Parhaps, I cannot yet say. Thank you for the link.

Does anybody know why the question was voted down? Is it too simple (should rather be posted on math.stackexchange)? Or never be posted?

Thanks for the answer. I am still processing it. When you say the unit roots, you mean $\exp(2\pi i j/n)$, for $j\in[n]$ and where $i$ is the imaginary unit, right?

What is a rotation? Take an undirected chain and a lazy directed chain. I flip a coin: either (a) both chains go clockwise, or (b) one goes counterclockwise and the other stays put? Oh... that's right! Now on to the even case...