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  "body": "\n## Summary\n\nAn irreducible representation is a nonzero [[group representation|WYGRXCG2RN]] with no proper nonzero [[invariant subspace|0ACX5QT8TE]].\nIrreducible representations are the building blocks of representation theory.\nFor a finite group over the complex numbers, every finite-dimensional representation is a direct sum of irreducible ones, by Maschke's theorem.\n\n## Explanations\n\n### L1 human\n\nA representation is irreducible when it cannot be cut into smaller pieces that the group preserves.\nThink of it as an atom: other representations are built from it, but it is not built from smaller ones.\nEvery one-dimensional representation is irreducible, because a line has no smaller nonzero subspace.\nThe rotations of the plane by multiples of $90$ degrees give an irreducible representation of the cyclic group of order $4$ over $\\R$.\nIndeed, a $90$ degree rotation preserves no line through the origin.\n",
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