Symbolic integration with respect to the Haar measure on the unitary group

TL;DR

The paper introduces IntU, a Mathematica package that performs symbolic integration of polynomial functions over the unitary group using Haar measure, with optimizations for specific cases and practical examples.

Contribution

It provides a new computational tool for symbolic integration over the unitary group, enhancing efficiency and usability in mathematical and physical applications.

Findings

Efficient symbolic integration of polynomials over the unitary group.

Special cases optimized for faster calculations.

Practical examples demonstrating package usage.

Abstract

We present IntU package for Mathematica computer algebra system. The presented package performs a symbolic integration of polynomial functions over the unitary group with respect to unique normalized Haar measure. We describe a number of special cases which can be used to optimize the calculation speed for some classes of integrals. We also provide some examples of usage of the presented package.