I'm confronted once more with a problem on representation theory which I cannot fully solve (Problem 3.17 http://math.mit.edu/~etingof/replect.pdf):
Let $G$ be the group of symmetries of a regular N-gon (it has 2N elements).
(a) Describe all irreducible complex representations of this group (consider the case of odd and even N).
(b) Let $V$ be the 2-dimensional complex representation of $G$ obtained by complexification of the standard represeation on the real plane (the plane of the polygon). Find the decomposition of $V \otimes V$ in a direct sum of irreducible representations.
I've solved part (a), and I'm left with the second part. To find the irreducible representations of $D_n$ I distinguished between two cases (n odd or even):
n=odd:
1-dimensional irreducible representations: the trivial representation and the sign representation.
2-dimensional irreducible representations: They are given by $\rho_k: D_n \rightarrow End(\mathbb{C})$ such that
$r \mapsto \rho_k(r)= \left( \begin{array}{ccc} e^{\frac{2\pi i}{n}k} & 0 \\ 0 & e^{-\frac{2\pi i}{n}k}\\ \end{array} \right) $ and $s \mapsto \rho_k(s)= \left( \begin{array}{ccc} 0 & 1 \\ 1 & 0\\ \end{array} \right) $, $k=1, \ldots \frac{n-1}{2}$
n=even:
1-dimensional irreducible representations: there are four in this case, given by the four combinations of the following actions: $r \mapsto \pm 1, s\mapsto \pm 1$.
2-dimensional irreducible representations: given by $\rho_k: D_n \rightarrow End(\mathbb{C})$ such that
$r \mapsto \rho_k(r)= \left( \begin{array}{ccc} e^{\frac{2\pi i}{n}k} & 0 \\ 0 & e^{-\frac{2\pi i}{n}k}\\ \end{array} \right) $ and $ s \mapsto \rho_k(s)= \left( \begin{array}{ccc} 0 & 1 \\ 1 & 0\\ \end{array} \right) $, $k=1, \ldots \frac{n}{2}-1$
I'm assuming that to solve this last part I might have to make a computation using the characters, but once more I don't know where to start.
Thank you very much.