So I'm pretty new into Representation Theory having so far covered only a couple of example sheets. I'm thinking about the following question:
Suppose we have the group $D_{2n}$ (for clarity this is the dihedral group of order $2n$, as notation can differ between texts).
We can describe this group as follows:
$$\langle \sigma, \tau | \sigma ^n=1, \tau ^2=1, \tau \sigma \tau = \sigma ^{-1} \rangle$$
We know this is isomorphic to the symmetries of the regular $n$-gon.
Question How can we construct a two-dimensional representation $\psi: D_{2n} \rightarrow \text{GL}_2(\mathbb{R})$?
Thoughts Do we simply set up a map that takes $\sigma$ to the standard $2 \times 2$ rotation matrix, with $\theta$ being $\frac{2 \pi}{n}$ and $\tau$ to something like the matrix $\left( \begin{array}{ccc} 0 & 1 \\ 1 & 0 \end{array} \right)$? If this is the case, is there any simple way to show it is two-dimensional?