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

Oeding, Luke

doi:10.2140/ant.2011.5.75

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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

DOI: 10.2140/ant.2011.5.75

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 $SL {(2)}^{\times n} ⋉ S_{n}$ . We describe an irreducible $G$ -module of degree-four 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.

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

Milestones

Received: 25 January 2010

Revised: 1 November 2010

Accepted: 5 December 2010

Published: 22 August 2011

Proposed: David Eisenbud

Authors

Luke Oeding
Dipartimento di Matematica “U. Dini” Università degli Studi di Firenze Viale Morgagni 67/A 50134 Firenze, Italy	Department of Mathematics University of California, Berkeley 970 Evans Hall #3840 Berkeley, CA 94720-3840 United States

Vol. 5, No. 1, 2011