{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:58f1c3d5-9aa5-45b5-b01b-8f998de65e40", "labels": { "zh-CN": "Weyl 特征标公式", "en": "Weyl character formula" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "b56ed226688a5ea7c5f6559e8dcfc572ba1122d3645551e4dca2f45e25b40962", "occurrences": [ { "concept_id": "urn:uuid:58f1c3d5-9aa5-45b5-b01b-8f998de65e40", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-09/04-tensor-products-of-representations", "anchor": "first-2aeebe1b55db59905bc6-zh", "chapter": "9", "section": "9.4", "offset": 3444, "range": [ 10528, 11005 ], "source_revision": "1d4dd7dd9c43a9dd38d15e6377795a8ce87e81c38569e13b0b5f9485d63303a1", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "theorem", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "books/woit-qtgr/2025-10-20/zh/chapter-09/04-tensor-products-of-representations", "lang": "zh-CN", "section": "9.4", "signal": "named-theorem", "label": "Weyl 特征标公式", "english": "", "offset": 3161 } }, "label": "> [!theorem] 定理（Weyl 特征标公式）", "context": { "range": [ 10528, 11005 ], "quote": "\n\n> [!theorem] 定理（Weyl 特征标公式）\n> 对于$\\mathrm{SU}(2)$的不可约表示$(\\pi_n,V^n)$以及下面的对角元素，\n>\n> $$\n> \\chi_{V^n}\\!\\left(\\begin{pmatrix}e^{i\\theta}&0\\\\0&e^{-i\\theta}\\end{pmatrix}\\right)=\\frac{e^{i(n+1)\\theta}-e^{-i(n+1)\\theta}}{e^{i\\theta}-e^{-i\\theta}}=\\frac{\\sin((n+1)\\theta)}{\\sin\\theta}\n> $$\n\n> [!proof]- 证明\n> 由下列恒等式和方程（9.2）可得此公式：\n>\n> $$\n> \\bigl(e^{in\\theta}+e^{i(n-2)\\theta}+\\cdots+e^{-i(n-2)\\theta}+e^{-in\\theta}\\bigr)(e^{i\\theta}-e^{-i\\theta})=e^{i(n+1)\\theta}-e^{-i(n+1)\\theta}\n> $$\n", "quote_sha256": "4be3b7ce176244317dd1a495879c34aff798e082cca3ec7ed0c5edb6a1daae9c", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-09/04-tensor-products-of-representations", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-4432011026dc45d3dba09b0f" }, { "concept_id": "urn:uuid:58f1c3d5-9aa5-45b5-b01b-8f998de65e40", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-09/04-tensor-products-of-representations", "anchor": "first-3031bbfcb3e29807b3b5-en", "chapter": "9", "section": "9.4", "offset": 5681, "range": [ 11927, 12665 ], "source_revision": "4aa1a1d7f3fcb427daecdc46e53ab72991dfbeec261c3e7e58a5d923efb74b3f", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "theorem", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "books/woit-qtgr/2025-10-20/en/chapter-09/04-tensor-products-of-representations", "lang": "en", "section": "9.4", "signal": "named-theorem", "label": "Weyl character formula", "english": "Weyl character formula", "offset": 5398 } }, "label": "> [!theorem] Theorem (Weyl character formula)", "context": { "range": [ 11927, 12665 ], "quote": "\n\n> [!theorem] Theorem (Weyl character formula)\n> For the irreducible representation $(\\pi_n,V^n)$ of $\\mathrm{SU}(2)$ and the diagonal element below,\n>\n> $$\n> \\chi_{V^n}\\!\\left(\\begin{pmatrix}e^{i\\theta}&0\\\\0&e^{-i\\theta}\\end{pmatrix}\\right)=\\frac{e^{i(n+1)\\theta}-e^{-i(n+1)\\theta}}{e^{i\\theta}-e^{-i\\theta}}=\\frac{\\sin((n+1)\\theta)}{\\sin\\theta}.\n> $$\n>\n\n> [!proof]- Proof\n> The formula follows from the identity\n>\n> $$\n> \\bigl(e^{in\\theta}+e^{i(n-2)\\theta}+\\cdots+e^{-i(n-2)\\theta}+e^{-in\\theta}\\bigr)(e^{i\\theta}-e^{-i\\theta})=e^{i(n+1)\\theta}-e^{-i(n+1)\\theta},\n> $$\n>\n> together with equation (9.2).\n", "quote_sha256": "57238ede40bbcb2455f2c57362399fda5d548433bd16b41b753539a5f43c2355", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-09/04-tensor-products-of-representations", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; 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