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

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.

28 pages. Updated with referee's suggestions

References in corpus (6)

• Eynard-Mehta theorem, Schur process, and their pfaffian analogs
• Hyperdeterminantal relations among symmetric principal minors
• Open problems on GKK tau-matrices
• Set-theoretic defining equations of the tangential variety of the Segre variety
• Not all GKK $\tau$-matrices are stable
• The Ring of Graph Invariants - Graphic Values

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• Blowup algebras of rational normal scrolls
• Lagrangian Grassmannians, CKP hierarchy and hyperdeterminantal relations
• A Notable Relation between $N$-Qubit and $2^{N-1}$-Qubit Pauli Groups via Binary $\mathrm{LGr}(N,2N)$
• Symmetrization of Principal Minors and Cycle-Sums
• Tau functions, infinite Grassmannians and lattice recurrences
• Lagrangian Grassmannians and Spinor Varieties in Characteristic Two
• Principal Minor Assignment, Isometries of Hilbert Spaces, Volumes of Parallelepipeds and Rescalling of Sesqui-holomorphic Functions
• The Kashaev Equation and Related Recurrences
• Likelihood Geometry of Determinantal Point Processes
• Graph States and the Variety of Principal Minors
• Polynomial relations among principal minors of a 4x4-matrix
• Principal Minor Ideals and Rank Restrictions on their Vanishing Sets