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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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    If you actually want to find a cycle decomposition, this probably isn't the way to do it. What is the intended application?2011-02-15
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    @Qiaochu Yuan: I know. I'm not actually interested in finding the dcd, I just want to prove a relation between two different constructions and it seems that expressing permutations as matrices may make things easier in this case.2011-02-15

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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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    @Mariano: I guess the multiplicity of 1 as an eigenvalue is the number of *fixed points* (and not the number of cycles). If I am correct, I will delete this comment after you modify your answer.2011-02-15
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    @Didier: pick an $n$-cycle in $S_n$ and compute the eigenvalues of its corresponding permutation matrix. What is the multiplicity of $1$?2011-02-15
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    An example is the shortest way from a guess to a statement :)2011-02-15
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    @Mariano: *Touché*. What is more, this also stems from the formula for $\mu_1$ you provide next. Well, I guess the hour is too late for me...2011-02-15
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    Thanks, I actually started thinking about the characteristic polynomial shortly after submitting the question. This seems to fit my purposes, but I still need to verify it.2011-02-15
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    @Anthony: to check it, notice that a permutation matrix is a diagonal block matrix with blocks corresponding to the cycles, and that $\mu_n$ and $c_\ell$ are «additive» with respect to *that*, so that in essence it is enough to check this when the permutation is just a cycle. That is a simple matter, for you can do everything completely explicitly.2011-02-15