# Finite free convolutions via Weingarten calculus

## TL;DR

This paper introduces a new combinatorial approach using Weingarten calculus to analyze finite free convolutions of polynomials, providing an alternative proof of their equivalence with matrix operations.

## Contribution

It offers an innovative method based on Weingarten calculus to establish the equivalence of different descriptions of finite free convolutions, highlighting a quadrature property of certain subgroup series.

## Key findings

- Established a combinatorial Weingarten approach for finite free convolutions
- Identified a quadrature property for specific subgroup series of unitary groups
- Provided explicit convolution formulas via this new method

## Abstract

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 \emph{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.