{"id":949,"date":"2020-02-17T17:55:01","date_gmt":"2020-02-17T09:55:01","guid":{"rendered":"http:\/\/singvision.net\/?p=949"},"modified":"2020-02-18T10:26:23","modified_gmt":"2020-02-18T02:26:23","slug":"notes-on-representation-theory-of-finite-groups","status":"publish","type":"post","link":"https:\/\/mindspectrum.xyz\/en\/2020\/02\/17\/notes-on-representation-theory-of-finite-groups\/","title":{"rendered":"Notes on Representation Theory of Finite Groups"},"content":{"rendered":"

\"\"<\/a><\/p>\n

2020\u8fd9\u4e2a\u4e00\u8a00\u96be\u5c3d\u7684\u5f00\u5c40\uff0c\u4e00\u4e2a\u6708\u57fa\u672c\u90fd\u5728\u5237\u624b\u673a\u548c\u5931\u7720\u91cc\u5ea6\u8fc7\u3002\u540e\u6765weibo\u70b8\u53f7\uff0c\u5fc3\u91cc\u53cd\u800c\u8f7b\u677e\u4e86\u4e0d\u5c11\uff0c\u8bf4\u51fa\u6765\u6ca1\u7528\u7684\u8bdd\uff0c\u4e0d\u8bf4\u4e5f\u7f62\u3002
\n\u6211\u4eec\u6765\u4ef0\u671b\u661f\u7a7a\u5427\u3002
\n\u7fa4\u8868\u793a\u8bba\u662f\u7269\u7406\u7cfb\u5b66\u751f\u80fd\u63a5\u89e6\u5230\u7684\u6570\u5b66\u91cc\u6700\u7f8e\u7684\u5206\u652f\u4e4b\u4e00\u3002\u8bfb\u7814\u65f6\u4e00\u4e2a\u665a\u4e0a\u7a81\u7136\u53d1\u73b0\u7269\u7406\u5b66\u91cc\u90a3\u4e9b\u5e38\u7528\u7684\u6b63\u4ea4\u51fd\u6570\u65cf\u3001Fourier\u53d8\u6362\u4e0e\u7fa4\u8868\u793a\u8bba\u95f4\u6709\u975e\u5e38\u6f02\u4eae\u7684\u8054\u7cfb\uff0c\u5927\u6982\u662f\u8bfb\u4e66\u65f6\u5370\u8c61\u6700\u6df1\u523b\u7684\u4e00\u6b21\u987f\u609f\u3002\u8fd9\u91cc\u8d34\u51fa\u4e00\u7bc7\u8fc7\u53bb\u7684\u6709\u9650\u7fa4\u8868\u793a\u8bba\u7b14\u8bb0\uff0c\u8c28\u4ee5\u6700\u7f8e\u597d\u7684\u6570\u5b66\u6765\u5bf9\u6297\u8fd9\u6bb5\u7cdf\u7cd5\u7684\u65e5\u5b50\u3002<\/p>\n

\\section{\u6709\u9650\u7fa4\u8868\u793a\u8bba}
\n\u672c\u7ae0\u8bd5\u56fe\u5bf9\u6709\u9650\u7fa4\u8868\u793a\u8bba\u7ed9\u51fa\u4e00\u4e2a\u7b80\u7ea6\u6e05\u6670\u7684\u4ecb\u7ecd\u3002<\/p>\n

\u6709\u9650\u7fa4\u8868\u793a\u8bba\u7684\u4e00\u4e2a\u57fa\u672c\u95ee\u9898\u662f\u6709\u9650\u7fa4\u8868\u793a\u7684\u5206\u7c7b\u4e0e\u6c42\u89e3\u3002\u8fd9\u5c06\u4f1a\u5bfc\u81f4\u5982\u4e0b\u7ecf\u5178\u7ed3\u679c\uff1a<\/p>\n

\\begin{enumerate}
\n\\item \u6709\u9650\u7fa4\u8868\u793a\u662f\u5b8c\u5168\u53ef\u7ea6\u7684\uff0c\u4e14\u53ef\u9149\u5316;
\n\\item \u6b63\u5219\u8868\u793a\u4e2d\u5305\u542b\u4e86\u5168\u90e8\u4e0d\u53ef\u7ea6\u8868\u793a\uff0c\u4e14\u5176\u4e2d\u5168\u90e8\u4e0d\u53ef\u7ea6\u8868\u793a\u7684\u6570\u76ee\u7b49\u4e8e\u5176\u7b49\u4ef7\u7c7b\u6570\u76ee,\u91cd\u6570\u7b49\u4e8e\u5176\u7ef4\u6570;
\n\\item Schur\u5f15\u7406\u3002\u4e0d\u53ef\u7ea6\u6a21\u95f4\u6ca1\u6709\u975e\u5e73\u51e1\u975e\u540c\u6784\u7684\u6a21\u540c\u6001;
\n\\item \u6b63\u4ea4\u5173\u7cfb\u7684\u5bfc\u51fa;
\n\\item \u7531\u7279\u5f81\u6807\u6b63\u4ea4\u5173\u7cfb\u5f97\u51fa\u7684\u4e0d\u53ef\u7ea6\u8868\u793a\u5224\u5b9a\u4e0e\u53ef\u7ea6\u8868\u793a\u7ea6\u5316\u3002
\n\\end{enumerate}<\/p>\n

\\subsection{\u4e3b\u8981\u6982\u5ff5}<\/p>\n

\\begin{definition}[\u7fa4\u8868\u793a] \u7fa4\"G\"\u7684\\textbf{\u8868\u793a}\u4e3a\u4e00\u4e2a\u540c\u6001\"\rho:G\to GL(V)\",\u5176\u4e2d\"V\"\u4e3a\u57df\"F\"\u4e0a\u7684\u6709\u9650\u7ef4\u5411\u91cf\u7a7a\u95f4\uff0c\u8fd9\u91cc\u82e5\u65e0\u7279\u6b8a\u8bf4\u660e\u6211\u4eec\u53d6\"F=C\".\"\dim_{F} V\"\u79f0\u4f5c\\textbf{\u8868\u793a\u7684\u7ef4\u6570};
\n\u82e5\"\ker \rho\"\u5e73\u51e1\uff0c\u5219\u79f0\u8be5\u8868\u793a\u662f\\textbf{\u5fe0\u5b9e\u7684};\u82e5\u5bf9\u6240\u6709\"g\in G,\rho(g)=\mathcal{E}\",\u5219\u79f0\u8868\u793a\u4e3a\\textbf{\u5e73\u51e1\u7684}\u3002
\n\u7fa4\u8868\u793a\u53ef\u770b\u4f5c\"V\"\u4e0a\u7684\"G\"-\u6a21\u3002
\n\\end{definition}<\/p>\n

