Polynomial Ensembles and P\'olya Frequency Functions

TL;DR

This paper introduces Pólya ensembles, a class of polynomial ensembles with closure properties under convolution across various matrix spaces, unifying and extending previous studies and connecting to Pólya frequency functions.

Contribution

It defines Pólya ensembles on multiple matrix spaces, explores their convolution properties, and establishes identities for group integrals, broadening the understanding of polynomial ensembles.

Findings

Pólya ensembles are closed under specific convolutions.

Derived identities for group integrals similar to Harish-Chandra-Itzykson-Zuber.

Connected Pólya ensembles to Pólya frequency functions.

Abstract

We study several kinds of polynomial ensembles of derivative type which we propose to call P\'olya ensembles. These ensembles are defined on the spaces of complex square, complex rectangular, Hermitian, Hermitian anti-symmetric and Hermitian anti-self-dual matrices, and they have nice closure properties under the multiplicative convolution for the first class and under the additive convolution for the other classes. The cases of complex square matrices and Hermitian matrices were already studied in former works. One of our goals is to unify and generalize the ideas to the other classes of matrices. Here we consider convolutions within the same class of P\'olya ensembles as well as convolutions with the more general class of polynomial ensembles. Moreover, we derive some general identities for group integrals similar to the Harish-Chandra-Itzykson-Zuber integral, and we relate P\'olya ensembles to P\'olya frequency functions. For illustration we give a number of explicit examples for our results.