Weingarten Calculus and the IntHaar Package for Integrals over Compact Matrix Groups

TL;DR

This paper introduces a unified approach using Weingarten calculus for integrating polynomials over compact matrix groups, simplifies formulas, and implements an optimized Maple package called IntHaar for practical computations.

Contribution

It provides a new uniform formula for polynomial integration over compact groups and implements an optimized Maple package for these calculations.

Findings

Unified formula for integrals over unitary, orthogonal, and symplectic groups

Simplified and optimized integration formulas

Practical Maple package with examples

Abstract

In this paper, we present a uniform formula for the integration of polynomials over the unitary, orthogonal, and symplectic groups using Weingarten calculus. From this description, we further simplify the integration formulas and give several optimizations for various cases. We implemented the optimized Weingarten calculus in Maple in a package called IntHaar for all three compact groups. Here we will discuss its functions, provide instructions for the package, and produce some examples of computed integrals.