The variety of principal minors of symmetric matrices, denoted , is invariant under the action of a group isomorphic to . We describe an irreducible -module of degree-four polynomials constructed from Cayley’s hyperdeterminant and show that it cuts out 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 -varieties.
"Set-theoretic defining equations of the variety of principal minors of symmetric matrices." Algebra Number Theory 5 (1) 75 - 109, 2011. https://doi.org/10.2140/ant.2011.5.75