# Rotating three (or $n$) times to come back to itself

Hello I'm a quantum mechanics newbie.

I learned about spinors, and how they are different from vectors because unlike vectors, rotating them once does not give the original spinor, but the negative of it.(Although while proving this, we only learned the definition of rotation operators and didn't learn how they were derived or what they meant (the professor said we will learn it in our senior years).

Anyways, this made me wonder.... are there things that, instead of spinors, has to be rotated 3 times, or 4 times, ... n times to come back to what it was? (will it be something like, if it comes back to what it was after n rotations, something like $$e^{2\pi/n}$$ multiplied after one rotation? (So that it comes back to its position after n rotations))

The reason why the spatial dimension is important is because the rotation group has a different topological structure in two dimensions than in more dimensions. The rotation group $$\mathrm{SO}(2)$$ is the circle and has infinite fundamental group, while the higher orthogonal groups $$\mathrm{SO}(n)$$ have only the two-element group as their fundamental group. So passing to the universal cover when seeking the quantum representations of the rotation group only "doubles" the available representations from the vectorial ones for higher dimensions, but yields infinitely many anyonic representations between two vectorial representations in two dimensions.