Monomial integrals on the classical groups

TL;DR

This paper introduces a recursive, efficient method for integrating monomials over classical groups, extending previous work on orthogonal groups to unitary and symplectic groups, with formulas suitable for computer algebra implementation.

Contribution

It develops a unified recursive integration method for classical groups, enabling analytical expressions with matrix dimension as a parameter, applicable to unitary and symplectic groups.

Findings

Derived recursive integration formulas for unitary and symplectic groups

Formulas are similar in structure across different classical groups

Method is easily implemented in computer algebra systems

Abstract

This paper presents a powerfull method to integrate general monomials on the classical groups with respect to their invariant (Haar) measure. The method has first been applied to the orthogonal group in [J. Math. Phys. 43, 3342 (2002)], and is here used to obtain similar integration formulas for the unitary and the unitary symplectic group. The integration formulas turn out to be of similar form. They are all recursive, where the recursion parameter is the number of column (row) vectors from which the elements in the monomial are taken. This is an important difference to other integration methods. The integration formulas are easily implemented in a computer algebra environment, which allows to obtain analytical expressions very efficiently. Those expressions contain the matrix dimension as a free parameter.