Composite parameterization and Haar measure for all unitary and special unitary groups

TL;DR

This paper extends the composite parameterization and Haar measure formulation from the unitary group to the special unitary group, providing concise formulas useful for high-order integral computations.

Contribution

It introduces a unified parameterization and Haar measure expression for both U(d) and SU(d), enabling efficient high-order integral calculations over these groups.

Findings

Derived a concise formula for Haar measure on U(d) and SU(d)

Unified parameterization for U(d) and SU(d) groups

Facilitated computation of high-order integrals over unitary groups

Abstract

We adopt the concept of the composite parameterization of the unitary group U(d) to the special unitary group SU(d). Furthermore, we also consider the Haar measure in terms of the introduced parameters. We show that the well-defined structure of the parameterization leads to a concise formula for the normalized Haar measure on U(d) and SU(d). With regard to possible applications of our results, we consider the computation of high-order integrals over unitary groups.