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  "body": "\n## Summary\n\nA representation of a [[group|G5S1M9]] $G$ on a [[vector space|000AYPA9AJ]] $V$ is a [[group homomorphism|0JYTG5K9Y4]] $G \\to GL(V)$.\nHere $GL(V)$ is the group of invertible [[linear maps|0NRYHJPQG2]] $V \\to V$.\nIt lets each element of $G$ act on $V$ as a linear transformation, compatibly with the group operation.\nRepresentations turn questions about groups into questions of linear algebra, where matrices, traces and eigenvalues are available.\nThey are central in group theory, number theory, physics and chemistry.\n\n## Explanations\n\n### L1 human\n\nA representation lets a group act by matrices.\nEach group element becomes an invertible matrix, and multiplying group elements corresponds to multiplying the matrices.\nFor example, the cyclic group of order $4$ can be represented on the plane $\\R^2$ by sending a generator to the rotation by $90$ degrees.\nThen the four group elements become the four rotations by $0$, $90$, $180$ and $270$ degrees.\n",
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