{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $Q$ be a finite [[quiver|RE9YFFS3TS]].\nA vertex $i$ is a *sink* if no arrow has source $i$, and a *source* if no arrow has target $i$.\nLet $\\sigma_i Q$ be the quiver obtained from $Q$ by reversing every arrow incident to $i$; write $a^*$ for the reverse of $a$.\nLet $i$ be a sink of $Q$ and $V$ a [[representation|TXVWFWP6VQ]] of $Q$ over $k$.\nThe representation $S_i^+ V$ of $\\sigma_i Q$ is defined by:\n\n1. $(S_i^+ V)_j = V_j$ and $(S_i^+ V)_b = V_b$ for vertices $j \\ne i$ and arrows $b$ not incident to $i$;\n2. $(S_i^+ V)_i = \\ker \\big( \\bigoplus_{a \\colon j \\to i} V_j \\xrightarrow{(V_a)} V_i \\big)$, the sum taken over the arrows $a$ ending at $i$;\n3. for an arrow $a \\colon j \\to i$, the map $(S_i^+ V)_{a^*} \\colon (S_i^+ V)_i \\to V_j$ is the inclusion of the kernel followed by the projection onto the summand $V_j$ of index $a$.\n\nWith the evident action on morphisms, $S_i^+$ is a [[functor|28NQ0KRHD3]].\nDually, for a source $i$, $S_i^-$ replaces $V_i$ by the cokernel of $V_i \\to \\bigoplus_{a \\colon i \\to j} V_j$.\n\n## Remarks\n\nLet $S_i$ be the simple representation at $i$.\nIf $V$ is [[indecomposable|E2P29GCD9T]] and $V \\not\\cong S_i$, then $S_i^+ V$ is indecomposable, $S_i^- S_i^+ V \\cong V$, and the dimension vector changes by $\\dim (S_i^+ V)_i = \\sum_{a \\colon j \\to i} \\dim V_j - \\dim V_i$.\nAlso $S_i^+ S_i = 0$.\nExample: for $Q = (1 \\xrightarrow{a} 2)$, the vertex $2$ is a sink.\nFor $V = (k \\xrightarrow{\\id} k)$, the kernel of $\\id \\colon k \\to k$ is $0$, so $S_2^+ V$ is the representation $k \\leftarrow 0$ of $1 \\leftarrow 2$.\nIts dimension vector $(1, 0)$ is the image of $(1, 1)$ under the reflection $x_2 \\mapsto x_1 - x_2$.\nNon-example: $S_1^+$ is not defined on $Q$, because vertex $1$ is not a sink.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "MG4RNMA8AR",
    "kind": "definition",
    "msc": [
      "16G20"
    ],
    "name": "Bernstein–Gelfand–Ponomarev reflection functors",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Gabriel%27s_theorem"
      }
    ],
    "tag": "8QTP2EFA92",
    "uses": [
      "RE9YFFS3TS",
      "TXVWFWP6VQ",
      "04C5ZBG1WT",
      "28NQ0KRHD3",
      "E2P29GCD9T"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/8QTP2EFA92/",
  "links": [
    {
      "field": "defines",
      "target": "MG4RNMA8AR"
    },
    {
      "field": "uses",
      "target": "RE9YFFS3TS"
    },
    {
      "field": "uses",
      "target": "TXVWFWP6VQ"
    },
    {
      "field": "uses",
      "target": "04C5ZBG1WT"
    },
    {
      "field": "uses",
      "target": "28NQ0KRHD3"
    },
    {
      "field": "uses",
      "target": "E2P29GCD9T"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/8QTP2EFA92.md",
  "msc": [
    "16G20"
  ],
  "name": "Bernstein–Gelfand–Ponomarev reflection functors",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/8QT/8QTP2EFA92.md",
  "status": null,
  "tag": "8QTP2EFA92",
  "uri": "m2://8QTP2EFA92",
  "url": "https://tamnd.github.io/math2/wiki/bernstein-gelfand-ponomarev-reflection-functors-definition/",
  "writing": {
    "tag": "8QTP2EFA92",
    "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
      }
    ]
  }
}