Suppose I want to compute a quantity of the type:
$\mathbb{E}\mathrm{tr}(AUBU^{\ast})$
where averaging is over Haar measure on the unitary group $\mathcal{U}(n)$ (one can of course consider higher order polynomials or other matrix ensembles etc.) and $A$, $B$ are some fixed matrices. Is there any standard technique for computing such averages? I'd guess people in random matrix theory or free probability compute such traces all the time, but I've been unable to find a reference. If it makes matters easier, I'm really interested in computing something for random projections (e.g. something of the form $\mathbb{E}\mathrm{tr}(APBP)$, where $P$ is a projection onto a random subspace), which of course reduces to computation of polynomials in $U$.
Best Answer
For the unitary group, the first paper I am aware of to do these sorts of averages is:
http://link.aip.org/link/JMAPAQ/v21/i12/p2695/s1
An early paper of Collins' in 2003 expresses such averages in terms of Weingarten functions, which are usually expressed as character expansions over $U(N)$, $O(N)$ or $Sp(N)$.
http://arxiv.org/abs/math-ph/0205010
Some early papers calculating these character expansions are:
http://jmp.aip.org/resource/1/jmapaq/v25/i6/p2028_s1
http://jmp.aip.org/resource/1/jmapaq/v43/i1/p604_s1