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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $F$ be a field.\nLet $V$ be a [[vector space|000AYPA9AJ]] over $F$, and let $GL(V)$ be the group of invertible [[linear maps|0NRYHJPQG2]] $V \\to V$ under composition.\nA *representation* of $G$ on $V$ is a [[group homomorphism|0JYTG5K9Y4]] $\\rho \\colon G \\to GL(V)$.\nThat is, $\\rho(gh) = \\rho(g) \\circ \\rho(h)$ for all $g, h \\in G$.\nThe dimension of $V$ is the degree of $\\rho$.\nTwo representations $\\rho \\colon G \\to GL(V)$ and $\\sigma \\colon G \\to GL(W)$ are isomorphic if there is a linear isomorphism $T \\colon V \\to W$ with $T \\circ \\rho(g) = \\sigma(g) \\circ T$ for all $g \\in G$.\n\n## Remarks\n\nExample: let $G = \\Z / n\\Z$ and $V = \\C$.\nThe map $\\rho(k) = e^{2\\pi i k / n}$, viewed as multiplication on $\\C$, is a representation of degree $1$, because $e^{2\\pi i (k + l)/n} = e^{2\\pi i k/n} e^{2\\pi i l/n}$, and the value does not depend on the representative $k$ of its class.\nNon-example: on $G = \\Z / 2\\Z$, the map sending $0$ to $\\id$ and $1$ to $2 \\cdot \\id$ on $\\C$ is not a representation, because $\\rho(1 + 1) = \\rho(0) = \\id$ but $\\rho(1)\\rho(1) = 4 \\cdot \\id$.\nWhen $V = F^n$, the group $GL(V)$ is the [[general linear group|0FCHE902D7]] $GL_n(F)$, and a representation is a matrix-valued homomorphism.\nEquivalently, a representation is a linear action of $G$ on $V$, or a module over the [[group algebra|MNP2HBWB5Q]] $F[G]$.\n",
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