ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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The Second Main Theorem at u=1 is block-diagonal decomposition

Example

Assume the Axiom of Choice. Let $(K, O, k)$ be a splitting $p$ -modular system for a finite group $G$ , with $k$ algebraically closed. At $u = 1$ one has $C_{G} (u) = G, d_{χ, φ}^{1} = d_{χ, φ}, c^{G} = c$ for every ordinary irreducible $χ$ , irreducible Brauer character $φ$ , and block $c$ of $k G$ . Consequently Brauer's Second Main Theorem specializes to $d_{χ, φ} = 0 unless χ and φ belong to the same block,$ which is precisely block diagonality of the ordinary decomposition matrix.

Facts & Assumptions

Given: AC and the modular system and characters in the Example.

[F1]

The explicit generalized-decomposition formula is Generalized decomposition numbers.

[F2]

For $u = 1$ , the subsection convention uses block induction from $G$ to itself (Brauer subsections and B-subsections).

[F3]

Brauer's Second Main Theorem gives generalized-decomposition support (Brauer's Second Main Theorem), under the algebraically closed residue-field and AC hypotheses (An algebraically closed field: every nonconstant polynomial has a root in the field and The Axiom of Choice).

[F4]

Ordinary decomposition matrices are independently known to be block diagonal (After block ordering, the decomposition matrix is block diagonal).

Verification

1.1

For $u = 1$ , restriction from $G$ to $C_{G} (1) = G$ leaves $χ$ as its sole ordinary constituent with multiplicity one. The scalar by which $1$ acts is $1$ . Substitution in F1 gives $d_{χ, φ}^{1} = d_{χ, φ} .$

2.1

Block induction from a group to itself is the identity in F2, so $c^{G} = c$ . Apply F3: if $d_{χ, φ} = d_{χ, φ}^{1}$ is nonzero, the local block $c$ of $φ$ equals the global block of $χ$ . This is exactly the off-block vanishing in F4.

F2F3F4step 1.1

3.1

Conversely, F4 verifies this boundary specialization independently of the Second Main Theorem. The calculation does not say that every within-block entry is nonzero. Algebraic closedness and AC are used only to invoke F3; steps 1.1–2.1 themselves are finite substitutions.

F3F4∎

Depends on

Used by

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Sources

Meierfrankenfeld, MTH 912 Class Notes, Theorem 6.7.15 and Corollary 6.7.16, pp. 169–171 (standard reference, not scraped)
Craven, The Brauer Correspondence, section 1.5, pp. 13–14 (standard reference, not scraped)