ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Grassmannians and flag manifolds as homogeneous spaces

Example

Assume ${A C}_{ω}$ . For $0 \leq k \leq n$ ,

${Gr}_{k} (R^{n}) ≅ O (n) / (O (k) \times O (n - k)),$

${Gr}_{k} (C^{n}) ≅ U (n) / (U (k) \times U (n - k)) .$

More generally, if positive block sizes $n_{1}, \dots, n_{s}$ sum to $n$ , the corresponding complete or partial real and complex flag manifolds are

$O (n) / (O (n_{1}) \times \dots \times O (n_{s})),$

$U (n) / (U (n_{1}) \times \dots \times U (n_{s})),$

as smooth homogeneous spaces.

Facts & Assumptions

Given: ${A C}_{ω}$ , the standard inner products on $R^{n}$ and $C^{n}$ , and the indicated Grassmannians and flag manifolds with their standard smooth structures.

[A1]

A smooth transitive action of a Lie group identifies the manifold with the quotient by the stabilizer. The Axiom of Countable Choice ( ${A C}_{ω}$ ), Transitive smooth actions identify M with G/H.

Verification

technique · choose adapted orthonormal bases and read off block stabilizers

1.1

Given two $k$ -planes, choose orthonormal bases for each and extend them to orthonormal bases of the ambient space. The orthogonal or unitary map between the adapted bases carries one plane to the other, proving transitivity. The stabilizer of the coordinate $k$ -plane preserves it and its orthogonal complement, hence is exactly the block diagonal subgroup $O (k) \times O (n - k)$ or $U (k) \times U (n - k)$ . Conversely every such block matrix stabilizes the plane.

givenalgebra

1.2

For a flag with successive quotient dimensions $n_{1}, \dots, n_{s}$ , choose an orthonormal basis adapted to all members of the flag. Mapping one adapted basis to another proves transitivity. A unitary or orthogonal transformation fixes the coordinate flag exactly when it preserves each successive orthogonal block, which gives the stated product block subgroup. These actions are smooth in the usual graph-coordinate charts for subspaces.

givenalgebra

2.1

Apply [A1] to steps 1.1 and 1.2. This gives all displayed equivariant diffeomorphisms. The cases $k = 0$ or $k = n$ have stabilizer the whole group and quotient a point. Repeated or zero flag blocks are omitted because they do not change a flag; partial flags correspond to any positive composition of $n$ . Countable choice is used through [A1].

A1step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)
Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I (standard reference, not scraped)