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

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

2008-09-24

arxiv.org/abs/0809.4236v3

Algebra and Number Theory, Vol. 5 (2011), No. 1, 75--109

Mathematics

Algebraic Geometry

28 pages. Updated with referee's suggestions

Scientific paper

10.2140/ant.2011.5.75

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.

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