Moment Methods on compact groups: Weingarten calculus and its applications

TL;DR

This paper surveys recent developments in Weingarten calculus, a systematic method for computing moments of Haar measures on compact groups and quantum groups, with applications across mathematics and physics.

Contribution

It provides a comprehensive overview of Weingarten calculus, highlighting its theoretical properties and diverse applications in various scientific fields.

Findings

Systematic computation of moments for Haar measures.

Connections between Weingarten functions and random matrix theory.

Applications in quantum probability and operator algebras.

Abstract

A fundamental property of compact groups and compact quantum groups is the existence and uniqueness of a left and right invariant probability -- the Haar measure. This is a natural playground for classical and quantum probability, provided it is possible to compute its moments. Weingarten calculus addresses this question in a systematic way. The purpose of this manuscript is to survey recent developments, describe some salient theoretical properties of Weingarten functions, as well as applications of this calculus to random matrix theory, quantum probability, and algebra, mathematical physics and operator algebras.