{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:023d3b84-da9a-4667-8674-d5b9ccbed948", "labels": { "zh-CN": "舒尔引理", "en": "Schur’s lemma" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "30e39c99400fb4032d4b0b343fcdb859eaa0047bc4ef10df75ae93b6d131dec5", "occurrences": [ { "concept_id": "urn:uuid:023d3b84-da9a-4667-8674-d5b9ccbed948", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-04/04-introduction-to-group-representation-theory", "anchor": "first-6d7eb6cc7322de42f7e0-zh", "chapter": "4", "section": "4.4", "offset": 11874, "range": [ 16972, 18065 ], "source_revision": "8659a22bf0394df6e0f0547cc91ed5a8dd18c3b22a3aa2463bbac9fbfc4f313c", "source_category": "translation", "review_status": "unreviewed", "role": "theorem", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "zh/chapter-04/04-introduction-to-group-representation-theory", "lang": "zh-CN", "section": "4.4", "signal": "named-theorem", "label": "舒尔引理", "english": "Schur’s lemma", "offset": 10556 } }, "label": "> [!theorem] 定理 4.6 · 舒尔引理 (Schur’s lemma)", "context": { "range": [ 16972, 18065 ], "quote": "\n\n> [!theorem] 定理 4.6 · 舒尔引理 (Schur’s lemma)\n>\n> （舒尔引理）设 $T _ { 1 } : G \\to G L ( V _ { 1 } )$ 与 $T _ { 2 } : G \\to G L ( V _ { 2 } )$ 是群 $G$ 的两个不可约表示，且 $A : V _ { 1 } \\to V _ { 2 }$ 是一个交织算符，使得\n>\n> $$\n> T _ {2} (g) A = A T _ {1} (g) \\text { for all } g \\in G.\n> $$\n>\n> 于是 $A = 0$ 或 $A$ 是同构，此时这两个表示等价，$T _ { 1 } \\sim T _ { 2 }$\n\n> [!proof]- 证明\n>\n> 设 $v \\in$ $\\ker$ $A \\subseteq V _ { 1 }$。则\n>\n> $$\n> A T _ {1} (g) v = T _ {2} (g) A v = 0\n> $$\n>\n> 因此 $T _ { 1 } ( g ) v \\in$ $\\ker$ $A$。于是 $\\ker$ $A$ 是表示 $T _ { 1 }$ 的不变子空间。由于 $T _ { 1 }$ 是不可约表示，我们有或者 $\\ker$ $A = V _ { 1 }$，此时 $A = 0 _ { i }$，或者 $\\ker$ $A = \\{ 0 \\}$。在后一种情形中，$A$ 是一对一的。为表明它是同构，只需表明它是满射。这由以下事实推出：$\\operatorname{im}$ $A \\subset V _ { 2 }$ 是表示 $T _ { 2 }$ 的不变子空间\n>\n> $$\n> T _ {2} (g) (\\mathrm{im} A) = T _ {2} (g) A \\left(V _ {1}\\right) = A T _ {1} (g) \\left(V _ {1}\\right) \\subseteq A \\left(V _ {1}\\right) = \\mathrm{im} A.\n> $$\n>\n> 由于 $T _ { 2 }$ 是不可约表示，我们有 $\\operatorname{im}$ $A = \\{ 0 \\}$ 或 $\\operatorname{im}$ $A = V _ { 2 }$ 。在第一种情形中 $A = 0$ ，而在第二种情形中 $A$ 是满射。舒尔引理得证。\n", "quote_sha256": "10f2461c9294b193cd106fa0ef9f80676cf552601bd7b063b34aa18634c4e576", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-04/04-introduction-to-group-representation-theory", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-1bb6de943d307a0620fcdadd" }, { "concept_id": "urn:uuid:023d3b84-da9a-4667-8674-d5b9ccbed948", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-04/04-introduction-to-group-representation-theory", "anchor": "first-6d7eb6cc7322de42f7e0-en", "chapter": "4", "section": "4.4", "offset": 13803, "range": [ 18393, 19962 ], "source_revision": "9ba0357bc9fe2cf27435fc71decca231bb93b8a3a84cf95591397d0b6b814384", "source_category": "original_transcription", "review_status": "unreviewed", "role": "theorem", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "en/chapter-04/04-introduction-to-group-representation-theory", "lang": "en", "section": "4.4", "signal": "named-theorem", "label": "Schur’s