Invariant and Group Theoretical Integrations over the U(n) Group

TL;DR

This paper compares invariant and group-theoretical methods for integrating over the U(n) group, introduces a hybrid approach, and illustrates their applications through various examples.

Contribution

It introduces a hybrid method combining invariant and group-theoretical techniques for U(n) integrals, enhancing computational efficiency and understanding.

Findings

The hybrid method outperforms individual methods in certain cases.

Comparison reveals strengths and weaknesses of each approach.

Numerous examples demonstrate practical applications.

Abstract

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.