This should be both well-known and probably easy, but I was wondering if the following is known (and, if so, how to easily calculate the thing or where to read about how to calculate it):


what is $$\int\_{\mathrm{SU}(n)} \mathrm{tr}(U^k) dU?$$ (Here by "$dU$" I mean normalized Haar measure.)


Of course for $k$ not a multiple of $n$ the integral is zero. (It's always zero on $\mathrm{U}(n)$ by the same argument.) Also by Weyl's integration formula (i.e. averaging over conjugates of $U$) one immediately reduces to performing the integral over diagonal matrices, but I didn't see a way to get a clean answer out.


Apologies if this is easy! I just couldn't find a good reference.


Thanks!



Moments of random special unitary matrices
I should have stared at the Weyl integration formula longer --- it's clear from that that, once $k\gg\_n 1$, the integral is zero. In fact [this article](http://ac.els-cdn.com/S0021869307003420/1-s2.0-S0021869307003420-main.pdf?_tid=485ff84e-8af5-11e4-895a-00000aab0f6c&acdnat=1419374861_b065384d2f6ee608801b6f5a9ee645ff) shows that the integral vanishes once $k\geq 2n-1$ (it already vanishes when $k$ is not a multiple of $n$). Hence the only case left is $k=n$, and, if I've interpreted their result correctly, the answer is $(-1)^{n-1}$. This could almost certainly have been read off from the formula in this special case, of course.


[To explain the title, note that the element $\int\_{\mathrm{SU}(n)} U^n dU$ commutes with all of $\mathrm{SU}(n)$ and must therefore be a scalar, so it suffices to know its trace.]


