{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a group and let $F$ be a field.\nLet $\\rho \\colon G \\to GL(V)$ be a [[representation|WYGRXCG2RN]] of $G$ on a vector space $V$ over $F$ of finite [[dimension|03GSZ8RTZJ]].\nThe *character* of $\\rho$ is the function $\\chi_\\rho \\colon G \\to F$ given by\n$$\\chi_\\rho(g) = \\tr \\rho(g),$$\nwhere $\\tr$ is the [[trace|0XC1RDBY7A]] of a linear map.\nThe degree of $\\chi_\\rho$ is $\\dim V$.\nWhen $F$ has characteristic $0$, the degree equals $\\chi_\\rho(1)$; in characteristic $p$ the value $\\chi_\\rho(1)$ is only $\\dim V$ modulo $p$.\nA character is irreducible if $\\rho$ is [[irreducible|FX3NEFYFFN]].\n\n## Remarks\n\nExample: let $S_3$ act on $\\C^3$ by permuting coordinates.\nThe matrix of a permutation $\\sigma$ has a $1$ on the diagonal exactly at the points fixed by $\\sigma$, so $\\chi(\\sigma)$ is the number of fixed points of $\\sigma$.\nThis gives $\\chi(1) = 3$, $\\chi(\\text{transposition}) = 1$ and $\\chi(\\text{3-cycle}) = 0$.\nNon-example: let $\\Z$ act by shifting on the space of finitely supported sequences.\nThis space has infinite dimension, so the trace is not defined, and the representation has no character.\nEvery character satisfies $\\chi(hgh^{-1}) = \\chi(g)$, so it is constant on each [[conjugacy class|0DEBXAJ5SA]]; such functions are called class functions.\nIsomorphic representations have the same character.\nFor a finite group over $\\C$, the converse holds as well.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "RMWMYBY8TC",
    "kind": "definition",
    "msc": [
      "20C15"
    ],
    "name": "Character of a representation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Character_theory"
      }
    ],
    "tag": "Z4PGTJFGYE",
    "uses": [
      "WYGRXCG2RN",
      "0XC1RDBY7A",
      "0DEBXAJ5SA",
      "03GSZ8RTZJ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/Z4PGTJFGYE/",
  "links": [
    {
      "field": "defines",
      "target": "RMWMYBY8TC"
    },
    {
      "field": "uses",
      "target": "WYGRXCG2RN"
    },
    {
      "field": "uses",
      "target": "0XC1RDBY7A"
    },
    {
      "field": "uses",
      "target": "0DEBXAJ5SA"
    },
    {
      "field": "uses",
      "target": "03GSZ8RTZJ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/Z4PGTJFGYE.md",
  "msc": [
    "20C15"
  ],
  "name": "Character of a representation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Z4P/Z4PGTJFGYE.md",
  "status": null,
  "tag": "Z4PGTJFGYE",
  "uri": "m2://Z4PGTJFGYE",
  "url": "https://math2.tamnd.com/wiki/character-of-a-representation-definition/",
  "writing": {
    "tag": "Z4PGTJFGYE",
    "kind": "definition",
    "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
      }
    ]
  }
}