Characterizing principal minors of symmetric matrices via determinantal multiaffine polynomials

TL;DR

This paper characterizes the image of the principal minor map of symmetric matrices over various rings using determinantal polynomials and group actions, extending known results and providing new algebraic criteria.

Contribution

It introduces a novel characterization of the principal minor map via Rayleigh differences and group orbits, generalizing previous complex and real cases.

Findings

Characterization of principal minors via Rayleigh differences as squares.

Use of Cayley's hyperdeterminant orbit under group actions.

Over reals, a single quadratic inequality suffices to describe the image.

Abstract

Here we consider the image of the principal minor map of symmetric matrices over an arbitrary unique factorization domain $R$. By exploiting a connection with symmetric determinantal representations, we characterize the image of the principal minor map through the condition that certain polynomials coming from so-called Rayleigh differences are squares in the polynomial ring over $R$. In almost all cases, one can characterize the image of the principal minor map using the orbit of Cayley's hyperdeterminant under the action of $(SL_2(R))^{n} \rtimes S_{n}$. Over the complex numbers, this recovers a characterization of Oeding from 2011, and over the reals, the orbit of a single additional quadratic inequality suffices to cut out the image. Applications to other symmetric determinantal representations are also discussed.