{"paper":{"title":"Set Theoretic Defining Equations of the Variety of Principal Minors of Symmetric Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Luke Oeding","submitted_at":"2008-09-24T18:16:03Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0809.4236","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}