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

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Inducing a nontrivial character of a three-cycle subgroup of $S_{3}$ gives an irreducible degree-two character

Example

Let $H = A_{3} = ⟨ (123) ⟩ \leq S_{3}$ , let $ω = e^{2 π i / 3}$ , and let $θ$ be the nontrivial linear character of $H$ with $θ ((123)) = ω$ and $θ ((132)) = ω^{2}$ . Then ${Ind}_{A_{3}}^{S_{3}} θ$ has values

(2, 0, - 1)

on the conjugacy classes ${e}$ , the transpositions, and the $3$ -cycles respectively. Its self-inner-product is $1$ , so it is irreducible of degree $2$ .

Facts & Assumptions

Given: The subgroup $A_{3} = ⟨ (123) ⟩ \leq S_{3}$ and the nontrivial character $θ$ defined in the Example.

[F1]

The induced character is computed by Frobenius' formula (Frobenius' formula for the character of an induced representation).

[F2]

A complex character is irreducible if and only if its self-inner-product is $1$ (A complex character is irreducible if and only if its self-inner-product is $1$ ).

[F3]

The notation ${Ind}_{A_{3}}^{S_{3}} θ$ is the induced character from The induced character ${Ind}_{H}^{G} χ$ of a complex character.

Verification

technique · direct

1.1

Since $[S_{3} : A_{3}] = 2$ , Frobenius' formula [F1] at the identity gives ${Ind}_{A_{3}}^{S_{3}} θ (e) = 2$ . If $g$ is a transposition, no conjugate of $g$ lies in $A_{3}$ , so Frobenius' formula gives ${Ind}_{A_{3}}^{S_{3}} θ (g) = 0$ .

F1F3given

1.2

If $g$ is a $3$ -cycle, then $A_{3}$ is normal in $S_{3}$ , so every $x \in S_{3}$ satisfies $x^{- 1} g x \in A_{3}$ . Exactly three of those conjugates equal $(123)$ and three equal $(132)$ , so [F1] gives ${Ind}_{A_{3}}^{S_{3}} θ (g) = (3 ω + 3 ω^{2}) / 3 = ω + ω^{2} = - 1$ .

F1givenalgebra

2.1

Therefore the induced character has values $(2, 0, - 1)$ on the three class types of $S_{3}$ . Its self-inner-product is $(1 / 6) (2^{2} + 3 \cdot 0^{2} + 2 \cdot (- 1)^{2}) = 1$ , so [F2] makes it irreducible; the value at $e$ shows that its degree is $2$ .

F2step 1.1step 1.2algebra∎

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Sources

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

Inducing a nontrivial character of a three-cycle subgroup of S3S3​ gives an irreducible degree-two character