LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Exact symmetric-minor and alternating-Pfaffian rank ideals

Statement

Assume the Axiom of Choice (The Axiom of Choice).

Let $k$ be an algebraically closed field of characteristic zero and let $t \geq 1$ . For each integer $0 \leq m \leq t$ , in the polynomial ring of a generic symmetric $t \times t$ matrix $Z$ , the ideal $I_{m + 1}$ generated by all $(m + 1)$ -minors is prime and its zero set is the locus $rank Z \leq m$ ; $k [Z] / I_{m + 1}$ is normal. For each even integer $0 \leq m \leq t$ , in the polynomial ring of a generic alternating $t \times t$ matrix $A$ , the ideal $J_{m + 2}$ generated by the principal Pfaffians of size $m + 2$ is prime and its zero set is the locus $rank A \leq m$ ; $k [A] / J_{m + 2}$ is normal. When the indicated generator size exceeds $t$ , the corresponding ideal is zero and the quotient is polynomial.

Facts & Assumptions

Given: The Axiom of Choice and the straightening and normality supplier obligations named in the Statement and in Gram-map image, full-rank fibers, and rank-stratum dimensions.

[F1]

Straightening basis for generic symmetric determinantal rings and Normality of the symmetric determinantal ring: the symmetric standard basis and normality argument are required inputs.

[F2]

Straightening basis for generic Pfaffian rank rings and Normality of the Pfaffian rank ring: the Pfaffian standard basis and normality argument are required inputs.

Proof

1.1givenalgebra

A matrix has rank at most $m$ exactly when all $(m + 1)$ -minors vanish, by Gaussian elimination. Thus the symmetric zero locus is precisely the asserted set. For an alternating matrix use the congruence normal form in Gram-map image, full-rank fibers, and rank-stratum dimensions: the coefficients of the $(n + 1)$ st exterior power of its two-form are $(n + 1)!$ times its principal $(2 n + 2)$ -Pfaffians. The factorial is nonzero in characteristic zero. In the canonical form consisting of $r / 2$ symplectic blocks, this exterior power is zero exactly when $r \leq 2 n = m$ . Consequently the Pfaffian zero locus is precisely the rank- $\leq m$ set, including $m = 0$ and the case of no possible generators.

2.1F1F2step 1.1algebra∎

The normality suppliers in [F1] and [F2] assert, with the same variables and ideals as here, that each indicated quotient is an integrally closed domain. Therefore their defining ideals are prime. Combined with step 1.1, their zero sets are exactly the stated rank-bounded loci. If the generator size exceeds $t$ , the ideal is empty-generated, hence zero, and its quotient is the polynomial ring.

Depends on

Used by

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Sources

Goodman, Multiplicity-Free Spaces and Schur-Weyl-Howe Duality, §§6.4–6.5 (standard reference, not scraped)
Lakshmibai, Raghavan, Sankaran, and Shukla, Standard Monomial Bases, Moduli Spaces of Vector Bundles & Invariant Theory, §§1.7 and 2 (standard reference, not scraped)
Linshaw and Song, Standard Monomials and Invariant Theory of Arc Spaces II: Symplectic Group, §1 (standard reference, not scraped)