DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-29

How statement and proof provenance work

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The fixed subspace $V^{G}$ of a representation

Definition

Let $ρ : G \to GL (V)$ be a representation of a group $G$ over a field $k$ (A finite-dimensional representation $ρ : G \to GL (V)$ over a field, and its degree). The fixed subspace of $V$ is

$V^{G} : = {v \in V : g \cdot v = v for every g \in G} .$

The set $V^{G}$ is a linear subspace of $V$ (Linear subspace of a vector space): it contains $0_{V}$ because every operator $ρ (g)$ is linear, it is closed under addition and scalar multiplication for the same reason, and the defining equations are the required pointwise identities. When $G$ is finite and $k = C$ , The averaging operator projects onto the fixed subspace shows that an explicit projection of $V$ has image exactly $V^{G}$ .

Depends on

Used by

Dependency tree · two levels

11 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, Section 3.2 (standard reference, not scraped)

The fixed subspace VGVG of a representation