False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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FALSE: reduction mod p of an ordinary irreducible is always irreducible

Statement

If $χ$ is an ordinary irreducible character, then its reduction modulo $p$ is always irreducible.

Facts & Assumptions

Given: A primitive cube root $ζ_{3}$ , the local cyclotomic triple $(K, O, k) = (Q_{3} (ζ_{3}), Z_{3} [ζ_{3}], F_{3}),$ and the standard $O S_{3}$ -lattice $L = {(a, b, c) \in O^{3} : a + b + c = 0},$ whose scalar extension to $K$ affords the ordinary standard irreducible representation of $S_{3}$ .

[F1]

Reduction modulo $p$ is recorded by the decomposition map (Decomposition map from ordinary to modular Grothendieck groups).

[F2]

Decomposition numbers describe the simple factors of that reduction (Decomposition numbers and the decomposition matrix).

Refutation

technique · direct

1.1

By the given realization, the ordinary character afforded by $K \otimes_{O} L$ is irreducible.

given

2.1

By [F1], reducing modulo the maximal ideal gives the $k S_{3}$ -module $\overline{L} = L / (1 - ζ_{3}) L .$ The nonzero vector $(1, 1, 1) \in \overline{L}$ is fixed by every permutation matrix and belongs to $\overline{L}$ because it is the reduction of $(1, 1, - 2) \in L$ . Hence $\overline{L}$ has a nontrivial proper invariant line and is reducible.

F1step 1.1algebra

3.1

Thus an ordinary irreducible representation can have reducible reduction modulo $p$ . In the decomposition process recorded by [F2], this reduction is therefore not irreducible. The statement is false.

F2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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)