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Everything the deals with properties and definitions of Haar measure, as well as related fields when the question relies heavily on the notion of haar measure - group harmonic analysis, group ergodic theory etc.

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Question about squaring a Haar random unitary

Consider an $n \times n$ unitary $U$, drawn from the Haar measure. I'm trying to find the distribution for $U^{2}$. Is it true that $U^{2}$ is also Haar random? Note that for any fixed unitary $V$, $V …