general linear group GL(n,C)
E595246
The general linear group GL(n,ℂ) is the Lie group consisting of all invertible n×n complex matrices under matrix multiplication, fundamental in linear algebra and representation theory.
All labels observed (3)
| Label | Occurrences |
|---|---|
| GL(n,ℂ) | 3 |
| GLₙ(ℂ) | 1 |
| general linear group GL(n,C) canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6456477 — 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: general linear group GL(n,C) Context triple: [Lie group, hasExample, general linear group GL(n,C)]
-
A.
general linear group GL(n,R)
The general linear group GL(n,ℝ) is the Lie group consisting of all invertible n×n real matrices under matrix multiplication, fundamental in linear algebra and differential geometry.
-
B.
special linear group SL(n,R)
The special linear group SL(n,ℝ) is the Lie group of all n×n real matrices with determinant 1, fundamental in linear algebra and differential geometry as the group of volume-preserving linear transformations.
-
C.
SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
-
D.
special orthogonal group SO(n)
The special orthogonal group SO(n) is the group of all n×n real rotation matrices with determinant 1, representing orientation-preserving isometries of n-dimensional Euclidean space that fix the origin.
-
E.
Lie group
A Lie group is a mathematical structure that is both a smooth manifold and a group, where the group operations are differentiable and used to study continuous symmetries.
- 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: general linear group GL(n,C) Target entity description: The general linear group GL(n,ℂ) is the Lie group consisting of all invertible n×n complex matrices under matrix multiplication, fundamental in linear algebra and representation theory.
-
A.
general linear group GL(n,R)
The general linear group GL(n,ℝ) is the Lie group consisting of all invertible n×n real matrices under matrix multiplication, fundamental in linear algebra and differential geometry.
-
B.
special linear group SL(n,R)
The special linear group SL(n,ℝ) is the Lie group of all n×n real matrices with determinant 1, fundamental in linear algebra and differential geometry as the group of volume-preserving linear transformations.
-
C.
SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
-
D.
special orthogonal group SO(n)
The special orthogonal group SO(n) is the group of all n×n real rotation matrices with determinant 1, representing orientation-preserving isometries of n-dimensional Euclidean space that fix the origin.
-
E.
Lie group
A Lie group is a mathematical structure that is both a smooth manifold and a group, where the group operations are differentiable and used to study continuous symmetries.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
Lie group
ⓘ
complex Lie group ⓘ linear algebraic group ⓘ matrix group ⓘ topological group ⓘ |
| actsOn | ℂⁿ by left multiplication ⓘ |
| appearsIn | classification of complex representations of finite groups via embeddings ⓘ |
| center | {λIₙ : λ ∈ ℂˣ} ⓘ |
| centerIsomorphicTo | ℂˣ ⓘ |
| conditionForMembership | det(A) ≠ 0 ⓘ |
| containsSubgroup |
Borel subgroup of upper triangular invertible matrices
ⓘ
linked to:
Borel subgroup
SL(n,ℂ) ⓘ
linked to:
special linear group SL(n,C)
U(n) ⓘ |
| definedAs | group of all invertible n×n complex matrices ⓘ |
| definedByPolynomialCondition | det(A) ≠ 0 ⓘ |
| determinantMap | det : GL(n,ℂ) → ℂˣ ⓘ |
| determinantMapIs | surjective group homomorphism ⓘ |
| dimensionAsComplexLieGroup | n² ⓘ |
| dimensionAsRealManifold | 2n² ⓘ |
| fundamentalGroup | ℤ ⓘ |
| hasDeterminantCharacter | det : GL(n,ℂ) → ℂˣ ⓘ |
| hasMaximalTorus | diagonal invertible matrices ⓘ |
| identityElement | n×n identity matrix ⓘ |
| inverseOperation | matrix inverse ⓘ |
| isAlgebraicGroupOver | ℂ ⓘ |
| isConnected | true ⓘ |
| isNonAbelian | true ⓘ |
| isOpenSubsetOf | Mₙ(ℂ) with respect to standard topology ⓘ |
| isReductive | true ⓘ |
| isSimplyConnected | false ⓘ |
| isSolvable | false ⓘ |
| kernelOfDeterminant |
SL(n,ℂ)
ⓘ
linked to:
special linear group SL(n,C)
|
| LieAlgebra | 𝔤𝔩(n,ℂ) ⓘ |
| LieAlgebraDefinedAs | all n×n complex matrices with usual commutator bracket ⓘ |
| maximalCompactSubgroup | U(n) ⓘ |
| notationVariant |
GLₙ(ℂ)
ⓘ
linked to:
general linear group GL(n,C)
|
| overField | ℂ ⓘ |
| parameter | n ∈ ℕ, n ≥ 1 ⓘ |
| quotientByCenter | PGL(n,ℂ) ⓘ |
| rank | n ⓘ |
| roleInMathematics |
basic example in Lie theory
ⓘ
basic example in algebraic geometry ⓘ central in representation theory ⓘ fundamental in linear algebra ⓘ |
| standardRepresentation | action on ℂⁿ ⓘ |
| subgroupDefinedAs | SL(n,ℂ) = {A ∈ GL(n,ℂ) : det(A) = 1} ⓘ |
| underOperation | matrix multiplication ⓘ |
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: general linear group GL(n,C) Description of subject: The general linear group GL(n,ℂ) is the Lie group consisting of all invertible n×n complex matrices under matrix multiplication, fundamental in linear algebra and representation theory.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
general linear group GL(n,C)
linked to: general linear group GL(n,C)
subject linked to:
special linear group SL(n,C)
linked to: general linear group GL(n,C)
subject linked to:
special linear group SL(n,C)
linked to: general linear group GL(n,C)
subject linked to:
special linear group SL(n,C)
linked to: general linear group GL(n,C)