基于正交关系的Weingarten演算：新的应用
Weingarten calculus via orthogonality relations: new applications
摘要：
Weingarten演算是计算矩阵代数的紧子群上Haar测度的矩的一种完全通用的显式方法。这种演算的特殊情况是由理论物理学家发起的-包括Weingarten，在他之后，第一作者在系统地调查之后创造了这种演算。第二作者及其同事基于表示论和组合技术取得了实质性进展。所有的公式'Weingarten演算'的精神Weingarten的开创性文件[W78]。然而，现代的证明是非常不同的Weingarten的最初想法。在本文中，我们重温Weingarten的初始证明，并通过揭示两个新的重要应用来说明它的力量：（i）Weingarten函数的统一界，包含现有的统一界，并且是最优的多项式因子，以及（ii）Weingarten演算到对称空间的扩展和第二作者建立的恒等式的概念证明。
英文摘要：
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.
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