Factoring determinants and applications to number theory

TL;DR

This paper proves factorization formulas for averages of ratios of shifted characteristic polynomials over classical compact groups, using operator theory and Fredholm determinants, providing new proofs and insights relevant to number theory.

Contribution

It introduces a novel operator-theoretic approach to factorize these averages, offering new proofs and connecting them to random matrix theory and number theory.

Findings

Averages factor into polynomials in matrix size over symplectic and orthogonal groups.

Fredholm expansions reveal swap terms characteristic of number theoretic averages.

Provides the fourth known proof of these average formulas, linking multiple mathematical methods.

Abstract

Products of shifted characteristic polynomials, and ratios of such products, averaged over the classical compact groups are of great interest to number theorists as they model similar averages of L-functions in families with the same symmetry type as the compact group. We use Toeplitz and Toeplitz plus Hankel operators and the identities of Borodin - Okounkov, Case - Geronimo, and Basor - Erhardt to prove that, in certain cases, these unitary averages factor as polynomials in the matrix size into averages over the symplectic group and the orthogonal group. Building on these identities we present new proofs of the exact formulas for these averages where the ``swap'' terms that are characteristic of the number theoretic averages occur from the Fredholm expansions of the determinants of the appropriate Hankel operator. This is the fourth different proof of the formula for the averages of ratios of products of shifted characteristic polynomials; the other proofs are based on supersymmetry; symmetric function theory, and orthogonal polynomial methods from Random Matrix Theory.