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2011 Set-theoretic defining equations of the variety of principal minors of symmetric matrices
Luke Oeding
Algebra Number Theory 5(1): 75-109 (2011). DOI: 10.2140/ant.2011.5.75


The variety of principal minors of n×n symmetric matrices, denoted Zn, is invariant under the action of a group G GL(2n) isomorphic to SL(2)×nSn. We describe an irreducible G-module of degree-four polynomials constructed from Cayley’s 2×2×2 hyperdeterminant and show that it cuts out Zn 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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Luke Oeding. "Set-theoretic defining equations of the variety of principal minors of symmetric matrices." Algebra Number Theory 5 (1) 75 - 109, 2011.


Received: 25 January 2010; Revised: 1 November 2010; Accepted: 5 December 2010; Published: 2011
First available in Project Euclid: 20 December 2017

zbMATH: 1238.14035
MathSciNet: MR2833786
Digital Object Identifier: 10.2140/ant.2011.5.75

Primary: 14M12
Secondary: 13A50 , 14L30 , 15A29 , 15A69 , 15A72 , 20G05

Keywords: determinant , G-module , G-variety , hyperdeterminant , hyperdeterminantal module , principal minors , relations among minors , representation theory , symmetric matrices , variety of principal minors

Rights: Copyright © 2011 Mathematical Sciences Publishers


Vol.5 • No. 1 • 2011
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