Group representation theory MOC

Equivalence of representations

Two group representations $𝔛, ˜ 𝔛$ of $𝐺$ carried by $𝑉$ and $𝑊$ respectively are equivalent, written $𝔛 ≃ ˜ 𝔛$ , iff the following interchangeable conditions hold #m/def/rep2

there exists a Natural isomorphism between $𝑆 : 𝔛 \Rightarrow ˜ 𝔛 : 𝐺 \to 𝖵 𝖾 𝖼 𝗍 𝕂$ ;
there exists a $𝕂$ -linear isomorphism $𝑆 : 𝑉 \to 𝑊$ or intertwiner such that $𝔛 (𝑔) = 𝑆 - 1 ˜ 𝔛 (𝑔) 𝑆$ for all $𝑔 \in 𝐺$ ;
$𝑉$ and $𝑊$ are isomorphic as $𝐺$ -modules, written $𝑉 ≅ 𝕂 [𝐺] 𝑊$ .

Properties

If $𝑆$ is unitary then it is a Unitary equivalence of representations
Every finite complex representation of a compact group is equivalent to a unitary representation

#state/tidy | #lang/en | #SemBr