CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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An ordinary irreducible representation can have reducible reduction modulo p

Statement refuted

Reducing an ordinary irreducible lattice modulo $p$ always preserves irreducibility.

Facts & Assumptions

Given: The standard $O S_{3}$ -lattice $L = {(a, b, c) \in O^{3} : a + b + c = 0}$ with $O = Z_{(3)}$ .

[F1]

Reduction modulo the maximal ideal produces a $k G$ -module (An OG-lattice is a finite free module over the valuation ring with G-action, and reduction modulo the maximal ideal produces a kG-module).

[L1]

The defining-characteristic page route allows reducibility after reduction (When the characteristic divides the group order, Maschke can fail and kG need not be semisimple).

[L2]

The reduced lattice $\overline{L}$ for this $S_{3}$ example is reducible (Reducing a standard integral lattice for S3 modulo 3 produces a reducible kS3-module).

Counterexample

technique · direct

1.1

Over characteristic $0$ , the lattice $L$ affords the standard $2$ -dimensional irreducible representation of $S_{3}$ .

givenalgebra

2.1

By [F1], reducing $L$ modulo $3$ gives a $k S_{3}$ -module $\overline{L}$ . The example [L2] shows that $\overline{L}$ is reducible, and [L1] explains why this does not contradict the modular route of the page.

F1L1L2step 1.1

3.1

Therefore an ordinary irreducible representation can have reducible reduction modulo $p$ , refuting the statement.

step 1.1step 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

Peter Webb, A Course in Finite Group Representation Theory (23 Feb 2016 draft) (standard reference, not scraped)