Invariant and group theoretical integrations over the U(n) group

In a previous article, an "invariant method" to calculate monomial integrals over the U(n) group was introduced. In this paper, we study the more traditional group-theoretical method, and compare its strengths and weaknesses with those of the invariant method. As a result, we are able to introduce a "hybrid method" which combines the respective strengths of the other two methods. There are many examples in the paper illustrating how each of these methods works. (C) 2004 American Institute of Physics.