 2:29
• Nitpicking: direct sum of two trivial representations is also two-dimensional, which is inequivalent to $\rho$. You need an extra adjective irreducible.
– Orat
Mar 20, 2019 at 6:25
• @Orat I have edited the question to specify that the representations should be simple. Thanks for your detailed and intuitive answer! Mar 20, 2019 at 13:16

Consider the following assignment: $$\sigma \colon r \mapsto \begin{bmatrix}-1 & 0\\ 1 & 1\end{bmatrix},\qquad s \mapsto \begin{bmatrix}1 & 1\\ 0 & -1\end{bmatrix}.$$ You can check that it is indeed a representation of $$D_3$$ that is equivalent to $$\rho$$ by comparing their characters.
Geometric meaning. Let $$e_1, e_2, e_3$$ be the standard basis of $$\mathbb{R}^3$$. Take $$\alpha = e_1 - e_2, \beta = e_2 - e_3$$ and consider $$2$$-dimensional subspace $$V = \operatorname{Span}_\mathbb{R}\{\alpha, \beta\} = \{\, x_1e_1 + x_2e_2 + x_3e_3 \mid x_1 + x_2 + x_3 = 0 \,\}$$. Define a linear transformation $$\sigma_\alpha$$ on $$V$$ as a reflection with respect to $$\alpha^\perp = \{\, v \in V \mid \langle \alpha, v \rangle = 0 \,\}$$: $$\sigma_\alpha(v) = v - 2\frac{\langle \alpha, v \rangle}{\langle \alpha, \alpha\rangle}\alpha \qquad (v \in V).$$ The representation matrix of $$\sigma_\alpha$$ is $$\left[\begin{smallmatrix}-1 & 0\\ 1 & 1\end{smallmatrix}\right]$$ as $$\sigma_\alpha\begin{bmatrix}\alpha\\ \beta\end{bmatrix} =\begin{bmatrix}-1 & 0\\ 1 & 1\end{bmatrix}\begin{bmatrix}\alpha\\ \beta\end{bmatrix}.$$ Similarly, the representation matrix of $$\sigma_\beta$$ is $$\left[\begin{smallmatrix}1 & 1\\ 0 & -1\end{smallmatrix}\right]$$.
Basically, they are symmetry of an equilateral triangle which $$S_3$$ also acts on.