Weingarten Calculus

Georg Köstenberger

We consider the problem of computing the integral $${\int }_{\mathcal{U}(d)}{u}_{i_1{j}_1}\cdots u_{i_n{j}_n}{\bar{u}}_{i_1^{\prime }{j}_1^{\prime }}\cdots {\bar{u}}_{i_{n^{\prime }}^{\prime }{j}_{n^{\prime }}^{\prime }}dU,$$ where the integrati…