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I could find many resources on generating random unitary matrices, usually citing F. Mezzadri, Notices of the AMS 54 (2007), 592-604 for a method which generates unitaries random with respect to the Haar measure, but none of them mentioned the special unitary subgroup. In particular, I'm not sure about one thing: say that we take random $n\times n$ unitaries $U_n$ generated this way and construct special unitary matrices simply by $U_n / \det(U_n)^{1/n}$.

Are special unitary matrices generated this way also random with respect to the Haar measure?

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yes, this normalization will produce a uniform distribution in ${\rm SU}(N)$, however, it might be more efficient to generate directly random matrices with unit determinant (you'll need one fewer parameter and no need to calculate the determinant), as explained in Composite parameterization and Haar measure for all unitary and special unitary groups

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