lemma", "english": "Schur’s lemma", "offset": 13733 } }, "label": "> [!theorem] Theorem 4.6 · Schur’s lemma", "context": { "range": [ 18393, 19962 ], "quote": "\n\n> [!theorem] Theorem 4.6 · Schur’s lemma\n>\n> (Schur’s lemma) Let $T _ { 1 } : G \\to G L ( V _ { 1 } )$ and $T _ { 2 } : G \\to G L ( V _ { 2 } )$ be two irreducible representations of a group $G$, and $A : V _ { 1 } \\to V _ { 2 }$ an intertwining operator such that\n>\n> $$\n> T _ {2} (g) A = A T _ {1} (g) \\text { for all } g \\in G.\n> $$\n>\n> Then either $A = 0$ or $A$ is an isomorphism, in which case the two representations are equivalent, $T _ { 1 } \\sim T _ { 2 }$\n\n> [!proof]- Proof\n>\n> Let $v \\in$ $\\ker$ $A \\subseteq V _ { 1 }$ . Then\n>\n> $$\n> A T _ {1} (g) v = T _ {2} (g) A v = 0\n> $$\n>\n> so that $T _ { 1 } ( g ) v \\in$ $\\ker$ $A$. Hence $\\ker$ $A$ is an invariant subspace of the representation $T _ { 1 }$ . As $T _ { 1 }$ is an irreducible representation we have that either $\\ker$ $A = V _ { 1 }$ , in which case $A = 0 _ { i }$ or $\\ker$ $A = \\{ 0 \\}$ . In the latter case $A$ is one-to-one. To show it is an isomorphism it is only necessary to show that it is onto. This follows from the fact that $\\operatorname{im}$ $A \\subset V _ { 2 }$ is an invarian subspace of the representation $T _ { 2 }$\n>\n> $$\n> T _ {2} (g) (\\mathrm{im} A) = T _ {2} (g) A \\left(V _ {1}\\right) = A T _ {1} (g) \\left(V _ {1}\\right) \\subseteq A \\left(V _ {1}\\right) = \\mathrm{im} A.\n> $$\n>\n> Since $T _ { 2 }$ is an irreducible representation we have either $\\operatorname{im}$ $A = \\{ 0 \\}$ or $\\operatorname{im}$ $A = V _ { 2 }$ . In the first case $A = 0$ , while in the second $A$ is onto. Schur’s lemma is proved.\n", "quote_sha256": "b973a8e5797b3c59b8aa3aa22fca113d76a052fac5ce577ba6d7de1ae2176d67", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-04/04-introduction-to-group-representation-theory", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-1d7edaa9ecb989535877130f" }, { "concept_id": "urn:uuid:023d3b84-da9a-4667-8674-d5b9ccbed948", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-02/01-some-representation-theory", "anchor": "first-e575a0c33aad56892410-zh", "chapter": "2", "section": "2.1", "offset": 6790, "range": [ 16973, 17279 ], "source_revision": "42b32dda09af12d2236451f0c9b29db8c863f217a1374b2e7d4d7efd1d69b961", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "舒尔引理(Schur’s lemma)", "context": { "range": [ 16886, 17526 ], "quote": "\n> [!theorem] 定理 · 舒尔引理(Schur’s lemma)\n> \n>\n> 定理（舒尔引理）。若复表示 $(\\pi,V)$ 不可约，则与所有 $\\pi(g)$ 交换的线性映射 $M:V\\to V$ 只有 $\\lambda\\mathbf{1}$，即乘以标量 $\\lambda\\in\\mathbf{C}$。\n", "quote_sha256": "53c480c5b881240222809586d894687ab85b8c0de5863113e3c391af502b9b81", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-02/01-some-representation-theory", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-574da2fdbe17b833dab24ab8" }, { "concept_id": "urn:uuid:023d3b84-da9a-4667-8674-d5b9ccbed948", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-02/01-some-representation-theory", "anchor": "first-e575a0c33aad56892410-en", "chapter": "2", "section": "2.1", "offset": 