DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Decomposition map from ordinary to modular Grothendieck groups

Definition

Let $R_{K} (G)$ be the Grothendieck group of finite-dimensional $K G$ -modules and $R_{k} (G)$ the Grothendieck group of finite-dimensional $k G$ -modules. If $V$ is a finite-dimensional $K G$ -module and $L \subseteq V$ is a $G$ -stable $O G$ -lattice, define

$d ([V]) : = [L / m L] \in R_{k} (G) .$

The map $d : R_{K} (G) \to R_{k} (G)$ is called the decomposition map. Its well-definedness with respect to the chosen stable lattice is proved in The decomposition map is independent of the stable lattice ↗.

If $χ$ is the ordinary character of $V$ , then the restriction $χ^{0}$ to $G^{0}$ is the Brauer character of the modular class $d ([V])$ .

Depends on

Used by

Decomposition numbers and the decomposition matrix Definition
The decomposition matrix of S3 in characteristic two Example
FALSE: reduction mod p of an ordinary irreducible is always irreducible False statement
Irreducible Brauer characters form a basis of the p-regular class functions Theorem
The decomposition map is independent of the stable lattice Theorem

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)