According to Bacher's article, the eigenvalues of the adjacency matrix of Cayley graph of the symmetric group of order $n$ are $2-2\cos(\pi/n)$; my question is:
If we know that this adjacency matrix is comprised of a direct sum of matrices $B^{\lambda}$, where $\lambda$ is a partition of $n$, and we also know that the eigenvalues of a direct sum of matrices are the union of the eigenvalues of each of the matrices comprising the direct sum, then how do we assign Bacher's eigenvalues to each of the $B^{\lambda}$ matrices? How do we know each of the matrices' eigenvalues? Thanks a lot