False statementConstruction: AI-adaptedVerification: AI-generatedverified 2026-09-26 (gpt-6-sol)

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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: A splitting $2$ -modular system for the cyclic group $C_{2} = ⟨ g ∣ g^{2} = 1 ⟩$ , with characteristic-zero fraction field $K$ and residue field of characteristic $2$ .

[F1]

The decomposition matrix has one row per ordinary irreducible character and one column per irreducible Brauer character (Decomposition numbers and the decomposition matrix).

[F2]

The number of simple modular modules, hence of irreducible Brauer characters, is the number of $2$ -regular conjugacy classes (The number of simple kG-modules equals the number of p-regular conjugacy classes). An element is $2$ -regular exactly when its order is odd (p-regular and p-singular elements).

[F3]

The Cartan matrix satisfies $C = D^{T} D$ (The Cartan matrix is D^T D).

Refutation

technique · direct

1.1givenalgebra

The group $C_{2}$ has exactly two ordinary irreducible characters over $K$ : in any representation the operator representing $g$ satisfies $(T - 1) (T + 1) = 0$ , and these factors are coprime because $2$ is invertible in $K$ , so the representation splits into its $+ 1$ and $- 1$ eigenspaces. An irreducible representation is therefore one-dimensional, with $g$ acting by $+ 1$ or $- 1$ ; both occur and are distinct.

2.1F1F2F3step 1.1

The only $2$ -regular element of $C_{2}$ is the identity, so [F2] gives exactly one irreducible Brauer character. By [F1] and step 1.1, $D$ has two rows and one column; by [F3], $C = D^{T} D$ has one row and one column.

3.1step 2.1∎

A $2 \times 1$ matrix cannot equal a $1 \times 1$ matrix. Thus this finite group and prime refute the universal statement.

Depends on

Used by

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Sources

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