Vol. 5, No. 1, 2011

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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 Zn, is invariant under the action of a group G GL(2n) isomorphic to SL(2)×n Sn. 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.

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