[[Group representation theory MOC]]
# Irrep

An **irreducible representation** or **irrep** is a representation that is not [[Reducibility of representations|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 $\widehat{G}$, which is called the dual object.
If $G$ is abelian, then $\widehat{G}$ is a group (see [[1-dimensional irrep]]).
It is common to take a [[unitary representation|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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