TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-05

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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).

Proof

technique · direct

1.1

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

L1given

2.1

The Cartan entry $c_{φ ψ}$ is the multiplicity of $S_{ψ}$ in the projective cover $P_{φ}$ by [F1]. Reducing the expansion from step 1.1 modulo $p$ and using [L1] again, each copy of $χ$ contributes $d_{χ ψ}$ copies of $S_{ψ}$ . Hence $c_{φ ψ} = \sum_{χ} d_{χ φ} d_{χ ψ} .$

F1L1step 1.1algebra

3.1

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

step 2.1algebra∎

Depends on

Used by

A Cartan matrix computed from D^T D Example
FALSE: the Cartan matrix equals the decomposition matrix False statement

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)