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  "body": "\n## Summary\n\nIn the field of representation theory in mathematics, a **projective representation** of a group $G$ on a [[vector space|000AYPA9AJ]] $V$ over a [[field|04C5ZBG1WT]] $F$ is a [[group homomorphism|0HV9Y7WG5R]] from $G$ to the projective linear group\n\n$$\\mathrm{PGL}(V) = \\mathrm{GL}(V) / F^*,$$\n\nwhere GL(*V*) is the [[general linear group|0FCHE902D7]] of invertible linear transformations of $V$ over $F$, and $F^{\\ast}$ is the [[normal subgroup|0CB0DX5BVV]] of $\\mathrm{GL}(V)$ consisting of nonzero scalar multiples of the identity transformation (see Scalar transformation).\n",
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