ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-30

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Restricting that degree-two $S_{3}$ character to the three-cycle subgroup gives the two nontrivial linear characters

Example

Let $χ_{2}$ be the degree-two irreducible character of $S_{3}$ from the previous example. Then

{Res}_{A_{3}}^{S_{3}} χ_{2} = θ + \overline{θ},

the sum of the two nontrivial linear characters of $A_{3}$ .

Facts & Assumptions

Given: The degree-two character $χ_{2} = (2, 0, - 1)$ of $S_{3}$ from the previous example, and the two nontrivial linear characters $θ$ and $\overline{θ}$ of $A_{3}$ .

[F1]

The previous example identifies $χ_{2}$ with the induced character ${Ind}_{A_{3}}^{S_{3}} θ$ (Inducing a nontrivial character of a three-cycle subgroup of $S_{3}$ gives an irreducible degree-two character).

[F2]

Frobenius reciprocity identifies multiplicities before and after induction (Frobenius reciprocity for complex characters).

Verification

technique · direct

1.1

Restricting the values $(2, 0, - 1)$ from [F1] to $A_{3} = {e, (123), (132)}$ gives ${Res}_{A_{3}}^{S_{3}} χ_{2} = (2, - 1, - 1)$ .

F1given

2.1

Frobenius reciprocity [F2] gives $⟨ θ, {Res}_{A_{3}}^{S_{3}} χ_{2} ⟩_{A_{3}} = ⟨ {Ind}_{A_{3}}^{S_{3}} θ, χ_{2} ⟩_{S_{3}} = ⟨ χ_{2}, χ_{2} ⟩_{S_{3}} = 1$ , and the same computation with $\overline{θ}$ gives multiplicity $1$ for $\overline{θ}$ .

F1F2step 1.1algebra

3.1

The restriction has degree $2$ at the identity, while the two nontrivial linear characters already account for degree $1 + 1 = 2$ . Hence no trivial summand occurs, and ${Res}_{A_{3}}^{S_{3}} χ_{2} = θ + \overline{θ}$ .

step 1.1step 2.1algebra∎

Depends on

Used by

Induction followed by restriction is the identity on complex representations False statement
Restriction of an irreducible complex representation is always irreducible False statement

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

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

Restricting that degree-two S3S3​ character to the three-cycle subgroup gives the two nontrivial linear characters