Characterizing principal minors of symmetric matrices via determinantal multiaffine polynomials

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 $(S{L}_2(R){)}^n⋊S_n$. 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.

19 pages