Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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FALSE: the Cartan matrix equals the decomposition matrix

Statement

For every finite group and prime, the Cartan matrix equals the decomposition matrix.

Facts & Assumptions

Given: The decomposition matrix D and the Cartan matrix C.

[L1]

The Cartan matrix satisfies C=DTD (The Cartan matrix is D^T D).

Refutation

technique · direct
1.1

The matrices D and C have different meanings and usually different shapes: rows of D are indexed by ordinary irreducibles, while both rows and columns of C are indexed by Brauer irreducibles.

given
2.1

If they were always equal, then [L1] would force D=DTD for every group. That is impossible in general for a non-square or non-idempotent decomposition matrix.

L1step 1.1algebra
3.1

Hence the statement is false.

step 2.1∎

Depends on

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