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

The variety of principal minors of n x n symmetric matrices, denoted Z(n), is invariant under the action of a group G subset of GL(2(n)) isomorphic to SL(2)(xn) x S-n. We describe an irreducible G-module of degree-four polynomials constructed from Cayley's 2 x 2 x 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.