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

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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.
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Decomposition numbers are nonnegative integers

Statement

Every decomposition number $d_{χ φ}$ is a nonnegative integer.

Facts & Assumptions

Given: An ordinary irreducible character $χ$ and an irreducible Brauer character $φ$ .

[F1]

The decomposition numbers are the coefficients of the simple-module expansion of the reduction of a stable lattice (Decomposition numbers and the decomposition matrix).

Proof

technique · direct

1.1F1given

By [F1], $d_{χ φ}$ is the multiplicity of the simple module $S_{φ}$ among the composition factors of the reduced lattice representing $χ$ .

2.1step 1.1algebra

A composition multiplicity counts how many times a simple factor occurs, so it is an integer and cannot be negative. Hence $d_{χ φ} \in Z_{\geq 0}$ .

3.1step 2.1∎

Therefore all decomposition numbers are nonnegative integers.

Depends on

Decomposition numbers and the decomposition matrix

Used by

Brauer reciprocity Theorem

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)