[Paper Review] Weingarten calculus via orthogonality relations: new applications

Benoı̂t Collins, Sho Matsumoto|arXiv (Cornell University)|Jan 17, 2017

Algebraic structures and combinatorial models11 references3 citations

TL;DR

This paper revisits Weingarten's original calculus for computing moments of the Haar measure on compact matrix groups using orthogonality relations, providing a unified and conceptual framework that yields optimal uniform bounds on the Weingarten function and extends the calculus to symmetric spaces. The key contribution is a new, tight uniform bound for the unitary group, valid for $ d > \sqrt{6}k^{7/4} $, and a conceptual derivation of identities in symmetric space settings via path-based combinatorial expressions on Weingarten graphs.

ABSTRACT

Weingarten calculus is a completely general and explicit method to compute the moments of the Haar measure on compact subgroups of matrix algebras. Particular cases of this calculus were initiated by theoretical physicists -- including Weingarten, after whom this calculus was coined by the first author, after investigating it systematically. Substantial progress was achieved subsequently by the second author and coworkers, based on representation theoretic and combinatorial techniques. All formulas of `Weingarten calculus' are in the spirit of Weingarten's seminal paper [W78]. However, modern proofs are very different from Weingarten's initial ideas. In this paper, we revisit Weingarten's initial proof and we illustrate its power by uncovering two new important applications: (i) a uniform bound on the Weingarten function, that subsumes existing uniform bounds, and is optimal up to a polynomial factor, and (ii) an extension of Weingarten calculus to symmetric spaces and conceptual proofs of identities established by the second author.

Motivation & Objective

• To revisit Weingarten’s original orthogonality-based method for Haar measure integration, which had been supplanted by representation-theoretic techniques.
• To address the lack of uniform bounds and explicit convergence criteria in prior Weingarten calculus approaches.
• To extend the calculus to symmetric spaces such as A III-type, providing conceptual proofs for identities previously derived computationally.
• To unify and generalize existing results in unitary, orthogonal, and symplectic cases using elementary, combinatorial methods.

Proposed method

• Reconstructs Weingarten calculus using Weingarten’s original orthogonality relations and linear systems, avoiding advanced representation theory.
• Derives a uniform bound on the Weingarten function $ \operatorname{Wg}^\mathrm{U}(\sigma,d) $ via rescaling and upper-triangularization in the large $ d $ limit.
• Introduces the Weingarten graph $ \mathcal{G}^{\mathrm{A\,III}} $ with solid, dashed, and squiggled edges to encode path-based expansions of the A III Weingarten function.
• Uses path enumeration on the Weingarten graph to derive a new combinatorial formula: $ \operatorname{Wg}^{\mathrm{A\,III}}(\sigma,d,d^{-}) = \sum_{p:\sigma\to\emptyset} (-1)^{\ell_0(p)} (d^{-})^{\ell_1(p)} d^{-\ell(p)} $.
• Applies the method to symmetric spaces by adapting the orthogonality relations and path decomposition to the A III setting, yielding conceptual proofs of known identities.
• Establishes a connection between the Weingarten function and the Möbius function via asymptotic analysis and polynomial rescaling.

Experimental results

Research questions

• RQ1 Can Weingarten’s original orthogonality-based method be revived and refined to yield optimal uniform bounds on the Weingarten function?
• RQ2 Can the Weingarten calculus be systematically extended to symmetric spaces such as A III, providing conceptual explanations for known identities?
• RQ3 What is the precise range of $ d $ for which the Weingarten function admits a unique solution, and how can this be quantified uniformly in $ k $?
• RQ4 How do path-based combinatorial structures on Weingarten graphs encode the moments of Haar measure on symmetric spaces?

Key findings

• A new uniform bound for the unitary Weingarten function is established: for $ d > \sqrt{6}k^{7/4} $, the ratio $ \frac{d^{k+|\sigma|} \operatorname{Wg}^\mathrm{U}(\sigma,d)}{\mathrm{Moeb}(\sigma)} $ is bounded between $ \frac{1}{1 - \frac{k-1}{d^2}} $ and $ \frac{1}{1 - \frac{6k^{7/2}}{d^2}} $.
• The lower bound holds for all $ d \geq k $, and the bound is optimal up to a polynomial factor in $ k $.
• The Weingarten graph for A III-type symmetric spaces provides a complete path-based expansion of the Weingarten function, with contributions from solid, dashed, and squiggled edges.
• The path-based formula $ \operatorname{Wg}^{\mathrm{A\,III}}(\sigma,d,d^{-}) = \sum_{p:\sigma\to\emptyset} (-1)^{\ell_0(p)} (d^{-})^{\ell_1(p)} d^{-\ell(p)} $ offers a new combinatorial expression that avoids reliance on Schur functions and characters.
• The method provides conceptual derivations of identities previously obtained only by computation, such as those in Matsumoto (2013) and Zeng-Jiang (2010).
• The approach subsumes and improves upon earlier uniform bounds from [CGP13] and [Mo13], offering tighter and more general estimates.