---
tag: FX3NEFYFFN
kind: concept
name: Irreducible representation
msc: [20C15]
specializes: [WYGRXCG2RN]
sources:
  - url: https://en.wikipedia.org/wiki/Irreducible_representation
same_as:
  wikidata: Q13690522
  wikipedia: Irreducible representation
---

## Summary

An irreducible representation is a nonzero [[group representation|WYGRXCG2RN]] with no proper nonzero [[invariant subspace|0ACX5QT8TE]].
Irreducible representations are the building blocks of representation theory.
For a finite group over the complex numbers, every finite-dimensional representation is a direct sum of irreducible ones, by Maschke's theorem.

## Explanations

### L1 human

A representation is irreducible when it cannot be cut into smaller pieces that the group preserves.
Think of it as an atom: other representations are built from it, but it is not built from smaller ones.
Every one-dimensional representation is irreducible, because a line has no smaller nonzero subspace.
The rotations of the plane by multiples of $90$ degrees give an irreducible representation of the cyclic group of order $4$ over $\R$.
Indeed, a $90$ degree rotation preserves no line through the origin.