\\begin{definition}[\u53ef\u7ea6\u4e0e\u4e0d\u53ef\u7ea6] \u4e00\u4e2a\u8868\u793a\u662f\\textbf{\u4e0d\u53ef\u7ea6}\u7684\u5982\u679c\u5b83\u4e0d\u5177\u6709\u6070\u5f53\u7684\u4e0d\u53d8\u5b50\u7a7a\u95f4\uff08\u5373\u4e0d\u53ef\u7ea6\u5b50\u8868\u793a\uff09;\u4e00\u4e2a\u8868\u793a\u662f\\textbf{\u5b8c\u5168\u53ef\u7ea6\u7684}\u5982\u679c\u5b83\u53ef\u4ee5\u5206\u89e3\u4e3a\u4e00\u7cfb\u5217\u4e0d\u53ef\u7ea6\u5b50\u8868\u793a\u7684\u76f4\u548c\u3002
\n\\end{definition}<\/p>\n

\\begin{definition}[\u6b63\u5219\u8868\u793a] \u8003\u8651\"G\"\u4e0a\u590d\u51fd\u6570\u6784\u6210\u7684\u5411\u91cf\u7a7a\u95f4\"\mathbb{C}(G)\",\u5b83\u7684\u57fa\u5e95\u4e3a\"\{\mathrm{e}_{g}\}_{g\in G}\",\u5176\u4e2d\"\mathrm{e}_g\"\u8868\u793a\u4e00\u4e2a\u5728\"g\"\u4e0a\u503c\u4e3a\"1\"\uff0c\u4f59\u7686\u4e3a\"0\"\u7684\u51fd\u6570\u3002
\n\u5219\"G\"\u53ef\u4ee5\u901a\u8fc7\"\mathbb{C}(G)\"\u4e0a\u7684\u7ebf\u6027\u7b97\u5b50\"\lambda(g)\"\u8fd9\u6837\u4f5c\u7528\u5728\u57fa\u77e2\u4e0a<\/p>\n

(1) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5219\u79f0\"\lambda(g)\"\u4e3a\"G\"\u7684\\textbf{\u5de6\u6b63\u5219\u8868\u793a}\u3002
\n\\end{definition}<\/p>\n

\\begin{definition}[G-\u7ebf\u6027\u6620\u5c04] \u8bbe\"G\"\u5728\"U,V\"\u4e0a\u5206\u522b\u6709\u8868\u793a\"\rho_1:G\to GL(U),\rho_2:G\to GL(V)\",\u4e14\u5b58\u5728\u7ebf\u6027\u6620\u5c04\"\sigma:U\to V\"\uff0c\u6ee1\u8db3<\/p>\n

(2) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5219\u79f0\"\sigma\"\u4e3a\u4e00\u4e2a\\textbf{\"G\"-\u7ebf\u6027\u6620\u5c04}\uff0c\u6216\u79f0\\textbf{\"\sigma\"\u4e0e\"G\"\u53ef\u4ea4\u6362}.\u82e5\"\sigma\"\u53ef\u9006\uff0c\u5219\u4e0a\u5f0f\u53ef\u5199\u4e3a<\/p>\n

(3) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u6b64\u65f6\u79f0\"\sigma\"\u4e3a\\textbf{\u7fa4\u8868\u793a\u7684\u540c\u6784}\u3002
\n\u7528\u6a21\u7684\u8bed\u8a00\u6765\u8bf4\uff0cG-\u7ebf\u6027\u6620\u5c04\u6b63\u662f\u4e00\u4e2a\u6a21\u540c\u6001\u3002
\n\\end{definition}<\/p>\n

\\subsection{\u9149\u8868\u793a\u4e0e\u53ef\u7ea6\u6027}
\n\\begin{definition}[\u9149\u8868\u793a] \u5411\u91cf\u7a7a\u95f4\"V\"\u4e0a\u7684\u8868\u793a\"\rho\"\u88ab\u79f0\u4f5c\\textbf{\u9149}\u7684\uff0c\u5982\u679c\"V\"\u88c5\u5907\u4e86\u4e00\u4e2a\u5728\"G\"\u4f5c\u7528\u4e0b\u4e0d\u53d8\u7684\u5384\u7c73\u5185\u79ef\"\Braket{\vert}\"<\/p>\n

(4) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u8868\u793a\u79f0\u4f5c\\textbf{\u53ef\u9149\u5316\u7684}\u5982\u679c\"V\"\u53ef\u4ee5\u88c5\u5907\u8fd9\u6837\u4e00\u4e2a\u5185\u79ef\u3002<\/p>\n

\u4e00\u4e2a\u8868\u793a\u662f\u9149\u7684\u5f53\u4e14\u4ec5\u5f53\u540c\u6001\"\rho:G\to GL(V)\"\u843d\u5728\u9149\u7fa4\"U(V)\"\u4e4b\u4e2d\u3002\u8be5\u6761\u4ef6\u4e5f\u53ef\u4ee5\u8868\u8ff0\u4e3a\"\rho^{\dagger}(g)=\rho(g^{-1})\".
\n\\end{definition}<\/p>\n

\\begin{lemma}[] \u8bbe\"V\"\u4e3a\u7fa4\"G\"\u7684\u9149\u8868\u793a\u800c\"W\"\u4e3a\"V\"\u4e2d\u7684\u4e00\u4e2a\u4e0d\u53d8\u5b50\u7a7a\u95f4\uff0c\u5219\"W\"\u7684\u6b63\u4ea4\u8865\"W^{\bot}\"\u4e5f\u662f\u4e00\u4e2a\u4e0d\u53d8\u5b50\u7a7a\u95f4\u3002
\n\\end{lemma}
\n\\begin{proof}
\n\u7531\u4e8e\"G\"\u4fdd\u6301\u9149\u8868\u793a\u7a7a\u95f4\u4e2d\u7684\u5185\u79ef\u4e0d\u53d8\uff0c\u4e8e\u662f\u5bf9\u4e8e\"w^{\prime}\in W^{\bot}\",\u5b83\u88ab\u4efb\u610f\"g\in G\"\u4f5c\u7528\u540e\u7684\u7ed3\u679c\u6ee1\u8db3<\/p>\n

(5) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u7531\u4e8e\"\rho(g)(w)\in W\",\u4e8e\u662f\"\rho(g)(w^{\prime})\in W^{\bot}\".
\n\\end{proof}<\/p>\n

