Dual representation

Let $Γ : 𝐺 \to G L (𝑉)$ be a Group representation. The dual representation is a representation carried by the dual vector space $𝑉 *$ defined by rep

Γ * (𝑔) = Γ (𝑔 - 1) * .

In the case of a unitary representation $Γ : 𝐺 \to G L (𝑉)$ , the dual representation $Γ *$ may be identified with the complex conjugate.

Functorial construction

The representation $Γ$ may be conceived as a functor from a groups-as-category $Γ : 𝐺 \to 𝖵 𝖾 𝖼 𝗍 𝕂$ . We have two contravariant functors $i n v : 𝐺 𝐨 𝐩 \to 𝐺$ and $d u a l : 𝖵 𝖾 𝖼 𝗍 𝐨 𝐩 𝕂 \to 𝖵 𝖾 𝖼 𝗍 𝕂$ . The dual representation is thus the covariant functor
$Γ * = d u a l \circ Γ \circ i n v : 𝐺 \to 𝖵 𝖾 𝖼 𝗍 𝕂$

A representation $Γ$ is called self-dual iff it is equivalent to its dual $Γ *$ , and a group representation is self-dual iff it preserves a nondegenerate bilinear form.

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Dual representation

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