False statementConstruction: AI-adaptedVerification: AI-generatedprecheck 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.

FALSE: modular representations are determined by ordinary characters

Statement

Two finite-dimensional modular representations with the same ordinary-style character data must be isomorphic.

Facts & Assumptions

Given: The cyclic group $C_{p}$ over a field $k$ of characteristic $p$ .

[L1]

Equality of Brauer characters determines only the semisimplification (Brauer-Nesbitt determines semisimplifications).

Refutation

technique · direct

1.1

Let $V$ be the indecomposable $2$ -dimensional $k C_{p}$ -module with generator acting by $(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})$ , and let $W = k \oplus k$ be the direct sum of two trivial modules. These modules are not isomorphic because $V$ is indecomposable while $W$ is semisimple.

givenalgebra

2.1

Their only composition factors are two copies of the trivial module, so they have the same semisimplification. By [L1], they therefore have the same Brauer character.

L1step 1.1

3.1

Thus the character data do not recover the full modular representation, only its semisimplification. The statement is false.

step 1.1step 2.1∎

Depends on

Brauer-Nesbitt determines semisimplifications

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)