\\begin{theorem}[\u9149\u8868\u793a\u7684\u5b8c\u5168\u53ef\u7ea6\u6027] \u8bbe\"\rho:G\to GL(V)\"\u4e3a\u4efb\u610f\u7fa4\"G\"\u4e0a\u7684\u6709\u9650\u7ef4\u9149\u8868\u793a\uff0c\u5219\u8be5\u8868\u793a\u5b8c\u5168\u53ef\u7ea6\uff0c\u5e76\u4e14\u53ef\u7ea6\u5316\u4e3a\u4e00\u7cfb\u5217\u4e0d\u53ef\u7ea6\u9149\u8868\u793a\u7684\u76f4\u548c\u3002
\n\\end{theorem}
\n\\begin{proof}
\n\u9996\u5148\uff0c\u6839\u636e\u4e0a\u8ff0\u5f15\u7406\uff0c\"V\"\u53ef\u4ee5\u4e0d\u65ad\u8fed\u4ee3\u5730\u6b63\u4ea4\u5206\u89e3\u4e3a\u4e00\u7cfb\u5217\u4e0d\u53d8\u5b50\u7a7a\u95f4\u7684\u76f4\u548c\uff0c\u8fd9\u4e2a\u5206\u89e3\u5fc5\u7136\u7ec8\u6b62\u4e8e\u4e00\u7cfb\u5217\u4e0d\u53ef\u7ea6\u5b50\u7a7a\u95f4\u3002<\/p>\n

\u7136\u540e\uff0c\"V\"\u4e0a\u7684\u5384\u7c73\u5185\u79ef\u9650\u5236\u5728\u5404\u4e0d\u53ef\u7ea6\u5b50\u7a7a\u95f4\u4e0a\uff0c\u663e\u7136\"G\"\u4ecd\u7136\u4fdd\u6301\u8fd9\u4e9b\u5b50\u7a7a\u95f4\u4e2d\u7684\u5185\u79ef\u4e0d\u53d8\uff0c\u56e0\u800c\"V\"\u88ab\u5206\u89e3\u6210\u4e86\u4e00\u7cfb\u5217\u4e0d\u53ef\u7ea6\u9149\u8868\u793a\u7684\u76f4\u548c\u3002
\n\\end{proof}<\/p>\n

\\subsection{\u6709\u9650\u7fa4\u8868\u793a\u7684\u5b8c\u5168\u53ef\u7ea6\u6027}
\n\\begin{theorem}[\u6709\u9650\u7fa4\u8868\u793a\u7684\u9149\u6027] \u6709\u9650\u7fa4\u7684\u6709\u9650\u7ef4\u590d\u8868\u793a\u5fc5\u53ef\u9149\u5316\u3002
\n\\end{theorem}
\n\\begin{proof}
\n\u53ea\u9700\u627e\u5230\u4e00\u4e2a\"G\"-\u4e0d\u53d8\u7684\u5185\u79ef\u5373\u53ef\u3002<\/p>\n

\u4ee5\u4e0b\u5f0f\u4ece\"\Braket{\vert}\"\u5b9a\u4e49\u51fa\u4e00\u4e2a\u65b0\u7684\"G-\"\u4e0d\u53d8\u5185\u79ef\"\Braket{\vert}^{\prime}\"<\/p>\n

(6) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u663e\u7136\u5177\u6709\"G\"-\u4e0d\u53d8\u548c\u6b63\u5b9a\u6027\u8d28\uff0c\u56e0\u800c\u662f\u4e00\u4e2a\"G\"-\u4e0d\u53d8\u7684\u5185\u79ef\u3002
\n\\end{proof}<\/p>\n

\\begin{theorem}[\u6709\u9650\u7fa4\u8868\u793a\u7684\u5b8c\u5168\u53ef\u7ea6\u6027] \u6709\u9650\u7fa4\u7684\u6709\u9650\u7ef4\u590d\u8868\u793a\u5fc5\u5b8c\u5168\u53ef\u7ea6\u3002
\n\\end{theorem}
\n\\begin{proof}
\n\u6709\u9650\u7fa4\u7684\u6709\u9650\u7ef4\u590d\u8868\u793a\u5fc5\u53ef\u9149\u5316\uff0c\u800c\u9149\u8868\u793a\u5fc5\u5b8c\u5168\u53ef\u7ea6\u3002
\n\\end{proof}<\/p>\n

\u4e0b\u9762\u7ed9\u51fa\u4e0a\u8ff0\u5b9a\u7406\u7684\u7b2c\u4e8c\u79cd\u8bc1\u660e\uff0c\u4ee5\u671f\u4ece\u4e0d\u540c\u7684\u89d2\u5ea6\u6765\u7406\u89e3\u8fd9\u4e00\u7ed3\u679c\u3002
\n\\begin{lemma}[] \u8bbe\"V\"\u4e3a\u7fa4\"G\"\u5728\u7279\u5f81\u4e0d\u6574\u9664\"|G|\"\u7684\u57df\"F\"\u4e0a\u7684\u6709\u9650\u7ef4\u8868\u793a\uff0c\u5219\u6bcf\u4e2a\u4e0d\u53d8\u5b50\u7a7a\u95f4\"W\subset V\"\u90fd\u6709\u4e00\u4e2a\u4e0d\u53d8\u7684\u8865\u7a7a\u95f4\"W^{\prime}\"\uff0c\u4f7f\u5f97\"W^{\prime}\oplus W = V\".
\n\\end{lemma}
\n\\begin{proof}
\n\u8bbe\"\pi:V\to W\"\u4e3a\"V\"\u5230\"W\"\u7684\u6295\u5f71\u7b97\u5b50\uff0c\u4ecd\u7136\u901a\u8fc7\u5728\"G\"\u4e0a\u5e73\u5747\u7684\u624b\u6cd5\u6784\u9020\u4e00\u4e2a\u65b0\u7684\"G\"-\u4e0d\u53d8\u6295\u5f71\u7b97\u5b50\"\pi^{\prime}\"<\/p>\n

(7) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5bb9\u6613\u9a8c\u8bc1\"\pi^{\prime}\"\u6709\u5982\u4e0b\u6027\u8d28
\n\\begin{enumerate}
\n\\item \"\pi^{\prime} \circ h = h \circ \pi^{\prime},\forall h\in G\"
\n\\item \"\pi^{\prime}=\id \quad on \, W\"
\n\\item \"\img \pi^{\prime}=W\"
\n\\end{enumerate}
\n\u53ef\u89c1\"\pi^{\prime}\"\u662f\u4e00\u4e2a\"V\to W\"\u7684\"G\"-\u7ebf\u6027\u6620\u5c04\uff0c\u6216\u8005\u8bf4\u4e00\u4e2a\u6a21\u540c\u6001\u3002<\/p>\n

