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Artin induction for rational characters

Statement

Let $G$ be a finite group, and let $x \in R_{Q} (G)$ in the sense of The rational representation ring $R_{Q} (G)$ and rational-valued class functions. Then:

$x$ is a rational linear combination of characters induced from cyclic subgroups of $G$ .
Equivalently, the controlled multiple $∣ G ∣ x$ lies in the cyclic induction subgroup $I_{c y c} (G)$ .

Facts & Assumptions

Given: A finite group $G$ and an element $x \in R_{Q} (G)$ .

[F1]

The cyclic induction subgroup $I_{c y c} (G)$ is an ideal of $R (G)$ (The cyclic induction subgroup is an ideal of the character ring).

[F2]

The trivial character satisfies an Artin relation $∣ G ∣ 1_{G} = \sum_{i} a_{i} {Ind}_{C_{i}}^{G} 1_{C_{i}}$ with $C_{i} \leq G$ cyclic and $a_{i} \in Z$ (A positive integer multiple of the trivial character is an integral combination of cyclic permutation characters).

[F3]

The projection formula says $({Ind}_{H}^{G} χ) ψ = {Ind}_{H}^{G} (χ {Res}_{H}^{G} ψ)$ (Induction and restriction satisfy the projection formula on character rings).

[A1]

If a finite-dimensional representation is defined over $Q$ , then its restriction to a subgroup is again defined over $Q$ .

Proof

technique · direct

1.1

By [F2], choose cyclic subgroups $C_{i} \leq G$ and integers $a_{i}$ with $∣ G ∣ 1_{G} = \sum_{i} a_{i} {Ind}_{C_{i}}^{G} 1_{C_{i}}$ . Multiplying by $x$ in $R (G)$ gives $∣ G ∣ x = \sum_{i} a_{i} ({Ind}_{C_{i}}^{G} 1_{C_{i}}) x$ . Because [F1] makes $I_{c y c} (G)$ an ideal containing each ${Ind}_{C_{i}}^{G} 1_{C_{i}}$ , this already shows $∣ G ∣ x \in I_{c y c} (G)$ .

F1F2givenalgebra

2.1

Applying [F3] to each summand of step 1.1 yields $∣ G ∣ x = \sum_{i} a_{i} {Ind}_{C_{i}}^{G} ({Res}_{C_{i}}^{G} x)$ . By [A1], each ${Res}_{C_{i}}^{G} x$ again lies in the rational representation ring of the cyclic subgroup $C_{i}$ . This identity is an explicit expression of $∣ G ∣ x$ as a sum of characters induced from cyclic subgroups.

F3A1step 1.1algebra

3.1

Dividing the identity of step 2.1 by $∣ G ∣$ expresses $x$ as a rational linear combination of characters induced from cyclic subgroups of $G$ . This is claim 1.

step 2.1algebra

4.1

Conversely, if $x$ is any rational linear combination of characters induced from cyclic subgroups, then multiplying by a common positive denominator places that multiple of $x$ in $I_{c y c} (G)$ . Step 1.1 shows that one may take the specific controlled denominator $∣ G ∣$ , so claims 1 and 2 are equivalent.

F1step 1.1step 3.1algebra∎

Depends on

Used by

Dependency tree · two levels

15 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

Tammo tom Dieck, Representation Theory, Theorem (4.5.2) (standard reference, not scraped)
Janos Kramar, Artin's and Brauer's Theorems on Induced Characters, Theorem 1 (standard reference, not scraped)
Kay Yang, Rational Valued Characters, Theorem 3 (standard reference, not scraped)