Apologies if this is too basic, but given a permutation matrix $M$, is there any parameter or formula based on $M$ that gives the disjoint cycle decomposition, or at least the conjugacy class, of the corresponding permutation?
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I don't really see what exactly you expect, as «parameter or formula based on $M$» is a pretty vague description. Here is one option... You can compute the number of cycles of a specific length from the multiplicities of other eigenvalues---which are all roots of unity. For example, the multiplicity of $1$ as an eigenvalue is the number of cycles. More generally, if we call $c_\ell$ the number of cycles of length $\ell$ in the permutation, and $\mu_n$ the multiplicity of $e^{2\pi i/n}$ as an eigenvalue of the matrix, then $$\mu_n = \sum_{n\mid\ell}c_\ell.$$ This relation be inverted using Moebius inversion to a formula exactly expressing the $c_\ell$ in terms of the multiplicities $\mu_n$. |
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