I am working in a problem which turn out that in one of its limits I have to solve the eigenvalue problem for a random permutation matrix.
I would like to know therefore more about the symmetric group, and more specifically about spectral properties of permutation matrices. Where can I study the symmetric group and try to understand it from a pedagogical point of view?
Are you identifying the symmetric group $S_n$ with its regular representation (i.e. its permutation representation), by any chance? In other words, are you identifying e.g. the cycle $(123)\in S_3$ with the matrix $ \left( \begin{array}{ccc} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} \right) $, the matrix that implements a $(123)$ permutation of the three basis vectors of $\mathbb C^3$?
If so, there is an easy way to find the eigenvalues. In the example above for $(123)$ the eigenvalues are $1, \zeta_3, \zeta_3^2$, where $\zeta_3 = \exp(2\pi i /3)$.
The eigenvectors are $(1,1,1), (1,\zeta_3^2, \zeta_3), (1, \zeta_3, \zeta_3^2)$, which you can verify by substitution.
To give you another example, if you're given $(123)(45678) \in S_8$, then the eigenvalues are $1, \zeta_3, \zeta_3^2$ and $1, \zeta_5, \zeta_5^2, \zeta_5^3, \zeta_5^4$, i.e. you read them off from the individual cycles. Hopefully you can see why this is true and hopefully you can guess what the corresponding eigenvectors are by extrapolating the previous example.