On some properties of orthogonal Weingarten functions

TL;DR

This paper introduces a Fourier-type formula for orthogonal Weingarten functions, simplifying calculations and expanding understanding of their properties within the context of Haar measure integrals over classical groups.

Contribution

It provides the first Fourier-type formula for orthogonal Weingarten functions, utilizing Jack polynomials, and explores new properties and conjectures related to these functions.

Findings

Derived a Fourier-type formula for orthogonal Weingarten functions

Reduced computational complexity of Weingarten formulas

Presented new properties, a conjecture, and a value table for these functions

Abstract

We give a Fourier-type formula for computing the orthogonal Weingarten formula. The Weingarten calculus was introduced as a systematic method to compute integrals of polynomials with respect to Haar measure over classical groups. Although a Fourier-type formula was known in the unitary case, the orthogonal counterpart was not known. It relies on the Jack polynomial generalization of both Schur and zonal polynomials. This formula substantially reduces the complexity involved in the computation of Weingarten formulas. We also describe a few more new properties of the Weingarten formula, state a conjecture and give a table of values.