\u7531\u7b2c\u4e00\u6761\u6027\u8d28\u77e5\u9053\"W^{\prime\prime}\equiv \ker \pi^{\prime}\"\u662f\u4e00\u4e2a\u4e0d\u53d8\u5b50\u7a7a\u95f4\uff1b\u7531\u7b2c\u4e8c\u6761\u6027\u8d28\u77e5\u9053\"W^{\prime\prime} \bigcap W = 0\";\u7531\u7b2c\u4e09\u6761\u6027\u8d28\u5f97\"\dim W + \dim W^{\prime\prime} = \dim V\",
\n\u5373\"W^{\prime\prime}\oplus W = V\".
\n\\end{proof}<\/p>\n

\\subsection{Schur\u5f15\u7406}
\n\\begin{theorem}[Schur\u5f15\u7406] \u8bbe\"\rho_V,\rho_W\"\u4e3a\u7fa4\"G\"\u7684\u6709\u9650\u7ef4\u4e0d\u53ef\u7ea6\u8868\u793a\uff0c\u8bbe\"\sigma:V\to W\"\u662f\u4e00\u4e2a\"G\"-\u7ebf\u6027\u6620\u5c04\uff0c\u5219\u6709
\n\\begin{enumerate}
\n\\item \u82e5\u4e24\u8868\u793a\u4e0d\u7b49\u4ef7\uff0c\u5219\"\sigma=0\";
\n\\item \u5f53\"V=W,\rho_V=\rho_W\"\u65f6\uff0c\"\sigma=\lambda \id\"\u3002
\n\\end{enumerate}
\n\\end{theorem}\\label{SchursLemma}
\n\\begin{proof}
\n\"\sigma\"\u662f\u4e00\u4e2a\"G\"-\u7ebf\u6027\u6620\u5c04\uff0c\u56e0\u6b64\"\ker \sigma\"\u4e0e\"\img \sigma\"\u5747\u4e3a\u4e0d\u53d8\u5b50\u7a7a\u95f4\uff0c\u4f46\u8003\u8651\u5230\"V,W\"\u90fd\u662f\u4e0d\u53ef\u7ea6\u8868\u793a\uff0c\u4e0d\u5305\u542b\u4efb\u4f55\u975e\u5e73\u51e1\u5b50\u7a7a\u95f4\uff0c\u56e0\u6b64\u53ea\u6709\u5982\u4e0b\u4e24\u79cd\u53ef\u80fd\u6027
\n\\begin{enumerate}
\n\\item \"\ker \sigma=0,\img \sigma=W\",\"\sigma\"\u4e3a\u8868\u793a\u540c\u6784\uff1b
\n\\item \"\ker \sigma=V,\img \sigma=0\",\"\sigma\"\u4e3a\u96f6\u6620\u5c04\u3002
\n\\end{enumerate}
\n\u56e0\u6b64\u7b2c\u4e00\u6761\u5f97\u8bc1\u3002<\/p>\n

\u5f53\"V=W\"\u65f6\uff0c\u7531\u4e8e\"\sigma\"\u4e0e\"G\"\u5bf9\u6613\uff0c\u56e0\u6b64\"\sigma\"\u7684\u672c\u5f81\u5b50\u7a7a\u95f4\"E_{\sigma}\"\u662f\u4e0d\u53d8\u5b50\u7a7a\u95f4\uff0c\u53c8\u7531\"V\"\u7684\u4e0d\u53ef\u7ea6\u6027\u6709\"E_{\sigma}=V\",\u56e0\u6b64\"\sigma=\lambda \id\".
\n\\end{proof}<\/p>\n

\\subsection{\u7279\u5f81\u6807\u4e0e\u7b2c\u4e00\u6b63\u4ea4\u5173\u7cfb}
\n\\begin{theorem}[] \u8bbe\"\rho_V,\rho_W\"\u4e3a\u7fa4\"G\"\u7684\u6709\u9650\u7ef4\u4e0d\u53ef\u7ea6\u8868\u793a\uff0c\u8bbe\"\sigma:V\to W\"\u662f\u4e00\u4e2a\u7ebf\u6027\u6620\u5c04.\u6211\u4eec\u7531\"\sigma\"\u6784\u9020\u6620\u5c04\"\tilde{\sigma}\"<\/p>\n

(8) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5219\"\tilde{\sigma}\"\u6ee1\u8db3\u5982\u4e0b\u6027\u8d28
\n\\begin{enumerate}
\n\\item \u82e5\u4e24\u8868\u793a\u4e0d\u7b49\u4ef7\uff0c\u5219\"\tilde{\sigma}=0\";
\n\\item \u5f53\"V=W,\rho_V=\rho_W\"\u65f6\uff0c\"\sigma=\lambda \id,\lambda = \frac{\tr \sigma}{\dim V}\"\u3002
\n\\end{enumerate}
\n\\end{theorem}\\label{SchurOrth}
\n\\begin{proof}
\n\u6613\u8bc1<\/p>\n

(9) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5373\"\rho_W(g) \circ \tilde{\sigma} = \tilde{\sigma}\circ \rho_V\",\u53ef\u89c1\"\tilde{\sigma}\"\u662f\u4e2a\"G\"-\u7ebf\u6027\u6620\u5c04\uff0c\u4ece\u800c\u7531??\u5f97\u51fa\u4e24\u4e2a\u65ad\u8a00\u3002
\n\u5173\u4e8e\"\lambda\"\u7684\u53d6\u503c\uff0c\u53ef\u4ee5\u8fd9\u6837\u8ba1\u7b97\u51fa\u6765<\/p>\n

(10) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\\end{proof}<\/p>\n

\u8003\u8651\u7fa4\"G\"\u4e0a\u7684\u590d\u51fd\u6570\u7a7a\u95f4\uff0c\u6211\u4eec\u5728\u5176\u4e2d\u5b9a\u4e49\u4efb\u610f\u4e24\u4e2a\u590d\u51fd\u6570\"\phi,\psi\"\u7684\u5185\u79ef<\/p>\n

(11) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\\begin{definition}[\u7279\u5f81\u6807] \u4e00\u4e2a\u8868\u793a\"\rho\"\u7684\\textbf{\u7279\u5f81\u6807}\"\chi_{\rho}:G\to \mathbb{C}\"\u5b9a\u4e49\u4e3a<\/p>\n

(12) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\\end{definition}<\/p>\n

