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    "title": "Maschke: complete reducibility of finite complex linear groups",
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    "area": "Finite-group representations and invariant complements",
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    "year": 1899,
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        "title": "Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten, intransitiv sind",
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        "description": "The historical paper treats finite complex linear substitution groups using conjugation and a positive Hermitian form. Its preliminary block-triangular reductions can apply to infinite groups, but the invariant-complement conclusion uses finiteness. The common zero is across every group matrix in a fixed position, not a zero in just one element. This record does not attribute a modern averaging-projection proof, the arbitrary-field characteristic-not-dividing-order extension, uniqueness of irreducible multiplicities or infinite-dimensional compact-group theory to the 1899 body.",
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        {
          "citation": "Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten, intransitiv sind",
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      "statement": "# Maschke: complete reducibility of finite complex linear groups\n\nMaschke removed an earlier restriction from his criterion that a finite linear substitution group whose matrices have a fixed identically vanishing off-diagonal coefficient is intransitive, meaning transformable into separate blocks of variables. The decisive finite-group step starts with an invariant block and uses a positive invariant Hermitian form to eliminate the cross terms; the resulting unitary relations force the opposite off-diagonal block to vanish. Thus an invariant subspace in this complex finite-dimensional setting has an invariant complementary subspace, giving complete reducibility by iteration.\n\n## Scientific scope\n\nThe historical paper treats finite complex linear substitution groups using conjugation and a positive Hermitian form. Its preliminary block-triangular reductions can apply to infinite groups, but the invariant-complement conclusion uses finiteness. The common zero is across every group matrix in a fixed position, not a zero in just one element. This record does not attribute a modern averaging-projection proof, the arbitrary-field characteristic-not-dividing-order extension, uniqueness of irreducible multiplicities or infinite-dimensional compact-group theory to the 1899 body.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nHeinrich Maschke\n\nSource-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.\n\n\n\n## Original sources\n\n- [Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten, intransitiv sind](https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0052/LOG_0032.pdf)\n",
      "summary": "Maschke removed an earlier restriction from his criterion that a finite linear substitution group whose matrices have a fixed identically vanishing off-diagonal coefficient is intransitive, meaning transformable into separate blocks of variables. The decisive finite-group step starts with an invariant block and uses a positive invariant Hermitian form to eliminate the cross terms; the resulting unitary relations force the opposite off-diagonal block to vanish. Thus an invariant subspace in this complex finite-dimensional setting has an invariant complementary subspace, giving complete reducibility by iteration.",
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    "title": "Maschke: complete reducibility of finite complex linear groups",
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    "what_changed": "Maschke removed an earlier restriction from his criterion that a finite linear substitution group whose matrices have a fixed identically vanishing off-diagonal coefficient is intransitive, meaning transformable into separate blocks of variables. The decisive finite-group step starts with an invariant block and uses a positive invariant Hermitian form to eliminate the cross terms; the resulting unitary relations force the opposite off-diagonal block to vanish. Thus an invariant subspace in this complex finite-dimensional setting has an invariant complementary subspace, giving complete reducibility by iteration.",
    "scope": "Maschke removed an earlier restriction from his criterion that a finite linear substitution group whose matrices have a fixed identically vanishing off-diagonal coefficient is intransitive, meaning transformable into separate blocks of variables. The decisive finite-group step starts with an invariant block and uses a positive invariant Hermitian form to eliminate the cross terms; the resulting unitary relations force the opposite off-diagonal block to vanish. Thus an invariant subspace in this complex finite-dimensional setting has an invariant complementary subspace, giving complete reducibility by iteration. The historical paper treats finite complex linear substitution groups using conjugation and a positive Hermitian form. Its preliminary block-triangular reductions can apply to infinite groups, but the invariant-complement conclusion uses finiteness. The common zero is across every group matrix in a fixed position, not a zero in just one element. This record does not attribute a modern averaging-projection proof, the arbitrary-field characteristic-not-dividing-order extension, uniqueness of irreducible multiplicities or infinite-dimensional compact-group theory to the 1899 body.",
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        "title": "Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten, intransitiv sind",
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