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 rep2

• there exists a Natural isomorphism between 𝑆 :𝔛 ⇒˜𝔛 :𝐺 →𝖵𝖾𝖼𝗍𝕂;

• there exists a 𝕂-linear isomorphism 𝑆 :𝑉 →𝑊 or intertwiner such that

  𝔛(𝑔)=𝑆−1˜𝔛(𝑔)𝑆

  for all 𝑔 ∈𝐺;

• 𝑉 and 𝑊 are isomorphic as [[Module over a group|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

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tidy | en | SemBr