{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\R^n$ have the standard inner product $\\langle \\cdot, \\cdot \\rangle$.\nFor $\\alpha \\in \\R^n \\setminus \\set{0}$, let $\\sigma_\\alpha(x) = x - 2 \\frac{\\langle \\alpha, x \\rangle}{\\langle \\alpha, \\alpha \\rangle} \\alpha$ be the [[reflection|06BRDF3F0E]] in the hyperplane orthogonal to $\\alpha$.\nLet $R \\subset \\R^n \\setminus \\set{0}$ be a root system: a finite set such that $\\sigma_\\alpha(R) = R$ and $R \\cap \\R \\alpha = \\set{\\alpha, -\\alpha}$ for every $\\alpha \\in R$.\nLet $W$ be the finite [[reflection group|G6FND7X69J]] generated by the $\\sigma_\\alpha$, and let $R_+ \\subset R$ contain exactly one of $\\alpha, -\\alpha$ for each $\\alpha \\in R$.\nLet $k \\colon R \\to \\C$ be a multiplicity function, that is, $k(w \\alpha) = k(\\alpha)$ for all $w \\in W$ and $\\alpha \\in R$.\nFor $\\xi \\in \\R^n$, the *Dunkl operator* $T_\\xi$ acts on smooth functions $f$ on $\\R^n$ by\n$$T_\\xi f(x) = \\partial_\\xi f(x) + \\sum_{\\alpha \\in R_+} k(\\alpha)\\, \\langle \\alpha, \\xi \\rangle\\, \\frac{f(x) - f(\\sigma_\\alpha x)}{\\langle \\alpha, x \\rangle},$$\nwhere $\\partial_\\xi$ is the [[directional derivative|06KWGHDN2J]] along $\\xi$.\n\n## Remarks\n\nEach term of the sum is the same for $\\alpha$ and $-\\alpha$, so $T_\\xi$ does not depend on the choice of $R_+$.\nThe difference quotient extends smoothly across the hyperplane $\\langle \\alpha, x \\rangle = 0$, so $T_\\xi$ maps homogeneous polynomials of degree $d$ to homogeneous polynomials of degree $d - 1$.\nDunkl's theorem states that $T_\\xi T_\\eta = T_\\eta T_\\xi$ for all $\\xi, \\eta \\in \\R^n$.\nFor $k = 0$, $T_\\xi = \\partial_\\xi$.\n\nExample: $n = 1$, $R = \\set{1, -1}$, $W = \\set{x \\mapsto x, x \\mapsto -x}$ and $k(\\pm 1) = k$.\nThen $T f(x) = f'(x) + k \\frac{f(x) - f(-x)}{x}$.\nFor $f(x) = x^m$, the quotient is $(1 - (-1)^m) x^{m-1}$, so $T x^m = m x^{m-1}$ for even $m$ and $T x^m = (m + 2k) x^{m-1}$ for odd $m$.\n\nNon-example: the operator $f \\mapsto f'(x) + k \\frac{f(x)}{x}$ on $\\R$ is not a Dunkl operator for $k \\ne 0$.\nIt lacks the reflected term $f(-x)$, and it does not even map the constant $1$ to a polynomial.\n\nThe Dunkl Laplacian $\\sum_{i=1}^n T_{e_i}^2$ reduces to the Laplacian when $k = 0$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "GTVCECGH40",
    "kind": "definition",
    "msc": [
      "33C52",
      "20F55",
      "33C80"
    ],
    "name": "Dunkl operator",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dunkl_operator"
      }
    ],
    "tag": "ABJ88GR4Z0",
    "uses": [
      "06BRDF3F0E",
      "G6FND7X69J",
      "06KWGHDN2J"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/ABJ88GR4Z0/",
  "links": [
    {
      "field": "defines",
      "target": "GTVCECGH40"
    },
    {
      "field": "uses",
      "target": "06BRDF3F0E"
    },
    {
      "field": "uses",
      "target": "G6FND7X69J"
    },
    {
      "field": "uses",
      "target": "06KWGHDN2J"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/ABJ88GR4Z0.md",
  "msc": [
    "33C52",
    "20F55",
    "33C80"
  ],
  "name": "Dunkl operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ABJ/ABJ88GR4Z0.md",
  "status": null,
  "tag": "ABJ88GR4Z0",
  "uri": "m2://ABJ88GR4Z0",
  "url": "https://tamnd.github.io/math2/wiki/dunkl-operator-definition/",
  "writing": {
    "tag": "ABJ88GR4Z0",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}