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  "body": "\n## Definition\n\nLet $G$ be a group, let $H$ be a [[subgroup|0X1JXT47GA]] of finite [[index|0DTXBQPA2S]] $n = [G : H]$, and let $F$ be a field.\nLet $\\theta \\colon H \\to GL(W)$ be a [[representation|WYGRXCG2RN]] of $H$ on a vector space $W$ over $F$.\nChoose representatives $t_1, \\ldots, t_n$ of the [[left cosets|03DEH43F4H]] of $H$ in $G$.\nFor $g \\in G$ and each $i$ there are a unique $j$ and a unique $h_i \\in H$ with $g t_i = t_j h_i$.\nThe *induced representation* $\\operatorname{Ind}_H^G \\theta$ acts on $V = \\bigoplus_{i=1}^{n} t_i W$, a direct sum of $n$ copies of $W$, by\n$$g \\cdot (t_i w) = t_j \\, (\\theta(h_i) w) \\quad (w \\in W).$$\n\n## Remarks\n\nUp to isomorphism, the result does not depend on the choice of the $t_i$.\nEquivalently, $\\operatorname{Ind}_H^G W = F[G] \\otimes_{F[H]} W$, using the [[group algebra|MNP2HBWB5Q]].\nIts degree is $[G : H] \\cdot \\dim W$.\nFor finite $G$, its character is\n$$\\operatorname{Ind}_H^G \\chi(g) = \\frac{1}{\\abs{H}} \\sum_{x \\in G,\\ x^{-1} g x \\in H} \\chi(x^{-1} g x).$$\nExample: let $G = S_3$, $H = A_3$ and $\\theta$ trivial of degree $1$.\nThen $V$ has dimension $2$, and the formula gives the values $2$ at $1$, $0$ at transpositions and $2$ at $3$-cycles.\nThis is the sum of the trivial and sign characters.\nNon-example: for $H = \\set{0}$ in $G = \\Z$, the index is infinite, so the definition does not apply; the space $\\bigoplus_{t \\in \\Z} W$ with the shift action is a different construction, sometimes called compact induction.\n",
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