Exterior-square realization of the Spin(6) cover
A ↦ Λ²A restricted to the invariant real six-dimensional subspace of Λ²C⁴. The wedge pairing and invariant Hermitian structure determine this real form. Kernel: {±I} ≅ C₂.
A ↦ Λ²A restricted to the invariant real six-dimensional subspace of Λ²C⁴. The wedge pairing and invariant Hermitian structure determine this real form.
Definition sourcesThe group map
Reference properties
| Property | Value | Evidence |
|---|---|---|
| Domain | su-4 | Sources |
| Codomain | so-6 | Sources |
| Specified homomorphism | A ↦ Λ²A restricted to the invariant real six-dimensional subspace of Λ²C⁴. The wedge pairing and invariant Hermitian structure determine this real form. | Sources |
| Kernel | Sources | |
| Image | Sources | |
| Injective | No | Sources |
| Surjective | Yes | Sources |
| Covering degree | Sources |
Global and representation consequences
This identifies SU(4) with Spin(6). The central element iI acts by −I on the vector 6, while −I acts trivially; quotienting by the full C₄ center would give PSO(6). [sources]
Related objects and maps
- Domain group
- Codomain group
Claims and sources
Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.
Definition
A ↦ Λ²A restricted to the invariant real six-dimensional subspace of Λ²C⁴. The wedge pairing and invariant Hermitian structure determine this real form.
- Sporadic isogenies to orthogonal groupsSection 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴Open source
map:su-4-to-so-6#definitionDomain
su-4
- Sporadic isogenies to orthogonal groupsSection 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴Open source
map:su-4-to-so-6#domainCodomain
so-6
- Sporadic isogenies to orthogonal groupsSection 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴Open source
map:su-4-to-so-6#codomainSpecified homomorphism
A ↦ Λ²A restricted to the invariant real six-dimensional subspace of Λ²C⁴. The wedge pairing and invariant Hermitian structure determine this real form.
- Sporadic isogenies to orthogonal groupsSection 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴Open source
map:su-4-to-so-6#formulaKernel
{±I} ≅ C₂
- Sporadic isogenies to orthogonal groupsSection 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴Open source
map:su-4-to-so-6#kernelImage
SO(6)
- Sporadic isogenies to orthogonal groupsSection 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴Open source
map:su-4-to-so-6#imageInjective
No
- Sporadic isogenies to orthogonal groupsSection 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴Open source
map:su-4-to-so-6#injectiveSurjective
Yes
- Sporadic isogenies to orthogonal groupsSection 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴Open source
map:su-4-to-so-6#surjectiveCovering degree
2
- Sporadic isogenies to orthogonal groupsSection 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴Open source
map:su-4-to-so-6#covering-degreeGlobal and representation consequences
This identifies SU(4) with Spin(6). The central element iI acts by −I on the vector 6, while −I acts trivially; quotienting by the full C₄ center would give PSO(6).
- Sporadic isogenies to orthogonal groupsSection 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴Open source
- Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotientsOpen source
map:su-4-to-so-6#meaning-1Domain group
group:su-4
- Sporadic isogenies to orthogonal groupsSection 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴Open source
map:su-4-to-so-6#relation-1Codomain group
group:so-6
- Sporadic isogenies to orthogonal groupsSection 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴Open source
map:su-4-to-so-6#relation-2