\u53ef\u4ee5\u8bc1\u660e\u7279\u5f81\u6807\u662f\u4e00\u4e2a\u5b8c\u5907\u4e0d\u53d8\u91cf\uff0c\u53ef\u4ee5\u5728\u540c\u6784\u7684\u610f\u4e49\u4e0a\u5bf9\u4e0d\u53ef\u7ea6\u8868\u793a\u8fdb\u884c\u5206\u7c7b\u3002
\n\\begin{theorem}[\u7279\u5f81\u6807\u7684\u57fa\u672c\u6027\u8d28] \u7279\u5f81\u6807\u5177\u6709\u4ee5\u4e0b\u6027\u8d28
\n\\begin{enumerate}
\n\\item \u7279\u5f81\u6807\u662f\u7c7b\u51fd\u6570\uff0c\"\chi(g)=\chi(hgh^{-1}),\forall g,h \in G\"
\n\\item \"\chi(1)=\dim V\"
\n\\item \"\chi(g^{-1})=\bar{\chi}(g)\"
\n\\item \u5bf9\u4e8e\u4efb\u610f\u4e24\u4e2a\u8868\u793a\"V,W\",\"\chi_{V\oplus W}=\chi_V+\chi_W,\chi_{V\otimes W}=\chi_V * \chi_W\"
\n\\item \u5bf9\u4e8e\u5bf9\u5076\u8868\u793a\"V^{*}\",\u6709\"\chi_{V^{*}}(g)=\chi_V(g^{-1})\"
\n\\end{enumerate}
\n\\end{theorem}
\n\\begin{proof}
\n\u663e\u7136\u3002
\n\\end{proof}<\/p>\n

\\begin{theorem}[\u7b2c\u4e00\u6b63\u4ea4\u5173\u7cfb] \u7279\u5f81\u6807\u6ee1\u8db3\u5982\u4e0b\u6b63\u4ea4\u5173\u7cfb
\n\\begin{enumerate}
\n\\item \u82e5\"\rho_V\"\u662f\u4e0d\u53ef\u7ea6\u7684\uff0c\u5219\"\Vert\chi_{V}\Vert=1\";
\n\\item \u82e5\"\rho_V,\rho_W\"\u4e0d\u53ef\u7ea6\u4e14\u4e0d\u540c\u6784\uff0c\u5219\"\Braket{\chi_W\vert \chi_V}=0\".
\n\\end{enumerate}
\n\\end{theorem}
\n\\begin{proof}
\n\u4f5c\u5982\u4e0b\u8ba1\u7b97<\/p>\n

(13) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u7531??,\u4ee4\"\sigma = \ket{w_i} \bra{v_j}\"\uff0c\u53ef\u77e5\"V,W\"\u4e0d\u7b49\u4ef7\u65f6\u4e0a\u8ff0\u5185\u79ef\u4e3a\u96f6\uff0c\u82e5\"V,W\"\u7b49\u4ef7\u5219\u6709<\/p>\n

(14) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\\end{proof}<\/p>\n

\u4e8e\u662f\u6211\u4eec\u6709\u5982\u4e0b\u7ed3\u8bba\u3002
\n\\begin{theorem}[] \u5bf9\u4e8e\u6709\u9650\u7fa4\u7684\u590d\u8868\u793a\uff0c\u6211\u4eec\u6709
\n\\begin{enumerate}
\n\\item \u4e0d\u53ef\u7ea6\u8868\u793a\"V\"\u5728\u8868\u793a\"W\"\u4e2d\u7684\u91cd\u6570\u7b49\u4e8e\"\Braket{\chi_V\vert \chi_W}\";
\n\\item \u4e24\u8868\u793a\u540c\u6784\u5f53\u4e14\u4ec5\u5f53\u5b83\u4eec\u5177\u6709\u76f8\u540c\u7684\u7279\u5f81\u6807;
\n\\item \"\Vert \chi \Vert^2\"\u603b\u662f\u4e00\u4e2a\u6574\u6570\uff0c\"V\"\u4e0d\u53ef\u7ea6\u5f53\u4e14\u4ec5\u5f53\"\Vert \chi_V \Vert^2=1\";
\n\\end{enumerate}
\n\\end{theorem}
\n\\begin{proof}
\n\u8003\u8651\u8868\u793a\"W\"\u7684\u4e0d\u53ef\u7ea6\u5206\u89e3\uff0c\u6211\u4eec\u6700\u7ec8\u8981\u8bc1\u660e\u5982\u4e0b\u7b49\u4ef7\u5173\u7cfb<\/p>\n

(15) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\\label{RepRingCharCorrespondence}
\n\u8bc1\u660e\u4e0a\u5f0f\u7684\u6b63\u5411\u7bad\u5934\u662f\u663e\u7136\u7684\uff0c\u7531\u4e8e\u4e0d\u53ef\u7ea6\u8868\u793a\u7279\u5f81\u6807\u7684\u6b63\u4ea4\u6027\uff0c\u4e0d\u53ef\u7ea6\u8868\u793a\"V_k\"\u5728\"W\"\u4e2d\u7684\u91cd\u6570\u53ef\u8fd9\u6837\u8ba1\u7b97<\/p>\n

(16) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u8fd9\u5c31\u8bc1\u660e\u4e86\u7b2c\u4e8c\u6761.<\/p>\n

\u8003\u8651??\u7684\u9006\u5411\u7bad\u5934\uff0c\u5bf9\u4e8e\u4efb\u610f\u4e0d\u53ef\u7ea6\u8868\u793a\"V_i\",\u90fd\u53ef\u4ee5\u901a\u8fc7\"\Braket{\chi_{V_i}\vert \chi_W}\"\u8ba1\u7b97\u51fa\u5b83\u5728\"W\"\u4e2d\u7684\u91cd\u6570\uff0c\u4ece\u800c\u53ef\u4ee5\u6839\u636e\u7279\u5f81\u6807\u91cd\u5efa\u51fa\"W\"\u7684\u4e0d\u53ef\u7ea6\u5206\u89e3\uff0c\u56e0\u800c\u9006\u5411\u7bad\u5934\u6210\u7acb\uff0c\u7b2c\u4e8c\u6761\u547d\u9898\u4e5f\u5f97\u8bc1\u3002<\/p>\n

\u5bf9\u4e8e\u4e0d\u53ef\u7ea6\u8868\u793a\"V_i\",\u5fc5\u6709\"\Vert \chi_{V_i} \Vert = 1\".\u53ef\u4ee5\u7b97\u51fa<\/p>\n

