Representations of groups
E503510
Representations of groups is a branch of mathematics that studies how abstract groups can act as linear transformations on vector spaces, typically via homomorphisms into groups of matrices.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Lie group representation | 1 |
| Lie group representations | 1 |
| Lorentz group representations | 1 |
| Representation Theory of Lie Groups | 1 |
| Representations of groups canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5211818 — 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: Representations of groups Context triple: [The Classical Groups: Their Invariants and Representations, libraryOfCongressSubject, Representations of groups]
-
A.
Methods of Representation Theory
Methods of Representation Theory is a foundational multi-volume work in mathematics that systematically develops the theory of group and algebra representations, coauthored by Israel Gelfand and collaborators.
-
B.
Theory of Groups
Theory of Groups is a foundational textbook in abstract algebra that systematically develops the theory of groups and has been widely used for advanced mathematical study and research.
-
C.
Representation Theory and Automorphic Functions
"Representation Theory and Automorphic Functions" is a seminal mathematical work by Israel Gelfand that develops the connections between representation theory of groups and the theory of automorphic forms, with deep applications in number theory and harmonic analysis.
-
D.
The Classical Groups: Their Invariants and Representations
The Classical Groups: Their Invariants and Representations is a foundational mathematical monograph by Hermann Weyl that systematically develops the theory of classical Lie groups, their invariants, and their representation theory.
-
E.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
- 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: Representations of groups Target entity description: Representations of groups is a branch of mathematics that studies how abstract groups can act as linear transformations on vector spaces, typically via homomorphisms into groups of matrices.
-
A.
Methods of Representation Theory
Methods of Representation Theory is a foundational multi-volume work in mathematics that systematically develops the theory of group and algebra representations, coauthored by Israel Gelfand and collaborators.
-
B.
Theory of Groups
Theory of Groups is a foundational textbook in abstract algebra that systematically develops the theory of groups and has been widely used for advanced mathematical study and research.
-
C.
Representation Theory and Automorphic Functions
"Representation Theory and Automorphic Functions" is a seminal mathematical work by Israel Gelfand that develops the connections between representation theory of groups and the theory of automorphic forms, with deep applications in number theory and harmonic analysis.
-
D.
The Classical Groups: Their Invariants and Representations
The Classical Groups: Their Invariants and Representations is a foundational mathematical monograph by Hermann Weyl that systematically develops the theory of classical Lie groups, their invariants, and their representation theory.
-
E.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
area of representation theory
ⓘ
branch of mathematics ⓘ topic in abstract algebra ⓘ |
| appliesTo |
Lie groups
ⓘ
linked to:
Lie group
abelian groups ⓘ compact groups ⓘ finite groups ⓘ locally compact groups ⓘ non-abelian groups ⓘ |
| formalizedAs | functors from a group to the category of vector spaces ⓘ |
| goal |
classify representations up to equivalence
ⓘ
decompose representations into irreducible components ⓘ |
| hasSubfield |
modular representation theory
ⓘ
p-adic group representations ⓘ representation theory of Lie groups ⓘ representation theory of finite groups ⓘ unitary representation theory ⓘ |
| historicallyDevelopedBy |
Ferdinand Frobenius
ⓘ
linked to:
Ferdinand Georg Frobenius
Hermann Weyl ⓘ Issai Schur ⓘ |
| relatedTo |
Fourier analysis on groups
ⓘ
category theory ⓘ harmonic analysis ⓘ invariant theory ⓘ number theory ⓘ particle physics ⓘ quantum mechanics ⓘ symmetry ⓘ |
| studies |
group actions by linear transformations on vector spaces
ⓘ
group homomorphisms into general linear groups ⓘ group homomorphisms into matrix groups ⓘ |
| usesConcept |
Maschke's theorem
ⓘ
linked to:
Maschke’s theorem
Schur's lemma ⓘ
linked to:
Schur’s lemma
character of a representation ⓘ complete reducibility ⓘ general linear group ⓘ
linked to:
general linear group GL(n,R)
group ⓘ group algebra ⓘ group homomorphism ⓘ induced representation ⓘ invariant subspace ⓘ irreducible representation ⓘ linear transformation ⓘ matrix representation ⓘ module over a ring ⓘ reducible representation ⓘ regular representation ⓘ restriction of representations ⓘ unitary representation ⓘ vector space ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Representations of groups Description of subject: Representations of groups is a branch of mathematics that studies how abstract groups can act as linear transformations on vector spaces, typically via homomorphisms into groups of matrices.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
The Classical Groups: Their Invariants and Representations
→
libraryOfCongressSubject
→
Representations of groups
ⓘ
linked to: Representations of groups
linked to: Representations of groups
linked to: Representations of groups
Lie Groups: History, Frontiers and Applications (contributions)
→
relatedTo
→
Lie group representations
ⓘ
linked to: Representations of groups