ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02

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.

Finite-group counting model for induction

Example

Let $G$ be finite, $H \leq G$ , and let $σ : H \to U (V)$ be a unitary representation on a complex Hilbert space $V$ . Give both groups counting measure and take $ρ = 1$ . The induced Hilbert space consists of functions $F : G \to V$ satisfying $F (g h) = σ (h)^{- 1} F (g)$ , with squared norm $\sum_{x H \in G / H} ∥ F (x) ∥^{2}$ . The left action is $[Π (g) F] (x) = F (g^{- 1} x)$ . This is the published algebraic induced module with its invariant counting inner product.

Facts & Assumptions

Given: The finite groups $G, H$ , the Hilbert space $V$ , and the unitary $H$ -action $σ$ .

[F1]

The algebraic induced module consists of the right- $H$ -covariant functions $F (g h) = σ (h)^{- 1} F (g)$ with left action $[g \cdot F] (x) = F (g^{- 1} x)$ (The induced $R$ -linear $G$ -module ${Ind}_{H}^{G} W$ as $H$ -covariant functions on $G$ ).

Verification

technique · direct

1.1givenF1construct

Counting measure is left and right invariant on a finite group, so both modular functions are $1$ and $ρ = 1$ satisfies the rho covariance. The finite quotient formula is the partition of a finite sum into cosets: $\sum_{x \in G} a (x) = \sum_{x H \in G / H} \sum_{h \in H} a (x h)$ . Thus the quotient measure is counting measure, and the general covariance and left action reduce to [F1].

1.2givenF1

If $x$ is replaced by $x h_{0}$ , then $F (x h_{0}) = σ (h_{0})^{- 1} F (x)$ , so unitarity makes $∥ F (x h_{0}) ∥ = ∥ F (x) ∥$ . The stated norm and inner product are therefore independent of coset representatives.

2.1step 1.1step 1.2F1

Left multiplication permutes the finite set $G / H$ , and [F1] shows it preserves the covariance law; reindexing the finite sum proves $∥ Π (g) F ∥ = ∥ F ∥$ . The covariant subspace is closed in the complete finite product $V^{G}$ , so completion adds no vectors. This is exactly the algebraic induced module with the stated invariant inner product. ∎

Depends on

The induced $R$-linear $G$-module $\operatorname{Ind}_H^G W$ as $H$-covariant functions on $G$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Bekka–de la Harpe–Valette, Kazhdan’s Property (T), Appendices B and E (standard reference, not scraped)
Vogan, Unitary induced representations, §§1–4 (standard reference, not scraped)