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  "body": "\n## Summary\n\nA representation of a [[Lie group|7JBRGCVJRB]] $G$ is a continuous homomorphism from $G$ to the group of invertible linear maps of a vector space.\nIt lets the group act on vectors by linear maps, so that group theory can be studied with linear algebra.\nEach finite-dimensional representation has a derivative, a representation of the Lie algebra, which often determines it.\nRepresentations of Lie groups are central in harmonic analysis, number theory and quantum physics.\n\n## Explanations\n\n### L1 human\n\nA representation turns each element of a group into a matrix, so that composing elements becomes multiplying matrices.\nThe rotation group of space acts on vectors of $\\R^3$ by rotation matrices, and this is a representation.\nIt also acts on functions on the sphere, by rotating the function's graph, and this is a larger representation.\nThe two must be continuous: rotations by nearby angles give nearby matrices.\nBreaking a representation into irreducible pieces is like the Fourier series of a function.\n",
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