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

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FALSE: every block has one ordinary and one Brauer irreducible character

Statement

Every block contains exactly one ordinary irreducible character and exactly one irreducible Brauer character.

Facts & Assumptions

Given: The cyclic group $C_{p}$ at the prime $p$ .

[L1]

Blocks partition the ordinary and Brauer irreducible characters (Blocks partition the ordinary and Brauer irreducible characters).

[L2]

After ordering by blocks, the decomposition matrix is block diagonal (After block ordering, the decomposition matrix is block diagonal).

Refutation

technique · direct

1.1givenalgebra

The group algebra $k C_{p}$ has only one irreducible Brauer character, namely the trivial one: in characteristic $p$ , a generator acts on every simple module as $1$ , since $(g - 1)^{p} = 0$ . Over a splitting field of characteristic $0$ , $C_{p}$ has $p$ distinct ordinary irreducible characters, all linear. Each has a stable rank-one lattice, whose reduction is a one-dimensional $k C_{p}$ -module and hence the trivial Brauer character. Thus every ordinary character has decomposition number $1$ in the unique Brauer column.

2.1L1L2step 1.1

By [L1] and [L2], a nonzero decomposition number joins an ordinary character to a Brauer character in the same block. Step 1.1 therefore places all $p$ ordinary irreducibles in the single block containing the trivial Brauer character. For any prime $p \geq 2$ , that block contains more than one ordinary irreducible.

3.1step 2.1∎

Therefore the statement is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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)