AWhat 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.
Scientific 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.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. 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.
Contributors
- The named authors in the research record · SCIENCE: Source-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.
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