Some properties of angular integrals

TL;DR

This paper introduces new representations for angular integrals related to classical groups, providing recursive formulas and polynomial-exponential decompositions, enhancing understanding of their structure and computation.

Contribution

It presents novel representations and recursion formulas for angular integrals over classical groups, connecting them to polynomial and exponential functions.

Findings

Angular integrals can be expressed as linear combinations of exponentials with polynomial coefficients.

Derived recursion formulas simplify the computation of angular integrals for arbitrary group size n.

Established connections between angular integrals and moments of Shatashvili type.

Abstract

We find new representations for Itzykson-Zuber like angular integrals for arbitrary beta, in particular for the orthogonal group O(n), the unitary group U(n) and the symplectic group Sp(2n). We rewrite the Haar measure integral, as a flat Lebesge measure integral, and we deduce some recursion formula on n. The same methods gives also the Shatashvili's type moments. Finally we prove that, in agreement with Brezin and Hikami's observation, the angular integrals are linear combinations of exponentials whose coefficients are polynomials in the reduced variables (x_i-x_j)(y_i-y_j).