  The variety of principal minors of $n\times n$ symmetric matrices, denoted
$Z_{n}$, is invariant under the action of a group $G\subset \GL(2^{n})$
isomorphic to $\G$. We describe an irreducible $G$-module of degree $4$
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.

Algebraic Geometry