We consider the three finite free convolutions for polynomials studied in a recent paper by Marcus, Spielman and Srivastava. Each can be described either by direct explicit formulae or in terms of operations on randomly rotated matrices. We present an alternate approach to the equivalence between these descriptions, based on combinatorial Weingarten methods for integration over the unitary and orthogonal groups. A key aspect of our approach is to identify a certain quadrature property, which is satisfied by some important series of subgroups of the unitary groups (including the groups of unitary, orthogonal, and signed permutation matrices), and which yields the desired convolution formulae.

中文翻译：

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通过 Weingarten 演算进行有限自由卷积

我们考虑了 Marcus、Spielman 和 Srivastava 在最近的一篇论文中研究的多项式的三个有限自由卷积。每个都可以通过直接的显式公式或随机旋转矩阵上的操作来描述。我们提出了一种替代方法来处理这些描述之间的等价性，基于组合 Weingarten 方法，用于在单一和正交组上进行积分。我们方法的一个关键方面是识别特定的正交属性，它由酉群的一些重要子群系列（包括酉、正交和有符号置换矩阵的群）满足，并产生所需的卷积公式。