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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]], let $N$ be a [[normal subgroup|0CB0DX5BVV]] of $G$, and let $F$ be a field.\nLet $V$ be an [[irreducible|FX3NEFYFFN]] [[representation|WYGRXCG2RN]] of $G$ over $F$ of finite dimension.\nFor a representation $W$ of $N$ and $g \\in G$, the conjugate ${}^g W$ is the space $W$ with $n \\in N$ acting as $g^{-1} n g$ acts on $W$.\n\n1. The restriction $V|_N$ is a semisimple representation of $N$. There are an irreducible representation $W$ of $N$, an integer $e \\ge 1$ and elements $g_1, \\ldots, g_t \\in G$ with $V|_N \\cong e \\, ({}^{g_1}W \\oplus \\cdots \\oplus {}^{g_t}W)$. The ${}^{g_i}W$ are pairwise non-isomorphic, and every $G$-conjugate of $W$ is isomorphic to one of them.\n2. Let $T = \\set{g \\in G : {}^g W \\cong W}$, the inertia group of $W$, and let $U$ be the sum of the subrepresentations of $V|_N$ isomorphic to $W$. Then $U$ is invariant under $T$, $U$ is an irreducible representation of $T$, $t = [G : T]$, and $V \\cong \\operatorname{Ind}_T^G U$.\n\n## Remarks\n\nFor the proof of part 1, take an irreducible $N$-subrepresentation $W$ of $V$.\nEach $gW$ is again $N$-invariant and irreducible, and isomorphic to a conjugate of $W$.\nThe sum $\\sum_g gW$ is a nonzero $G$-invariant subspace, so it is $V$.\nThe elements of $G$ permute the isotypic components of $V|_N$ transitively, and $T$ is the [[stabiliser|0AYMP41THN]] of the component $U$.\nPart 2 follows, since a representation that is a direct sum of transitively permuted subspaces is [[induced|JJP4JFH3C6]] from a stabiliser.\nExample: let $G = S_3$, $N = A_3 \\cong \\Z/3$ and let $V$ be the irreducible complex representation of degree $2$.\nThen $V|_N$ is the sum of the two non-trivial characters of $A_3$, which conjugation by a transposition swaps.\nHere $e = 1$, $t = 2$, $T = N$, and $V = \\operatorname{Ind}_{A_3}^{S_3} \\omega$ for a non-trivial character $\\omega$ of $A_3$.\nThe theorem also holds when $G$ is any group and $N$ has finite index in $G$.\n\n## Explanations\n\n### L1 human\n\nClifford theory explains how an irreducible representation of a group looks when one only watches a normal subgroup.\nIt breaks into equally many copies of a few irreducible pieces, and those pieces are all twists of one another by the big group.\nMoreover the whole representation can be rebuilt by induction from the subgroup that fixes one piece.\n\n### L2 human\n\nThis is the basic tool for building representations of a group from those of a normal subgroup and of the quotient.\nIt applies well to solvable groups, which have chains of normal subgroups.\nIn character language it states that the irreducible constituents of $\\chi|_N$ form one $G$-orbit and all occur with the same multiplicity $e$.\nThe Clifford correspondence, part 2, gives a bijection between irreducible characters of $T$ lying over $W$ and irreducible characters of $G$ lying over $W$, by induction.\nIt is a standard step in inductive arguments in character theory, which pass from $G$ to a normal subgroup.\n",
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