False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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Induction followed by restriction is the identity on complex representations

Statement

False claim: for every subgroup $H \leq G$ , the composite ${Res}_{H}^{G} {Ind}_{H}^{G}$ is the identity on complex representations of $H$ .

Facts & Assumptions

Given: The subgroup $A_{3} \leq S_{3}$ and the nontrivial linear character $θ$ of $A_{3}$ .

[F1]

Inducing $θ$ to $S_{3}$ gives the irreducible degree-two character $χ_{2}$ (Inducing a nontrivial character of a three-cycle subgroup of $S_{3}$ gives an irreducible degree-two character).

[F2]

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

Refutation

technique · direct

1.1

By [F1], ${Ind}_{A_{3}}^{S_{3}} θ = χ_{2}$ .

F1given

2.1

Applying restriction and then [F2] gives ${Res}_{A_{3}}^{S_{3}} {Ind}_{A_{3}}^{S_{3}} θ = {Res}_{A_{3}}^{S_{3}} χ_{2} = θ + \overline{θ} \neq θ$ .

F2step 1.1

3.1

So induction followed by restriction is not the identity in general.

step 2.1∎

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

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