TheoremStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-26 (gpt-6-sol)

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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.

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The Cartan matrix is D^T D

Statement

If $D = (d_{χ φ})$ is the decomposition matrix and $C = (c_{φ ψ})$ is the Cartan matrix, then

$C = D^{T} D .$

Equivalently,

$c_{φ ψ} = \sum_{χ} d_{χ φ} d_{χ ψ} .$

Facts & Assumptions

Given: The decomposition matrix $D$ and the Cartan matrix $C$ of $G$ at $p$ .

[F1]

The Cartan invariants record composition multiplicities in projective covers (Projective indecomposable characters and Cartan invariants).

[L1]

Brauer reciprocity identifies $d_{χ φ}$ with the multiplicity of $χ$ in the projective indecomposable character $Φ_{φ}$ (Brauer reciprocity).

[F2]

The decomposition map is additive on ordinary representation classes, sends the class afforded by $χ$ to $\sum_{ψ} d_{χ ψ} [S_{ψ}]$ , and sends the projective lattice lift affording $Φ_{φ}$ to $[P_{φ}]$ (Decomposition numbers and the decomposition matrix, Projective indecomposable characters and Cartan invariants).

Proof

technique · direct

1.1L1given

Fix $φ$ . Decompose the projective indecomposable character as $Φ_{φ} = \sum_{χ} d_{χ φ} χ$ by [L1].

2.1F1F2step 1.1algebra

The Cartan entry $c_{φ ψ}$ is the multiplicity of $S_{ψ}$ in the projective cover $P_{φ}$ by [F1]. Apply the decomposition map [F2] to the ordinary-character expansion in step 1.1. Its left side becomes $[P_{φ}]$ , while the right side becomes $\sum_{χ} d_{χ φ} \sum_{ψ} d_{χ ψ} [S_{ψ}]$ . Comparing coefficients of $[S_{ψ}]$ gives $c_{φ ψ} = \sum_{χ} d_{χ φ} d_{χ ψ} .$

3.1step 2.1algebra∎

This is exactly the $(φ, ψ)$ entry of $D^{T} D$ , so $C = D^{T} D$ .

Depends on

Used by

A Cartan matrix computed from D^T D Example
FALSE: the Cartan matrix equals the decomposition matrix False statement
Weight-zero symmetric-group blocks are defect zero Proposition

Dependency tree · two levels

10 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)