Set-theoretic defining equations of the variety of principal minors of symmetric matrices

Abstract

The variety of principal minors of $n\times n$ symmetric matrices, denoted $Z_n$, is invariant under the action of a group $G\subset GL\left(2^n\right)$ isomorphic to $SL{\left(2\right)}^{\times n}⋉{\mathfrak{S}}_n$. We describe an irreducible $G$-module of degree-four polynomials constructed from Cayley’s $2\times 2\times 2$ hyperdeterminant and show that it cuts out $Z_n$ set-theoretically. This solves the set-theoretic version of a conjecture of Holtz and Sturmfels. Standard techniques from representation theory and geometry are explored and developed for the proof of the conjecture and may be of use for studying similar $G$-varieties.