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 (SL2(R))n⋊Sn. Over C, this recovers a characterization of Oeding from 2011, and over R, the orbit of a single additional quadratic inequality suffices to cut out the image. Applications to other symmetric determinantal representations are also discussed.