Set Theoretic Defining Equations of the Variety of Principal Minors of Symmetric Matrices

TL;DR

This paper characterizes the set of principal minors of symmetric matrices using specific degree 4 polynomials derived from hyperdeterminants, confirming a conjecture and advancing the understanding of related algebraic varieties.

Contribution

It provides a set-theoretic defining equation for the variety of principal minors of symmetric matrices, solving a conjecture by Holtz and Sturmfels using representation theory and geometry.

Findings

Identifies a degree 4 polynomial module that defines the variety set-theoretically.

Uses hyperdeterminants to construct defining equations.

Advances techniques in representation theory for algebraic geometry.

Abstract

The variety of principal minors of $n\times n$ symmetric matrices, denoted $Z_{n}$, is invariant under the action of a group $G\subset \GL(2^{n})$ 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.