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?
Conjugacy class of a permutation from its matrix representation
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matrices
permutations
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0@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
1 Answers
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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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0@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