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  "scope_note": "§2.6 and §4.4 discuss group homomorphisms into invertible linear transformations.",
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        "quote": "它被称为$G$的$n$维<u id=\"concept-2-representation\" class=\"concept-term\" data-concept-identity=\"urn:uuid:1273c312-ea76-4998-85e7-e75280ef3ebb\" data-concept-route=\"/concept-identities#concept-1273c312-ea76-4998-85e7-e75280ef3ebb\">表示(representation)</u>。类似地，同态$\\phi: G \\rightarrow$ $G L(n, \\mathbb{C})$被称为$G$的复$n$维表示。",
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        "quote": "群在物理学中最常通过其对向量空间的作用出现，这种作用称为表示。更具体地说，任意群 $G$ 在向量空间 $V$ 上的一个 <u id=\"concept-4-representation\" class=\"concept-term\" data-concept-identity=\"urn:uuid:1273c312-ea76-4998-85e7-e75280ef3ebb\" data-concept-route=\"/concept-identities#concept-1273c312-ea76-4998-85e7-e75280ef3ebb\">表示(representation)</u> 是同态 $T$ ，它将 $G$ 映入 $V$ 的线性自同构群，",
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      "label": "> [!definition] Definition · <span class=\"definition-name\">Representation</span>",
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        "quote": "\n\n> [!definition] Definition · <span class=\"definition-name\">Representation</span>\n> <span id=\"source-block-1085\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">Definition (<u class=\"concept-term\" data-concept-id=\"concept-1-representation\" data-concept-identity=\"urn:uuid:1273c312-ea76-4998-85e7-e75280ef3ebb\" data-concept-route=\"/concept-identities#concept-1273c312-ea76-4998-85e7-e75280ef3ebb\">Representation</u>). </span>A representation $( \\pi , V )$ of a group $G$ is a homomorphism\n>\n> <span id=\"source-block-1086\" class=\"block-anchor\"></span>\n>\n> $$\n> \\pi : g \\in G \\to \\pi (g) \\in GL(V)\n> $$\n>\n> <span id=\"concept-1-general-linear-group\" class=\"block-anchor\"></span>\n>\n> <span id=\"source-block-1087\" class=\"block-anchor\"></span>\n>\n> where $GL(V)$ denotes the [<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/general-linear-group\" data-concept-identity=\"urn:uuid:9a7432a7-4b27-41b4-83d5-d9b6ff0ce028\" data-concept-route=\"/concept-identities#concept-9a7432a7-4b27-41b4-83d5-d9b6ff0ce028\" id=\"first-d7a574bbe40adfb64e6d-en\">general linear group</u>](../../../../../concepts/linear-algebra/general-linear-group), the group of invertible linear maps $V\\to V$ for a vector space $V$.\n",
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      "label": "Representation",
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        "quote": "\n\n> [!definition] Definition · <span class=\"definition-name\">Representation</span>\n> <span id=\"source-block-1085\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">Definition (<u class=\"concept-term\" data-concept-id=\"concept-1-representation\" data-concept-identity=\"urn:uuid:1273c312-ea76-4998-85e7-e75280ef3ebb\" data-concept-route=\"/concept-identities#concept-1273c312-ea76-4998-85e7-e75280ef3ebb\">Representation</u>). </span>A representation $( \\pi , V )$ of a group $G$ is a homomorphism\n>\n> <span id=\"source-block-1086\" class=\"block-anchor\"></span>\n>\n> $$\n> \\pi : g \\in G \\to \\pi (g) \\in GL(V)\n> $$\n>\n> <span id=\"concept-1-general-linear-group\" class=\"block-anchor\"></span>\n>\n> <span id=\"source-block-1087\" class=\"block-anchor\"></span>\n>\n> where $GL(V)$ denotes the [<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/general-linear-group\" data-concept-identity=\"urn:uuid:9a7432a7-4b27-41b4-83d5-d9b6ff0ce028\" data-concept-route=\"/concept-identities#concept-9a7432a7-4b27-41b4-83d5-d9b6ff0ce028\" id=\"first-d7a574bbe40adfb64e6d-en\">general linear group</u>](../../../../../concepts/linear-algebra/general-linear-group), the group of invertible linear maps $V\\to V$ for a vector space $V$.\n",
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      "label": "群表示(representation)",
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        "quote": "我们可以从这个例子抽象出一般概念：群通过线性变换作用于向量空间，由此得到[<u class=\"concept-term\" data-concept-reference=\"concept-1-representation\" data-concept-identity=\"urn:uuid:1273c312-ea76-4998-85e7-e75280ef3ebb\" data-concept-route=\"/concept-identities#concept-1273c312-ea76-4998-85e7-e75280ef3ebb\" id=\"first-c96dd35415fecc4ceac3-zh\">群表示(representation)</u>](#concept-1-representation)：",
