Amplitude of superposition states I have a question about the following chart (for 3 and 4 particles case):

What does $P_{cycl}$ mean? How's that relevant to the amplitude of superposition states and their symmetricity?
 A: *

*The states given in your table seem to be states of a system of N spins in the $\hat{S}^z_{\text{tot}}$ and $\hat{S}^2_{\text{tot}}$ basis which are also eigenstates of the cyclic permutation operator $\hat{P}_{\text{cycl}}$ with $P_{\text{cycl}}$ being the eigenvalue of the cyclic permutation operator.


*An Example definition for the cyclic permutation Operator would be:
$$\hat{P}_{\text{cycl}}|s_1,...,s_n>=|s_n,s_1,...,s_{n-1}>$$


*You could calculate this explicitly for these states to convince yourself for example:
$$\hat{P}_{\text{cycl}}(|\uparrow\uparrow\downarrow>+e^{i\frac{2}{3}\pi}|\uparrow\downarrow\uparrow>+e^{-i\frac{2}{3}\pi}|\downarrow\uparrow\uparrow>)=
|\downarrow\uparrow\uparrow>+e^{i\frac{2}{3}\pi}|\uparrow\uparrow\downarrow>+e^{-i\frac{2}{3}\pi}|\uparrow\downarrow\uparrow>=e^{i\frac{2}{3}\pi}(|\uparrow\uparrow\downarrow>+e^{i\frac{2}{3}\pi}|\uparrow\downarrow\uparrow>+e^{-i\frac{2}{3}\pi}|\downarrow\uparrow\uparrow>)$$
$$\longrightarrow P_{\text{cycl}}=e^{i\frac{2}{3}\pi}$$


*The $P_{\text{cycl}}$ is just a third quantumnumber you could use to characterise these states. That means if you would have a N-particle Hamitlonian which commutes with the operators $\hat{S}^z_{\text{tot}}$, $\hat{S}^2_{\text{tot}}$ and $\hat{P}_{\text{cycl}}$ the usage of these states as a basis would probably reduce the complexity of your problem.


*To use these states as an orthonormal basis, you need to normalise them. Therefore, $P_{\text{cycl}}$ is in my opinion only relevant for the normalisation of the states concerning the "amplitudes".


*As already discussed these states are "symmetric" under the cyclic permutation operation with different eigenvalues.
I hope this helps you with your understanding.
Jan
