ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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The two-dimensional trivial representation of $C_{2}$ has many irreducible splittings but one isotypic component

Example

Let $k$ be a field, let $C_{2} = {e, t}$ , and let $V = k e_{1} \oplus k e_{2}$ with the trivial action of $C_{2}$ . Then

V = k e_{1} \oplus k e_{2} = k (e_{1} + e_{2}) \oplus k e_{2}

are two different decompositions into irreducible subrepresentations, but the unique isotypic component is all of $V$ .

Facts & Assumptions

Given: A field $k$ , the two-dimensional vector space $V = k e_{1} \oplus k e_{2}$ , and the trivial action of $C_{2}$ on $V$ .

[L1]

The one-dimensional trivial representation is the representation on $k$ in which every group element acts as the identity (The trivial representation, the regular representation, and permutation representations from finite $G$ -sets).

[L2]

A subrepresentation is an invariant subspace, and an irreducible representation is a nonzero representation with no proper nonzero subrepresentation (Subrepresentations, direct sums of representations, and irreducibility).

[L3]

The isotypic decomposition of a completely reducible representation is unique (The isotypic decomposition of a completely reducible representation is unique).

Verification

technique · direct

1.1

Every line in $V$ is invariant because the action is trivial. Any nonzero line is irreducible by [L2], since a one-dimensional vector space has only the subspaces $0$ and itself. Therefore both displayed decompositions are decompositions into irreducible subrepresentations, and they are different because $k e_{1} \neq k (e_{1} + e_{2})$ .

L2givenalgebra

2.1

On every nonzero line the action is trivial, so each line is equivalent to the one-dimensional trivial representation of [L1]. Hence every irreducible summand of $V$ has the same type, and the sum of all irreducible summands of that type is all of $V$ . By [L3], this is the unique isotypic component.

L1L3step 1.1givenalgebra∎

Depends on

Used by

False statement: a completely reducible representation has a unique decomposition into irreducible subrepresentations False statement

Dependency tree · two levels

12 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

Peter Webb, A Course in Finite Group Representation Theory, Chapter 1 Section 1.2 (standard reference, not scraped)

The two-dimensional trivial representation of C2C2​ has many irreducible splittings but one isotypic component