I'm studying Identical Particles in Quantum Mechanics and somewhere I saw that some particles, called Anyons, can generalize the concept of Fermions and Bosons, showing a symmetry under permutation expressed by $$ P\left | \psi \right > = e^{i\theta}\left| \psi \right > $$ where $\psi$ is the state of system (of $N$ particles) and $P$ is any $N$-particle permutation. The case where $\theta = 0$ express the Boson case and $\theta = p\pi$, where $p$ is the parity of $P$, is the Fermion case.
At first sight, it's plausible that such symmetry can occur, at least at a mathematical level. It's a consequence of the unitarity property of any permutation operator, and the fact that the equation above is a eigenvalue problem.
However, there is another property of Permutation Operators that turns it strange. Any permutation of $N$-particles can be decomposed as $N-1$ permutations of 2 particles. For example, if $P_{123}$ is a 3-particle permutation, we can write $P_{123}=P_{12}P_{23}$. It happens that 2-particle permutations are also Hermitian, so it's Eigenvalue will always be $\pm 1$. Considering this, how Anyons can be possible?
