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

Question

Using the following presentation of the dihedral group $D_{3}$ \begin{equation} D_{3} = \left\langle r,s \mid r^{2} = s^{2} = (rs)^{3} = e \right\rangle \end{equation} 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 $D_{3} \cong S_{3}$.

Answer 1

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.