(17) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u53ef\u89c1\"\Vert \chi \Vert^2\"\u5fc5\u4e3a\u6574\u6570\uff0c\"\Vert \chi \Vert = 1\"\u65f6\u5fc5\u7136\u610f\u5473\u7740\u8be5\u8868\u793a\u4e0d\u53ef\u7ea6\u3002\u7b2c\u4e09\u6761\u8bc1\u6bd5\u3002
\n\\end{proof}<\/p>\n

\\subsection{\u6b63\u5219\u8868\u793a}<\/p>\n

\\begin{proposition}[] \u5bf9\u4e8e\u6b63\u5219\u8868\u793a\"\lambda\",\u6211\u4eec\u6709\u5982\u4e0b\u7ed3\u679c
\n\\begin{enumerate}
\n\\item<\/p>\n

(18) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\\item \u4efb\u610f\u4e0d\u53ef\u7ea6\u8868\u793a\"\rho_V\"\u5728\"\lambda\"\u4e2d\u7684\u91cd\u6570\u7b49\u4e8e\"\dim V\";
\n\\item \u5168\u90e8\u4e0d\u53ef\u7ea6\u8868\u793a\u7684\u7ef4\u6570\u5e73\u65b9\u548c\u7b49\u4e8e\u7fa4\u7684\u9636\uff0c\u5373\"\sum_V \dim^2_V=|G|\".
\n\\end{enumerate}
\n\\end{proposition}
\n\\begin{proof}
\n\u7b2c\u4e00\u6761\uff1a \u6b63\u5219\u8868\u793a\u4e0b\u9664\u4e86\u5355\u4f4d\u5143\"e\"\uff0c\u4efb\u4f55\u7fa4\u5143\u90fd\u6ca1\u6709\u975e\u96f6\u5bf9\u89d2\u5143\uff1b<\/p>\n

\u7b2c\u4e8c\u6761\uff1a \"\Braket{\chi_V \vert \chi_{reg}}=\frac{1}{|G|}\chi_V(e)\chi_{reg}(e)=\chi_V(e)=\dim V\";<\/p>\n

\u7b2c\u4e09\u6761\uff1a \u663e\u7136\u3002
\n\\end{proof}<\/p>\n

\u73b0\u5728\u6211\u4eec\u5df2\u7ecf\u8bc1\u660e\u4e86\u4e0d\u53ef\u7ea6\u8868\u793a\u7279\u5f81\u6807\u662f\u4e00\u7ec4\u6b63\u4ea4\u5f52\u4e00\u5411\u91cf\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u53ef\u4ee5\u8fdb\u4e00\u6b65\u8bc1\u660e\u5b83\u4eec\u5728\u7c7b\u51fd\u6570\u7a7a\u95f4\u91cc\u662f\u5b8c\u5907\u7684\u3002
\n\u8bc1\u660e\u7684\u57fa\u672c\u601d\u8def\u5c31\u662f\u5047\u8bbe\u7c7b\u51fd\u6570\"\phi\"\u6b63\u4ea4\u4e8e\u6240\u6709\u7684\u4e0d\u53ef\u7ea6\u8868\u793a\u7279\u5f81\u6807\uff0c\u8bf4\u660e\u5b83\u5fc5\u4e3a\u96f6\uff0c\u4ece\u800c\u4e0d\u5b58\u5728\u8fd9\u6837\u975e\u5e73\u51e1\u7684\"\phi\"\u3002<\/p>\n

\u5bf9\u4e8e\u4efb\u610f\u51fd\u6570\"\phi:G\to \mathbb{C}\"\u548c\"G\"\u7684\u4efb\u610f\u8868\u793a\"\rho_V\"\uff0c\u5b9a\u4e49\u5982\u4e0b\"V\"\u7684\u81ea\u6620\u5c04\"\rho(\phi)\"<\/p>\n

(19) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u4e0a\u8ff0\u81ea\u6620\u5c04\u6709\u5982\u4e0b\u6027\u8d28
\n\\begin{lemma}[] \"\phi\"\u662f\u4e2a\u7c7b\u51fd\u6570\u5f53\u4e14\u4ec5\u5f53\u5728\u4efb\u4f55\u8868\u793a\u4e0b\"\rho(\phi)\"\u4e0e\"G\"\u5bf9\u6613\u3002
\n\\end{lemma}
\n\\begin{proof}
\n\u9996\u5148\u5047\u8bbe\"\phi\"\u662f\u4e2a\u7c7b\u51fd\u6570\uff0c\u5373\"\phi(h^{-1}gh)=\phi(g),\forall g,h\in G\",\u5219\u6709<\/p>\n

(20) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u7136\u540e\u5047\u8bbe\u4efb\u4f55\u8868\u793a\u4e0b\"\rho(\phi)\"\u4e0e\"G\"\u5bf9\u6613\uff0c\u53d6\"\rho_V\"\u4e3a\u6b63\u5219\u8868\u793a\"\lambda\",\u6709<\/p>\n

(21) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u7531\u5bf9\u6613\u5047\u8bbe\u77e5\u4e24\u5f0f\u76f8\u7b49\uff0c\u4ece\u800c\u6709\"\phi(h^{-1}g)=\phi(gh^{-1})\",\u8bc1\u6bd5\u3002
\n\\end{proof}<\/p>\n

\u5f53\"\phi\"\u662f\u7c7b\u51fd\u6570\u4e14\"\rho_V\"\u4e0d\u53ef\u7ea6\u65f6\uff0cSchur\u5f15\u7406\u544a\u8bc9\u6211\u4eec\"\rho(\phi)\"\u662f\u4e2a\u5e38\u6570\u6620\u5c04.\u6211\u4eec\u53d6\u8ff9\u4ee5\u8ba1\u7b97\u5b83<\/p>\n

(22) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\\begin{theorem}[] \u4e0d\u53ef\u7ea6\u8868\u5f81\u6807\u662f\u7c7b\u51fd\u6570\u7a7a\u95f4\u7684\u4e00\u7ec4\u5b8c\u5907\u57fa\u3002
\n\\end{theorem}
\n\\begin{proof}
\n\u5047\u8bbe\u7c7b\u51fd\u6570\"\phi\"\u6b63\u4ea4\u4e8e\u6240\u6709\u7684\u4e0d\u53ef\u7ea6\u8868\u793a\u7279\u5f81\u6807\uff0c\u5219\u5728\u4efb\u4f55\u4e0d\u53ef\u7ea6\u8868\u793a\u6216\u5b8c\u5168\u53ef\u7ea6\u8868\u793a\u4e0a\u6709\"\rho(\phi)=0\",<\/p>\n

