#### Vol. 5, No. 1, 2011

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

### Luke Oeding

Vol. 5 (2011), No. 1, 75–109
##### Abstract

The variety of principal minors of $n×n$ symmetric matrices, denoted ${Z}_{n}$, is invariant under the action of a group $G\subset GL\left({2}^{n}\right)$ isomorphic to $SL{\left(2\right)}^{×n}⋉{\mathfrak{S}}_{n}$. We describe an irreducible $G$-module of degree-four polynomials constructed from Cayley’s $2×2×2$ 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.

##### Keywords
principal minors, symmetric matrices, hyperdeterminant, G-variety, G-module, representation theory, hyperdeterminantal module, relations among minors, variety of principal minors, determinant
##### Mathematical Subject Classification 2000
Primary: 14M12
Secondary: 15A69, 15A29, 15A72, 20G05, 13A50, 14L30