Abstract.
In [9], it was shown that if U is a random n×n unitary matrix, then for any p≥n, the eigenvalues of U p 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 U p 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.
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Received: 19 December 2000 / Revised version: 7 August 2002 / Published online: 21 February 2003
Permanent address: Center for Communications Research, Princeton, NJ, USA. e-mail: rains@idaccr.org
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Rains, E. Images of eigenvalue distributions under power maps. Probab. Theory Relat. Fields 125, 522–538 (2003). https://doi.org/10.1007/s00440-002-0250-2
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DOI: https://doi.org/10.1007/s00440-002-0250-2