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arXiv · 0809.4236

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

Luke Oeding
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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.

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BibTeXRIS

Luke Oeding. 2010-11-11. Set Theoretic Defining Equations of the Variety of Principal Minors of Symmetric Matrices. https://doi.org/10.2140/ant.2011.5.75

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