Group representation theory MOC

Irrep

An irreducible representation or irrep is a representation that is not reducible. They play a fundamental role in representation theory, since in many cases a representation may be decomposed into irreducible representations (Maschke’s theorem).

For a given group 𝐺, the set of (equivalence classes of) irreps is ̂𝐺, which is called the dual object. If 𝐺 is abelian, then ̂𝐺 is a group (see 1-dimensional irrep). It is common to take a unitary irrep as a specimen of each equivalence class.

Properties

• Schur’s lemma
• Character irreducibility criterion
• Irreps of abelian groups are 1-dimensional
• Invariant subspaces of ∗-representations and unitary representations coïncide

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