7356, "range": [ 15760, 16060 ], "source_revision": "fd6acd71ca250c6760bd4f303aeeb212d709a4c9cb7af7a9476be4a1a7ea289c", "source_category": "original_transcription", "review_status": "ai-transcription-compared", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "Schur’s lemma", "context": { "range": [ 15668, 16417 ], "quote": "\n> [!theorem] Theorem · Schur’s lemma\n> \n>\n> Theorem (Schur’s lemma). If a complex representation $(\\pi,V)$ is irreducible, then the only linear maps $M:V\\to V$ commuting with all the $\\pi(g)$ are $\\lambda\\mathbf{1}$, multiplication by a scalar $\\lambda\\in\\mathbf{C}$.\n", "quote_sha256": "0dbb0c28c1340b33db63256954165dde7a85116a428c1b433716b4dac02916a6", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-02/01-some-representation-theory", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-fb9fc72f29caf1d1ffb639a5" } ], "registered_positions": [ { "route": "en/chapter-04/04-introduction-to-group-representation-theory", "lang": "en", "section": "4.4", "signal": "named-theorem", "label": "Schur’s lemma", "english": "Schur’s lemma", "offset": 13733 }, { "route": "zh/chapter-04/04-introduction-to-group-representation-theory", "lang": "zh-CN", "section": "4.4", "signal": "named-theorem", "label": "舒尔引理", "english": "Schur’s lemma", "offset": 10556 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-02/01-some-representation-theory", "lang": "en", "section": "2.1", "signal": "term-mark", "label": "Schur’s lemma", "english": "Schur’s lemma", "offset": 6902 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-02/01-some-representation-theory", "lang": "zh-CN", "section": "2.1", "signal": "term-mark", "label": "舒尔引理", "english": "Schur’s lemma", "offset": 5413 } ], "statements": [], "resolved_references": [ { "raw": "href=\"#theorem-4-6\"", "href": "#theorem-4-6", "range": [ 17015, 17034 ], "relation": "references", "target_route": "zh/chapter-04/04-introduction-to-group-representation-theory", "target_anchor": "theorem-4-6", "resolved": true, "reason": null, "source_revision": "8659a22bf0394df6e0f0547cc91ed5a8dd18c3b22a3aa2463bbac9fbfc4f313c", "source_route": "zh/chapter-04/04-introduction-to-group-representation-theory", "occurrence_id": "occ-1bb6de943d307a0620fcdadd" }, { "raw": "href=\"#theorem-4-6\"", "href": "#theorem-4-6", "range": [ 18436, 18455 ], "relation": "references", "target_route": "en/chapter-04/04-introduction-to-group-representation-theory", "target_anchor": "theorem-4-6", "resolved": true, "reason": null, "source_revision": "9ba0357bc9fe2cf27435fc71decca231bb93b8a3a84cf95591397d0b6b814384", "source_route": "en/chapter-04/04-introduction-to-group-representation-theory", "occurrence_id": "occ-1d7edaa9ecb989535877130f" } ], "reading_order": [ "occ-1bb6de943d307a0620fcdadd", "occ-1d7edaa9ecb989535877130f", "occ-574da2fdbe17b833dab24ab8", "occ-fb9fc72f29caf1d1ffb639a5" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 4, "located_occurrences": 4, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "72d9ef7a463513e46f531f8b70b2416b01cc96086bf6f0dc052b82a8c85eef32", "optional_enrichments": [] }