ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

How statement and proof provenance work

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.

A repeated root collides the generic one-box contents

Example

For $r = 2, n = 1$ , the two generic standard multitableaux have contents $u_{1}$ and $u_{2}$ , which are distinct in $K = Q (Q, u_{1}, u_{2})$ . Under the specialization $Q \mapsto 2$ and $u_{1}, u_{2} \mapsto 1$ into $Q$ , the contents coincide and the algebra becomes $Q [ε] / (ε^{2})$ with $ε = X - 1 \neq 0$ . The generic interpolation factor $(X - u_{2}) / (u_{1} - u_{2})$ therefore has no value at this specialization.

Facts & Assumptions

Given: The multiplicative content function for standard multitableaux and the rank-one level-two quotient.

[F1]

Multipartitions, standard multitableaux, and generic contents: ${res}_{t} (i) = u_{c} Q^{b - a}$ for a box $(c, a, b)$ , and the content field is $K = Q (Q, u_{1}, \dots, u_{r})$ .

[F2]

The rank-one cyclotomic quotient has a monic basis: when $r = 2$ and $u_{1} = u_{2} = c$ , the rank-one algebra is $F [ε] / (ε^{2})$ and $ε$ is nonzero.

Verification

Given: The level-two, rank-one, one-box content sequences and the parameter specialization stated in the example.

1.1F1algebra

There are exactly two multipartitions of $1$ with two components, one box in component $1$ or one box in component $2$ . In either tableau the box is $(c, 1, 1)$ , so [F1] gives content $u_{c} Q^{0} = u_{c}$ . The two generic values $u_{1}, u_{2}$ are unequal in the rational function field because they are distinct indeterminates.

2.1F2step 1.1algebra∎

Under $Q \mapsto 2$ , $u_{1}, u_{2} \mapsto 1$ , both contents become $1$ . By [F2] with $F = Q$ , the specialized algebra is $Q [ε] / (ε^{2})$ and $ε$ is nonzero. The denominator $u_{1} - u_{2}$ of the generic interpolation factor becomes zero, so that rational expression cannot be specialized to a projector. This exhibits the content collision without asserting semisimplicity or failure of the PBW basis.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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