False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-30

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Restriction of an irreducible complex representation is always irreducible

Statement

False claim: if a complex representation of $G$ is irreducible, then its restriction to every subgroup is irreducible.

Facts & Assumptions

Given: The irreducible degree-two character $χ_{2}$ of $S_{3}$ and the subgroup $A_{3} \leq S_{3}$ .

[F1]

The restriction of $χ_{2}$ to $A_{3}$ is $θ + \overline{θ}$ (Restricting that degree-two $S_{3}$ character to the three-cycle subgroup gives the two nontrivial linear characters).

Refutation

technique · direct

1.1

The character $χ_{2}$ is irreducible on $S_{3}$ by the example that constructs it.

givenalgebra

2.1

But [F1] writes its restriction to $A_{3}$ as the sum of two distinct nontrivial characters, so the restricted representation is reducible.

F1step 1.1

3.1

Therefore restriction does not preserve irreducibility in general.

step 2.1∎

Depends on

Restricting that degree-two $S_3$ character to the three-cycle subgroup gives the two nontrivial linear characters

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Pavel Etingof et al., Introduction to Representation Theory, Section 4.11 (standard reference, not scraped)