ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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 Cartan matrix computed from D^T D

Example

For $S_{3}$ in characteristic $2$ , the decomposition matrix

$D = (\begin{array}{cc} 1 & 0 \\ 1 & 0 \\ 0 & 1 \end{array})$

gives the Cartan matrix

$C = D^{T} D = (\begin{array}{cc} 2 & 0 \\ 0 & 1 \end{array}) .$

Facts & Assumptions

Given: The characteristic- $2$ decomposition matrix of $S_{3}$ .

[L1]

The Cartan matrix is $D^{T} D$ (The Cartan matrix is D^T D).

[L2]

The decomposition matrix of $S_{3}$ in characteristic $2$ is the displayed matrix (The decomposition matrix of S3 in characteristic two).

Verification

technique · direct

1.1

By [L2], $D^{T} = (\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 0 & 1 \end{array}) .$

L2givenalgebra

2.1

Multiplying gives $D^{T} D = (\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 0 & 1 \end{array}) (\begin{array}{cc} 1 & 0 \\ 1 & 0 \\ 0 & 1 \end{array}) = (\begin{array}{cc} 2 & 0 \\ 0 & 1 \end{array}) .$

step 1.1algebra

3.1

Therefore [L1] yields the displayed Cartan matrix.

L1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

J. Miquel Martinez, Modular Representation Theory of Finite Groups (standard reference, not scraped)
Tudor Ciurca, Representation Theory (standard reference, not scraped)