\u8003\u8651\u6b63\u5219\u8868\u793a\uff0c<\/p>\n

(23) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u8bc1\u6bd5\u3002
\n\\end{proof}<\/p>\n

\\subsection{\u7b2c\u4e8c\u6b63\u4ea4\u5173\u7cfb}
\n\u6211\u4eec\u8003\u8651\u7b2c\u4e00\u6b63\u4ea4\u5173\u7cfb\uff0c\u6709<\/p>\n

(24) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u53ef\u89c1\"T\"\u662f\u4e00\u4e2a\u9149\u77e9\u9635\uff0c\u7b2c\u4e00\u6b63\u4ea4\u5173\u7cfb\u6b63\u662f\u8bf4\u8be5\u77e9\u9635\u7684\u884c\u5411\u91cf\u76f8\u4e92\u6b63\u4ea4\u3002\u6211\u4eec\u77e5\u9053\u9149\u77e9\u9635\u7684\u884c\u4e0e\u5217\u5747\u6b63\u4ea4\uff0c\u4e8e\u662f\u53ef\u4ee5\u5f97\u51fa\u5217\u5411\u91cf\u7684\u6b63\u4ea4\u5173\u7cfb<\/p>\n

(25) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5176\u4e2d\"R\"\u4e3a\u6240\u6709\u7684\u4e0d\u53ef\u7ea6\u8868\u793a\u96c6\u5408\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

