CounterexampleConstruction: Literature-sourcedVerification: 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.

Ordinary trace on a p-singular unipotent element is not a Brauer-character value

Statement refuted

For every element of a modular representation, the ordinary matrix trace is the Brauer-character value.

Facts & Assumptions

Given: The cyclic group $C_{p} = ⟨ u ⟩$ over a field $k$ of characteristic $p$ , with $u$ acting on $k^{2}$ by $(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})$ .

[L1]

The false statement above is the claim to be refuted (FALSE: a Brauer character is defined on all elements by the usual trace).

[F1]

Brauer characters are defined only on $p$ -regular elements (Brauer character of a finite-dimensional kG-module).

Counterexample

technique · direct

1.1

The element $u$ has order $p$ , so it is $p$ -singular. Its action matrix still has ordinary trace $2$ .

givenalgebra

2.1

By [F1], the Brauer character of this module is not defined at $u$ , because $u \notin C_{p}^{0}$ . So the trace from step 1.1 cannot be a Brauer-character value.

F1step 1.1

3.1

This directly refutes the statement [L1].

L1step 2.1∎

Depends on

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

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