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An undergraduate is performing some computations, related to a Sato-Tate conjecture of $U(3)$ type (a curve over $Q$, for which the roots of local L-functions look like eigenvalues of a random matrix in $U(3)$ and their complex conjugates, scaled appropriately).

There are well-known results in the literature, on random matrices from $U(3)$, with respect to Haar measure. For example, the expected value of the trace is zero (obviously), and the expected value of the square of the trace is one (not-so-obviously, but this is what I recall). The Weyl integration formula allows one to express other expectations as definite integrals. Results of Diaconis and others give a few moments, but most results I find look at $U(N)$ as $N$ grows large.

Does anyone know a precise elementary formula for the distribution of traces of a random matrix in $U(3)$? Or a good reference?

For example, the trace of a random matrix $g$ in $SU(2)$ is between $-2$ and $2$. The probability distribution of $Tr(g) = t$ is given, up to some normalization, by: $$P(t) = \sqrt{1 - \left( \frac{t}{2} \right)^2 },$$

The real part of the trace of a random matrix $g$ in $U(3)$ is between $-3$ and $3$. Can anyone write the probability distribution of $Tr(g) = z$, in as simple a fashion? Of course, here the traces can be complex -- I'd be happy also to find the probability that $Re(Tr(g)) = t$, as an elementary function of $t$.

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I think that Rains (after Diaconis-Shashahani) found the expected value of $|Tr(U)|^{2k}$ for all integers k (not just those up to the dimension $N$), from which you can apply inversion, I guess? See: I got this from a paper of Tracy (see page 2), who notes work of Gessel. Tracy: Gessel: Sorry, that's the best I can do for now. – Junkie May 26 '10 at 7:39
I should have said all positive integers of course. The moments are given in Sloane as A005802. – Junkie May 26 '10 at 11:52
Thanks for the references! I hadn't found the Rains paper before, which sounds very promising. I'll take a look later today, but it sounds like exactly what I want. – Marty May 26 '10 at 14:15
I thought that the roots of local L-functions of curves seemed to be drawn from USp(2g), not just a unitary group...? – David Hansen May 27 '10 at 20:03
Correct, generically. But I'm having the undergraduate work with "Picard curves" -- certain genus three curves whose Jacobian has extra endomorphisms (by the ring of integers in $Q(\sqrt{-3})$). These extra endomorphisms cut down the image of the Galois representation from $USp(6)$ to $U(3)$ (at primes congruent to $1$ mod $3$), and make it an interesting case to study. It's a natural first choice to study, if one wants to go beyond the $SU(2)$ regime. – Marty May 27 '10 at 23:43

If the literature is such a poor state, I'm a little puzzled as to why.

Maybe for SU(3) it should be clear in principle what is happening? The trace is an invariant of conjugacy classes, and the Weyl integration formula should reduce any issue to an integral over a two-dimensional region. Namely the maximal torus T is acted on by the Weyl group W, and apart from stuff of Haar measure zero the quotient can be represented by an explicit fundamental domain. The push forward of Haar measure on SU(3) has a density with respect to Lebesgue measure on the fundamental domain that was written down by Weyl (if not before). The trace is also a function on this fundamental domain. Points in T can be represented by three angles summing to zero mod 2pi.

Am I being slow? The generic conjugacy class is not a difficult thing to understand, once having thrown out something of measure zero. This looks like an exercise in multiple integration over sets x + y + z = constant.

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You're definitely not being slow. I spent a little time mucking about with the appropriate integral on the space of angles, using Weyl integration. But it seemed quite messy -- I suppose I can go see what Mathematica spits out - I'll try it later today. To see what makes it messy - you have to integrate over "x+y+z=constant", also constrained by the x,y,z a point on the unit circle in C, and the integrand is a function that is a polynomial in x,y,z. – Marty May 26 '10 at 14:14

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