2020\u8fd9\u4e2a\u4e00\u8a00\u96be\u5c3d\u7684\u5f00\u5c40\uff0c\u4e00\u4e2a\u6708\u57fa\u672c\u90fd\u5728\u5237\u624b\u673a\u548c\u5931\u7720\u91cc\u5ea6\u8fc7\u3002\u540e\u6765weibo\u70b8\u53f7\uff0c\u5fc3\u91cc\u53cd\u800c\u8f7b\u677e\u4e86\u4e0d\u5c11\uff0c\u8bf4\u51fa\u6765\u6ca1\u7528\u7684\u8bdd\uff0c\u4e0d\u8bf4\u4e5f\u7f62\u3002 \u6211\u4eec\u6765\u4ef0\u671b\u661f\u7a7a\u5427\u3002 \u7fa4\u8868\u793a\u8bba\u662f\u7269\u7406\u7cfb\u5b66\u751f\u80fd\u63a5\u89e6\u5230\u7684\u6570\u5b66\u91cc\u6700\u7f8e\u7684\u5206\u652f\u4e4b\u4e00\u3002\u8bfb\u7814\u65f6\u4e00\u4e2a\u665a\u4e0a\u7a81\u7136\u53d1\u73b0\u7269\u7406\u5b66\u91cc\u90a3\u4e9b\u5e38\u7528\u7684\u6b63\u4ea4\u51fd\u6570\u65cf\u3001Fourier\u53d8\u6362\u4e0e\u7fa4\u8868\u793a\u8bba\u95f4\u6709\u975e\u5e38\u6f02\u4eae\u7684\u8054\u7cfb\uff0c\u5927\u6982\u662f\u8bfb\u4e66\u65f6\u5370\u8c61\u6700\u6df1\u523b\u7684\u4e00\u6b21\u987f\u609f\u3002\u8fd9\u91cc\u8d34\u51fa\u4e00\u7bc7\u8fc7\u53bb\u7684\u6709\u9650\u7fa4\u8868\u793a\u8bba\u7b14\u8bb0\uff0c\u8c28\u4ee5\u6700\u7f8e\u597d\u7684\u6570\u5b66\u6765\u5bf9\u6297\u8fd9\u6bb5\u7cdf\u7cd5\u7684\u65e5\u5b50\u3002 \\section{\u6709\u9650\u7fa4\u8868\u793a\u8bba} \u672c\u7ae0\u8bd5\u56fe\u5bf9\u6709\u9650\u7fa4\u8868\u793a\u8bba\u7ed9\u51fa\u4e00\u4e2a\u7b80\u7ea6\u6e05\u6670\u7684\u4ecb\u7ecd\u3002 \u6709\u9650\u7fa4\u8868\u793a\u8bba\u7684\u4e00\u4e2a\u57fa\u672c\u95ee\u9898\u662f\u6709\u9650\u7fa4\u8868\u793a\u7684\u5206\u7c7b\u4e0e\u6c42\u89e3\u3002\u8fd9\u5c06\u4f1a\u5bfc\u81f4\u5982\u4e0b\u7ecf\u5178\u7ed3\u679c\uff1a \\begin{enumerate} \\item \u6709\u9650\u7fa4\u8868\u793a\u662f\u5b8c\u5168\u53ef\u7ea6\u7684\uff0c\u4e14\u53ef\u9149\u5316; \\item \u6b63\u5219\u8868\u793a\u4e2d\u5305\u542b\u4e86\u5168\u90e8\u4e0d\u53ef\u7ea6\u8868\u793a\uff0c\u4e14\u5176\u4e2d\u5168\u90e8\u4e0d\u53ef\u7ea6\u8868\u793a\u7684\u6570\u76ee\u7b49\u4e8e\u5176\u7b49\u4ef7\u7c7b\u6570\u76ee,\u91cd\u6570\u7b49\u4e8e\u5176\u7ef4\u6570; \\item Schur\u5f15\u7406\u3002\u4e0d\u53ef\u7ea6\u6a21\u95f4\u6ca1\u6709\u975e\u5e73\u51e1\u975e\u540c\u6784\u7684\u6a21\u540c\u6001; \\item \u6b63\u4ea4\u5173\u7cfb\u7684\u5bfc\u51fa; \\item \u7531\u7279\u5f81\u6807\u6b63\u4ea4\u5173\u7cfb\u5f97\u51fa\u7684\u4e0d\u53ef\u7ea6\u8868\u793a\u5224\u5b9a\u4e0e\u53ef\u7ea6\u8868\u793a\u7ea6\u5316\u3002 \\end{enumerate} \\subsection{\u4e3b\u8981\u6982\u5ff5} \\begin{definition}[\u7fa4\u8868\u793a] \u7fa4\u7684\\textbf{\u8868\u793a}\u4e3a\u4e00\u4e2a\u540c\u6001,\u5176\u4e2d\u4e3a\u57df\u4e0a\u7684\u6709\u9650\u7ef4\u5411\u91cf\u7a7a\u95f4\uff0c\u8fd9\u91cc\u82e5\u65e0\u7279\u6b8a\u8bf4\u660e\u6211\u4eec\u53d6.\u79f0\u4f5c\\textbf{\u8868\u793a\u7684\u7ef4\u6570}; \u82e5\u5e73\u51e1\uff0c\u5219\u79f0\u8be5\u8868\u793a\u662f\\textbf{\u5fe0\u5b9e\u7684};\u82e5\u5bf9\u6240\u6709,\u5219\u79f0\u8868\u793a\u4e3a\\textbf{\u5e73\u51e1\u7684}\u3002 \u7fa4\u8868\u793a\u53ef\u770b\u4f5c\u4e0a\u7684-\u6a21\u3002 \\end{definition} \\begin{definition}[\u53ef\u7ea6\u4e0e\u4e0d\u53ef\u7ea6] \u4e00\u4e2a\u8868\u793a\u662f\\textbf{\u4e0d\u53ef\u7ea6}\u7684\u5982\u679c\u5b83\u4e0d\u5177\u6709\u6070\u5f53\u7684\u4e0d\u53d8\u5b50\u7a7a\u95f4\uff08\u5373\u4e0d\u53ef\u7ea6\u5b50\u8868\u793a\uff09;\u4e00\u4e2a\u8868\u793a\u662f\\textbf{\u5b8c\u5168\u53ef\u7ea6\u7684}\u5982\u679c\u5b83\u53ef\u4ee5\u5206\u89e3\u4e3a\u4e00\u7cfb\u5217\u4e0d\u53ef\u7ea6\u5b50\u8868\u793a\u7684\u76f4\u548c\u3002 \\end{definition} \\begin{definition}[\u6b63\u5219\u8868\u793a] \u8003\u8651\u4e0a\u590d\u51fd\u6570\u6784\u6210\u7684\u5411\u91cf\u7a7a\u95f4,\u5b83\u7684\u57fa\u5e95\u4e3a,\u5176\u4e2d\u8868\u793a\u4e00\u4e2a\u5728\u4e0a\u503c\u4e3a\uff0c\u4f59\u7686\u4e3a\u7684\u51fd\u6570\u3002 \u5219\u53ef\u4ee5\u901a\u8fc7\u4e0a\u7684\u7ebf\u6027\u7b97\u5b50\u8fd9\u6837\u4f5c\u7528\u5728\u57fa\u77e2\u4e0a (1)   \u5219\u79f0\u4e3a\u7684\\textbf{\u5de6\u6b63\u5219\u8868\u793a}\u3002 \\end{definition} \\begin{definition}[G-\u7ebf\u6027\u6620\u5c04] \u8bbe\u5728\u4e0a\u5206\u522b\u6709\u8868\u793a,\u4e14\u5b58\u5728\u7ebf\u6027\u6620\u5c04\uff0c\u6ee1\u8db3 (2)   \u5219\u79f0\u4e3a\u4e00\u4e2a\\textbf{-\u7ebf\u6027\u6620\u5c04}\uff0c\u6216\u79f0\\textbf{\u4e0e\u53ef\u4ea4\u6362}.\u82e5\u53ef\u9006\uff0c\u5219\u4e0a\u5f0f\u53ef\u5199\u4e3a (3)   \u6b64\u65f6\u79f0\u4e3a\\textbf{\u7fa4\u8868\u793a\u7684\u540c\u6784}\u3002 \u7528\u6a21\u7684\u8bed\u8a00\u6765\u8bf4\uff0cG-\u7ebf\u6027\u6620\u5c04\u6b63\u662f\u4e00\u4e2a\u6a21\u540c\u6001\u3002 \\end{definition} \\subsection{\u9149\u8868\u793a\u4e0e\u53ef\u7ea6\u6027} \\begin{definition}[\u9149\u8868\u793a] \u5411\u91cf\u7a7a\u95f4\u4e0a\u7684\u8868\u793a\u88ab\u79f0\u4f5c\\textbf{\u9149}\u7684\uff0c\u5982\u679c\u88c5\u5907\u4e86\u4e00\u4e2a\u5728\u4f5c\u7528\u4e0b\u4e0d\u53d8\u7684\u5384\u7c73\u5185\u79ef (4)   \u8868\u793a\u79f0\u4f5c\\textbf{\u53ef\u9149\u5316\u7684}\u5982\u679c\u53ef\u4ee5\u88c5\u5907\u8fd9\u6837\u4e00\u4e2a\u5185\u79ef\u3002 \u4e00\u4e2a\u8868\u793a\u662f\u9149\u7684\u5f53\u4e14\u4ec5\u5f53\u540c\u6001\u843d\u5728\u9149\u7fa4\u4e4b\u4e2d\u3002\u8be5\u6761\u4ef6\u4e5f\u53ef\u4ee5\u8868\u8ff0\u4e3a. \\end{definition} \\begin{lemma}[] \u8bbe\u4e3a\u7fa4\u7684\u9149\u8868\u793a\u800c\u4e3a\u4e2d\u7684\u4e00\u4e2a\u4e0d\u53d8\u5b50\u7a7a\u95f4\uff0c\u5219\u7684\u6b63\u4ea4\u8865\u4e5f\u662f\u4e00\u4e2a\u4e0d\u53d8\u5b50\u7a7a\u95f4\u3002 \\end{lemma} […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_mi_skip_tracking":false,"_exactmetrics_sitenote_active":false,"_exactmetrics_sitenote_note":"","_exactmetrics_sitenote_category":0,"footnotes":""},"categories":[141],"tags":[144,142,143],"class_list":["post-949","post","type-post","status-publish","format-standard","hentry","category-math","tag-144","tag-142","tag-143"],"translation":{"provider":"WPGlobus","version":"2.12.2","language":"en","enabled_languages":["zh","en"],"languages":{"zh":{"title":true,"content":true,"excerpt":false},"en":{"title":false,"content":false,"excerpt":false}}},"_links":{"self":[{"href":"https:\/\/mindspectrum.xyz\/en\/wp-json\/wp\/v2\/posts\/949","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mindspectrum.xyz\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mindspectrum.xyz\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mindspectrum.xyz\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mindspectrum.xyz\/en\/wp-json\/wp\/v2\/comments?post=949"}],"version-history":[{"count":14,"href":"https:\/\/mindspectrum.xyz\/en\/wp-json\/wp\/v2\/posts\/949\/revisions"}],"predecessor-version":[{"id":987,"href":"https:\/\/mindspectrum.xyz\/en\/wp-json\/wp\/v2\/posts\/949\/revisions\/987"}],"wp:attachment":[{"href":"https:\/\/mindspectrum.xyz\/en\/wp-json\/wp\/v2\/media?parent=949"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mindspectrum.xyz\/en\/wp-json\/wp\/v2\/categories?post=949"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mindspectrum.xyz\/en\/wp-json\/wp\/v2\/tags?post=949"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}