Anyon

Remarks

The generators $\sigma_i$ are all conjugate in $B_n$, so a one-dimensional representation sends every $\sigma_i$ to the same phase $e^{i\theta}$, the statistics angle. The angle $\theta = 0$ gives bosons and $\theta = \pi$ gives fermions; in both cases $\rho(\sigma_i)^2 = 1$.

Example: $\theta = \pi/3$ gives abelian anyons, since $\rho(\sigma_i)^2 = e^{2\pi i/3} \ne 1$; quasiparticles of the quantum Hall state at filling $1/3$ are predicted to have this angle.

Non-example: $\theta = \pi$ fails the defining condition, since $\rho(\sigma_i)^2 = e^{2\pi i} = 1$; these particles are fermions.

In $\R^d$ with $d \ge 3$ the fundamental group of the configuration space is $S_n$ itself, so $\rho(\sigma_i)^2 = \id$ always and anyons cannot occur.