Maschke’s theorem
E924213
Maschke’s theorem is a fundamental result in representation theory stating that every finite group representation over a field of characteristic not dividing the group order is completely reducible into a direct sum of irreducible representations.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Maschke’s theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11412034 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Maschke’s theorem Context triple: [Methods of Representation Theory, covers, Maschke’s theorem]
-
A.
Peter–Weyl theorem
The Peter–Weyl theorem is a fundamental result in representation theory and harmonic analysis that decomposes square-integrable functions on a compact topological group into a direct sum of finite-dimensional irreducible unitary representations.
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B.
Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
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C.
Jordan–Hölder theorem
The Jordan–Hölder theorem is a fundamental result in group theory stating that any two composition series of a finite group have the same length and the same (up to order and isomorphism) simple factor groups.
-
D.
Cayley–Hamilton theorem
The Cayley–Hamilton theorem is a fundamental result in linear algebra stating that every square matrix satisfies its own characteristic polynomial.
-
E.
Artin–Wedderburn theorem
The Artin–Wedderburn theorem is a fundamental result in ring theory that classifies all semisimple rings as finite direct products of matrix rings over division rings.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Maschke’s theorem Target entity description: Maschke’s theorem is a fundamental result in representation theory stating that every finite group representation over a field of characteristic not dividing the group order is completely reducible into a direct sum of irreducible representations.
-
A.
Peter–Weyl theorem
The Peter–Weyl theorem is a fundamental result in representation theory and harmonic analysis that decomposes square-integrable functions on a compact topological group into a direct sum of finite-dimensional irreducible unitary representations.
-
B.
Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
-
C.
Jordan–Hölder theorem
The Jordan–Hölder theorem is a fundamental result in group theory stating that any two composition series of a finite group have the same length and the same (up to order and isomorphism) simple factor groups.
-
D.
Cayley–Hamilton theorem
The Cayley–Hamilton theorem is a fundamental result in linear algebra stating that every square matrix satisfies its own characteristic polynomial.
-
E.
Artin–Wedderburn theorem
The Artin–Wedderburn theorem is a fundamental result in ring theory that classifies all semisimple rings as finite direct products of matrix rings over division rings.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem ⓘ theorem in representation theory ⓘ |
| appliesTo |
finite groups
ⓘ
group representations ⓘ linear representations of finite groups ⓘ |
| assumes |
characteristic of the field does not divide the group order
ⓘ
group order is finite ⓘ |
| concerns |
direct sum decompositions
ⓘ
finite-dimensional modules ⓘ irreducible representations ⓘ modules over group algebras ⓘ |
| concludes |
every finite-dimensional representation is completely reducible
ⓘ
every representation decomposes as a direct sum of irreducible representations ⓘ every subrepresentation has a complementary subrepresentation ⓘ |
| equivalentTo | semisimplicity of the group algebra over the field ⓘ |
| failsWhen | field characteristic divides the group order ⓘ |
| field | representation theory ⓘ |
| hasConsequence |
characters of irreducible representations form an orthonormal basis of class functions
ⓘ
every finite group representation over C is completely reducible ⓘ regular representation decomposes into a direct sum of irreducibles ⓘ |
| hasProofTechnique |
averaging over the group
ⓘ
construction of invariant complements ⓘ |
| holdsOver |
fields of characteristic p not dividing the group order
ⓘ
fields of characteristic zero ⓘ |
| implies |
category of finite-dimensional representations is semisimple
ⓘ
group algebra is semisimple ⓘ |
| isTaughtIn |
graduate algebra courses
ⓘ
representation theory courses ⓘ |
| namedAfter | Heinrich Maschke NERFINISHED ⓘ |
| originalLanguage | German NERFINISHED ⓘ |
| originallyPublishedIn | Mathematische Annalen NERFINISHED ⓘ |
| relatedTo |
Schur’s lemma
NERFINISHED
ⓘ
Wedderburn’s theorem NERFINISHED ⓘ complete reducibility ⓘ group algebra ⓘ semisimple module ⓘ |
| requires |
field whose characteristic does not divide the group order
ⓘ
finite group ⓘ |
| subfield | group representation theory ⓘ |
| usedIn |
Fourier analysis on finite groups
NERFINISHED
ⓘ
character theory of finite groups ⓘ classification of irreducible representations of finite groups ⓘ construction of projection operators onto irreducible components ⓘ decomposition of group representations ⓘ harmonic analysis on finite groups ⓘ |
| yearProved | 1899 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Maschke’s theorem Description of subject: Maschke’s theorem is a fundamental result in representation theory stating that every finite group representation over a field of characteristic not dividing the group order is completely reducible into a direct sum of irreducible representations.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.