The orthogonal Weingarten formula in compact form

TL;DR

This paper introduces a simplified and compact formulation of the orthogonal Weingarten formula, enabling easier analysis of integrals over orthogonal groups with applications to understanding their properties.

Contribution

It presents a new compact expression for the orthogonal Weingarten formula using matrix exponents, simplifying the analysis of related integrals.

Findings

Established vanishing conditions for integrals

Analyzed possible poles of the integrals

Described asymptotic behavior of the integrals

Abstract

We present a compact formulation of the orthogonal Weingarten formula, with the traditional quantity $I(i_1,...,i_{2k}:j_1,...,j_{2k}) = \int_{O_n}u_{i_1j_1} ... u_{i_{2k}j_{2k}} du$ replaced by the more advanced quantity $I(a)=\int_{O_n}\Pi u_{ij}^{a_{ij}} du$, depending on a matrix of exponents $a\in M_n(\mathbb N)$. Among consequences, we establish a number of basic facts regarding the integrals $I(a)$: vanishing conditions, possible poles, asymptotic behavior.