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# The Weingarten Calculus

Communicated by *Notices* Associate Editor Steven Sam

## 1. Introduction

Every compact topological group supports a unique translation invariant probability measure on its Borel sets — the *Haar measure*. The Haar measure was first constructed for certain families of compact matrix groups by Hurwitz in the nineteenth century in order to produce invariants of these groups by averaging their actions. Hurwitz’s construction has been reviewed from a modern perspective by Diaconis and Forrester, who argue that it should be regarded as the starting point of modern random matrix theory DF17. An axiomatic construction of Haar measures in the more general context of locally compact groups was published by Haar in the 1930s, with further important contributions made in work of von Neumann, Weil, and Cartan; see Bou04. The existence of recent works on the Haar measure, see, e.g., DS14 or Mec19, can be seen as a token of the timeliness of this object as a modern research topic.

Given a measure, one wants to integrate. The Bochner integral for continuous functions on a compact group taking values in a given Banach space is called the *Haar integral*; it is almost always written simply

with no explicit notation for the Haar measure. While integration on groups is a concept of fundamental importance in many parts of mathematics, including functional analysis and representation theory, probability, and ergodic theory, etc., the actual computation of Haar integrals is a problem which has received curiously little attention. As far as the authors are aware, it was first considered by theoretical physicists in the 1970s in the context of nonabelian gauge theories, where the issue of evaluating — or at least approximating — Haar integrals plays a major role. In particular, the physics literature on quantum chromodynamics, the main theory of strong interactions in particle physics, is littered with so-called “link integrals,” which are Haar integrals of the form

where

an evaluation which holds for all unitary groups of rank

The missing theory of Haar integrals began to take shape in the early 2000s, driven by an explosion of interest in random matrix theory. The basic Hilbert spaces of random matrix theory are *moment method* in random matrix theory, pioneered by Wigner (Wig58) in the 1950s, is an algebraic approach to this problem. The main idea is to adopt the algebra

It is straightforward to see that, in both of the above

where

and

In the Gaussian case, monomial scalar products can be computed systematically using a combinatorial algorithm which physicists call the “Wick formula” and statisticians call the “Isserlis theorem.” This device leverages independence together with the characteristic feature of centered normal distributions — vanishing of all cumulants but the second — to compute Gaussian expectations as polynomials in the variance parameter

The case of Haar unitary matrices is a priori more complicated: the random variables *Weingarten calculus*, a name chosen by Collins Col03 to honor the contributions of Donald Weingarten, a physicist whose early work in the subject is of foundational importance.

The Weingarten calculus has matured rapidly over the course of the past decade, and the time now seems right to give a pedagogical account of the subject. The authors are currently preparing a monograph intended to meet this need. In this article, we aim to provide an easily digestible and hopefully compelling preview of our forthcoming work, emphasizing the big picture but still providing some of the important details.

First and foremost, we wish to impart the insight that, like the calculus of Newton and Leibniz, the core of Weingarten calculus is a fundamental theorem which converts a computational problem into a symbolic problem: whereas the usual fundamental theorem of calculus converts the problem of integrating functions on the line into computing antiderivatives, the fundamental theorem of Weingarten calculus converts the problem of integrating functions on groups into computing certain matrices associated to tensor invariants. The fundamental theorem of Weingarten calculus is presented in detail in Section 2.

We then turn to examples illustrating the fundamental theorem in action. We present two detailed case studies: integration on the automorphism group

Section 5 gives a necessarily brief discussion of Weingarten calculus for the remaining classical groups, namely the orthogonal group

## 2. The Fundamental Theorem

Given a compact group

The *Weingarten integrals* of the unitary representation

where

The fundamental theorem of Weingarten calculus addresses this problem by linearizing it. The basic observation is that, for each

where

is the orthonormal basis of

are the matrix elements of the unitary operator

where

obtained by integrating the unitary operators

This is where the characteristic feature of Haar measure, the invariance

comes into play: it forces

Thus, we see that the basic problem of Weingarten calculus is in fact very closely related to the basic problem of invariant theory, which is to determine a basis for the space of

Indeed, suppose we have access to a basis

of degree

Then we have the matrix factorization

familiar from matrix analysis as the multidimensional generalization of the undergraduate “outer product divided by inner product” formula for orthogonal projection onto a line. The

of the basic

the *Weingarten matrix* of the invariants

Does Theorem 2.1 actually solve the basic problem of Weingarten calculus? Yes, insofar as the classical fundamental theorem of calculus solves the problem of computing definite integrals: it reduces a numerical problem to a symbolic problem. In order to apply the fundamental theorem of calculus to integrate a given function, one must find its antiderivative, and as every student of calculus knows this can be a wild ride. In order to use the fundamental theorem of Weingarten calculus to compute the Weingarten integrals of a given unitary representation, one must solve a souped-up version of the basic problem of invariant theory which involves not only finding basic tensor invariants, but computing their Weingarten matrices. Just like the computation of antiderivatives, this may prove to be a difficult task.

## 3. The Symmetric Group

In this Section, we consider a toy example. Fix

The permutation representation of

The corresponding system of matrix elements

We will evaluate the Weingarten integrals of

Each Weingarten integral

Thus,

The fiber fingerprint of the composite function

and so we have

Clearly, such a permutation exists if and only if the fibers of

where

in total, where

Let us now evaluate

where

corresponding to a function

which is clearly

so that the distinct invariants produced by symmetrization of the initial basis in

These tensors are pairwise orthogonal: for any

So, the Gram matrix of the basis