{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.\nFor a finite group $K$ and functions $\\alpha, \\beta \\colon K \\to \\C$ write $\\ip{\\alpha}{\\beta}_K = \\frac{1}{\\abs{K}} \\sum_{k \\in K} \\alpha(k) \\overline{\\beta(k)}$.\nLet $\\theta$ be a class function on $H$, let $\\psi$ be a class function on $G$, and let $\\operatorname{Ind}_H^G \\theta$ be the [[induced|JJP4JFH3C6]] class function.\nThen\n$$\\ip{\\operatorname{Ind}_H^G \\theta}{\\psi}_G = \\ip{\\theta}{\\psi|_H}_H,$$\nwhere $\\psi|_H$ is the restriction of $\\psi$ to $H$.\n\n## Remarks\n\nThe proof is a direct computation: insert the formula $\\operatorname{Ind}_H^G \\theta(g) = \\frac{1}{\\abs{H}}\\sum_{x \\in G,\\ x^{-1}gx \\in H} \\theta(x^{-1}gx)$, substitute $k = x^{-1} g x$, and use that $\\psi$ is constant on [[conjugacy classes|0DEBXAJ5SA]].\nThe module form holds over any field $F$: for an $F[H]$-module $W$ and an $F[G]$-module $V$, $\\Hom_G(\\operatorname{Ind}_H^G W, V) \\cong \\Hom_H(W, V|_H)$ as vector spaces.\nExample: for irreducible [[characters|RMWMYBY8TC]] $\\theta$ of $H$ and $\\psi$ of $G$, the multiplicity of $\\psi$ in $\\operatorname{Ind}_H^G \\theta$ equals the multiplicity of $\\theta$ in $\\psi|_H$.\nFor $H = \\set{1}$ and $\\theta$ trivial this gives that each irreducible $\\psi$ occurs $\\psi(1)$ times in the regular representation.\n\n## Explanations\n\n### L1 human\n\nInducing goes up from a subgroup to the group, and restricting goes down.\nFrobenius reciprocity says that these two moves are mirror images: counting how often one piece appears after going up gives the same number as counting after going down.\nThis turns hard questions about the big group into easier questions about a subgroup.\n\n### L2 human\n\nIn the language of categories, induction is left adjoint to restriction.\nThe formula is used constantly to decompose induced characters, for instance in the proofs of [[Brauer's theorem on induced characters|V8JVWE87TE]] and [[Artin's induction theorem|2CABBY4EEB]].\nIt also gives Mackey-type computations and is the basis of [[Clifford theory|SAH4PD2J9T]].\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "JJP4JFH3C6",
      "RMWMYBY8TC"
    ],
    "kind": "statement",
    "msc": [
      "20C15"
    ],
    "name": "Frobenius reciprocity",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q48998319",
      "wikipedia": "Frobenius reciprocity"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Frobenius_reciprocity"
      }
    ],
    "tag": "PAP76H0ND9",
    "uses": [
      "G5S1M9",
      "0X1JXT47GA",
      "JJP4JFH3C6",
      "RMWMYBY8TC",
      "0DEBXAJ5SA",
      "0FTH2E9TAS"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/PAP76H0ND9/",
  "links": [
    {
      "field": "about",
      "target": "JJP4JFH3C6"
    },
    {
      "field": "about",
      "target": "RMWMYBY8TC"
    },
    {
      "field": "uses",
      "target": "G5S1M9"
    },
    {
      "field": "uses",
      "target": "0X1JXT47GA"
    },
    {
      "field": "uses",
      "target": "JJP4JFH3C6"
    },
    {
      "field": "uses",
      "target": "RMWMYBY8TC"
    },
    {
      "field": "uses",
      "target": "0DEBXAJ5SA"
    },
    {
      "field": "uses",
      "target": "0FTH2E9TAS"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/PAP76H0ND9.md",
  "msc": [
    "20C15"
  ],
  "name": "Frobenius reciprocity",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PAP/PAP76H0ND9.md",
  "status": null,
  "tag": "PAP76H0ND9",
  "uri": "m2://PAP76H0ND9",
  "url": "https://math2.tamnd.com/wiki/frobenius-reciprocity/",
  "writing": {
    "tag": "PAP76H0ND9",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}