TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Irreducible Brauer characters form a basis of the p-regular class functions

Statement

Choose once and for all the standard complex realization of Brauer character values by identifying the prime-to- $p$ roots of unity in the splitting $p$ -modular system with the corresponding complex roots of unity. Under that identification, the irreducible Brauer characters of $G$ form a basis of the complex vector space of class functions on $G^{0}$ .

Facts & Assumptions

Given: A finite group $G$ , a prime $p$ , and a splitting $p$ -modular system for $G$ .

[L1]

Every Brauer character is a class function on $G^{0}$ (Brauer characters are class functions on p-regular elements).

[L2]

Brauer characters are additive on short exact sequences (Brauer characters are additive on short exact sequences).

[L3]

Ordinary irreducible complex characters form a basis of the space of class functions on $G$ (The irreducible complex characters form an orthonormal basis of $c f (G)$ ).

[F4]

For every ordinary irreducible character $χ$ , its restriction $χ^{0} : = χ ∣_{G^{0}}$ is the Brauer character of the modular reduction of a stable lattice affording $χ$ (Decomposition map from ordinary to modular Grothendieck groups).

[F5]

Proposition 1.5 of J. Miquel Martínez's cited course notes proves that the irreducible Brauer characters are linearly independent as functions on $G^{0}$ , by applying linear independence of modular trace functions and recovering their semisimple parts from $p$ -regular elements.

Proof

technique · direct

1.1

By [L1], every irreducible Brauer character is a class function on $G^{0}$ , so the family lies in the target vector space. By [F5], it is linearly independent.

L1F5given

1.2

Let $f : G^{0} \to C$ be a class function. Extend it to a class function $\tilde{f} : G \to C$ by setting $\tilde{f} (g) = 0$ on the $p$ -singular elements. Then [L3] gives $\tilde{f} = \sum_{χ} a_{χ} χ$ as a linear combination of ordinary irreducible characters. Restricting to $G^{0}$ yields $f = \sum_{χ} a_{χ} χ^{0} .$

L3givenalgebra

2.1

By [F4], each $χ^{0}$ is the Brauer character of some finite-dimensional $k G$ -module. A composition series for that module and [L2] write $χ^{0}$ as a linear combination of irreducible Brauer characters. Step 1.2 therefore expresses $f$ as a linear combination of irreducible Brauer characters.

L2F4step 1.2algebra

3.1

Steps 1.1 and 2.1 show that the irreducible Brauer characters are linearly independent and span the class functions on $G^{0}$ , hence form a basis.

step 1.1step 2.1∎

Depends on

Used by

The number of simple kG-modules equals the number of p-regular conjugacy classes Corollary
Decomposition numbers and the decomposition matrix Definition
Brauer characters of a p-group Example
Blocks partition the ordinary and Brauer irreducible characters Theorem
Brauer-Nesbitt determines semisimplifications Theorem

Dependency tree · two levels

14 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)