DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-29

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.

The dual or contragredient complex representation

Definition

Let $ρ : G \to GL (V)$ be a finite-dimensional complex representation of a group $G$ (A finite-dimensional representation $ρ : G \to GL (V)$ over a field, and its degree). Its dual or contragredient representation is the representation $ρ^{*}$ of $G$ on the algebraic dual space $V^{*}$ (Linear functionals and the algebraic dual $V^{*} = L (V, F)$ ) defined by

$(g \cdot f) (v) : = f (g^{- 1} \cdot v) (g \in G, f \in V^{*}, v \in V),$

equivalently $g \cdot f = f \circ ρ (g^{- 1})$ .

Each operator $ρ^{*} (g)$ is $C$ -linear because $f$ and $ρ (g^{- 1})$ are. The inverse appears so that composition is order-preserving: writing $g \cdot f = f \circ ρ (g^{- 1})$ ,

$(g h) \cdot f = f \circ ρ ((g h)^{- 1}) = f \circ ρ (h^{- 1}) \circ ρ (g^{- 1}) = g \cdot (h \cdot f),$

and $1 \cdot f = f \circ {id}_{V} = f$ . Hence $ρ^{*}$ is a group homomorphism $G \to GL (V^{*})$ , a representation of the same degree as $ρ$ .

Depends on

Used by

Dependency tree · two levels

10 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 3.1 (standard reference, not scraped)
Pavel Etingof et al., Introduction to Representation Theory, Section 3.4 (standard reference, not scraped)