Images of eigenvalue distributions under power maps

In [6], it was shown that if U is a random n×n unitary matrix, then for any p ≥ n, the eigenvalues of Up are i.i.d. uniform; similar results were also shown for general compact Lie groups. We study what happens when p < n instead. For the classical groups, we find that we can describe the eigenvalue distribution of Up in terms of the eigenvalue distributions of smaller classical groups; the earlier result is then a special case. The proofs rely on the fact that a certain subgroup of the Weyl group is itself a Weyl group. We generalize this fact, and use it to study the power-map problem for general compact Lie groups. In [6], it was shown that if a (uniformly) random n × n unitary matrix U is raised to a power p ≥ n, the eigenvalues of the resulting matrix are (exactly) independently distributed; this despite the rather complicated dependence between the eigenvalues of U itself. Our purpose in the present note is to extend this result to the case p < n. We find that the eigenvalue distribution of U can in that case be described in terms of a union of p independent distributions, each of which is itself the eigenvalue distribution of a random unitary matrix. More precisely, we have: U(n) ∼ ⊕ 0≤i<p U(⌈ n− i p ⌉). (1) That is, if we take the pth power of a uniformly distributed element of U(n), the resulting eigenvalue distribution is the same as if we took the union of the eigenvalues of p independent matrices. For p ≥ n, this reduces to the earlier result, since then we have U(n) ∼ ⊕