# Exact cubature rules for symmetric functions

## TL;DR

This paper develops exact multivariate cubature rules for symmetric functions over hypercubes, leveraging a multivariate Gauss quadrature extension, with applications to unitary random matrix ensembles and Bernstein-Szeg"o polynomials.

## Contribution

It introduces a multivariate extension of Gauss quadrature for symmetric functions, enabling exact integration with applications to matrix ensembles and special polynomials.

## Key findings

- Derived explicit cubature rules for symmetric functions.
- Enabled exact integration of rational functions with prescribed poles.
- Applied rules to unitary circular Jacobi distributions.

## Abstract

We employ a multivariate extension of the Gauss quadrature formula, originally due to Berens, Schmid and Xu [BSX95], so as to derive cubature rules for the integration of symmetric functions over hypercubes (or infinite limiting degenerations thereof) with respect to the densities of unitary random matrix ensembles. Our main application concerns the explicit implementation of a class of cubature rules associated with the Bernstein-Szeg\"o polynomials, which permit the exact integration of symmetric rational functions with prescribed poles at coordinate hyperplanes against unitary circular Jacobi distributions stemming from the Haar measures on the symplectic and the orthogonal groups.