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      "role_basis": {
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      "label": "群表示(representation)",
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        "quote": "\n> [!definition] 定义 · <span class=\"definition-name\"><u class=\"concept-term\" data-concept-id=\"concept-1-representation\" data-concept-identity=\"urn:uuid:1273c312-ea76-4998-85e7-e75280ef3ebb\" data-concept-route=\"/concept-identities#concept-1273c312-ea76-4998-85e7-e75280ef3ebb\">群表示(representation)</u></span>\n> <span id=\"source-block-1085\" class=\"block-anchor\"></span>\n>\n> <span class=\"author-definition-label\" aria-hidden=\"true\">定义（群表示(representation)）。</span>群 $G$ 的表示 $(\\pi,V)$ 是一个[<u class=\"concept-term\" data-concept-reference=\"concepts/group-theory/group-homomorphism\" data-concept-identity=\"urn:uuid:3e749de6-9d6f-43ea-a6de-ddff7c549a00\" data-concept-route=\"/concept-identities#concept-3e749de6-9d6f-43ea-a6de-ddff7c549a00\" id=\"first-da110d60b99705240778-zh\">群同态(group homomorphism)</u>](../../../../../concepts/group-theory/group-homomorphism)：\n>\n> <span id=\"source-block-1086\" class=\"block-anchor\"></span>\n>\n> $$\n> \\pi : g \\in G \\to \\pi (g) \\in GL(V)\n> $$\n>\n> <span id=\"concept-1-general-linear-group\" class=\"block-anchor\"></span>\n>\n> <span id=\"source-block-1087\" class=\"block-anchor\"></span>\n>\n> 其中，$GL(V)$ 是由向量空间 $V$ 到自身的可逆线性映射构成的[<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/general-linear-group\" data-concept-identity=\"urn:uuid:9a7432a7-4b27-41b4-83d5-d9b6ff0ce028\" data-concept-route=\"/concept-identities#concept-9a7432a7-4b27-41b4-83d5-d9b6ff0ce028\" id=\"first-d7a574bbe40adfb64e6d-zh\">一般线性群(general linear group)</u>](../../../../../concepts/linear-algebra/general-linear-group)。\n",
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      "chapter": "2",
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      "label": "群表示(group representation)",
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        "quote": "> [!recall]- 回顾：群表示 → [第1.3节](../chapter-01/03-unitary-group-representations#concept-1-representation) <span id=\"source-block-2011\" class=\"block-anchor\"></span>\n> 回顾[<u class=\"concept-term\" data-concept-reference=\"concepts/representation-theory/group-representation\" data-concept-identity=\"urn:uuid:1273c312-ea76-4998-85e7-e75280ef3ebb\" data-concept-route=\"/concept-identities#concept-1273c312-ea76-4998-85e7-e75280ef3ebb\" id=\"first-a634d2ae45b7c75364c7-zh\">群表示(group representation)</u>](../../../../../concepts/representation-theory/group-representation)的定义：\n>\n> <span id=\"source-block-2012\" class=\"block-anchor\"></span>\n>\n> 定义（表示）。群 $G$ 在复向量空间 $V$ 上的一个表示 $( \\pi , V )$（选定一组基，并据此将 $V$ 与 $\\mathbf{C}^n$ 识别后）是一个同态\n>\n> <span id=\"source-block-2013\" class=\"block-anchor\"></span>\n>\n> $$\n> \\pi : G \\to GL(n, \\mathbf {C})\n> $$\n>\n> <span id=\"source-block-2014\" class=\"block-anchor\"></span>\n>\n> 具体而言，它就是一族 $n\\times n$ 矩阵：每个群元素对应一个矩阵，并且这些矩阵满足与群乘法相容的乘法关系。$n$ 称为该表示的维数。",
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      "anchor": "source-block-5122",
      "chapter": "5",
      "section": "5.4",
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      "role": "occurrence",
      "role_basis": {
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      },
      "label": "群表示(group representation)",
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        "quote": "我们已经把[<u class=\"concept-term\" data-concept-reference=\"concepts/representation-theory/group-representation\" data-concept-identity=\"urn:uuid:1273c312-ea76-4998-85e7-e75280ef3ebb\" data-concept-route=\"/concept-identities#concept-1273c312-ea76-4998-85e7-e75280ef3ebb\">群表示(group representation)</u>](../../../../../concepts/representation-theory/group-representation)定义为一种[<u class=\"concept-term\" data-concept-reference=\"concepts/group-theory/group-homomorphism\" data-concept-identity=\"urn:uuid:3e749de6-9d6f-43ea-a6de-ddff7c549a00\" data-concept-route=\"/concept-identities#concept-3e749de6-9d6f-43ea-a6de-ddff7c549a00\">群同态(group homomorphism)</u>](../../../../../concepts/group-theory/group-homomorphism)，即保持群乘法的群映射。",
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