Let $\sigma$ be a permutation matrix of order $n$. What are all the invariant subspaces of $\sigma$?
(I can only find 1 and n-1 dimensional subspaces) Thanks in advance.
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3$\begingroup$ By "invariant", do you mean "mapped back to itself"? If so, since every permutation matrix is semisimple (at least in characteristic 0), then the invariant subspaces are direct sums of one-dimensional invariant subspaces. So if you know all the one-dimensional invariant subspaces, then you know all the invariant subspaces. $\endgroup$– Jason StarrCommented Oct 3, 2013 at 16:58
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$\begingroup$ "the invariant subspaces are direct sums of one-dimensional invariant subspaces" . Is this also true for fields that are not algebraically closed? $\endgroup$– JokerCommented Oct 3, 2013 at 19:13
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$\begingroup$ "Is this also true for fields that are not algebraically closed?" No, my comment was for algebraically closed fields. Over a non-algebraically closed field, you can try to use Galois descent: the eigenvalues of $\sigma$ are $n^{\text{th}}$ roots of unity. The Galois group acts on these roots. For each Galois orbit, the sum over that orbit of the projectors onto the corresponding eigenspaces will be Galois invariant. $\endgroup$– Jason StarrCommented Oct 3, 2013 at 22:50
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