lecture-notes
Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual Representations
Pavel Etingof
Use in this reference: Linked MIT OpenCourseWare notes. Numerical root-system facts are used with original explanations.
Relevant material: Sections 32.1–32.5, printed pages 172–178.
Dependent assertions
| Record | Assertion | Source locator |
|---|---|---|
| SU(2) | Representations | Section 32.5, Example 32.13, printed page 176: even n has real type and odd n has quaternionic type |
| SU(2) | In physics | Section 32.5, Example 32.13, printed page 176: even n has real type and odd n has quaternionic type |
| SO(3) | Representations | Section 32.5, Example 32.13: sl(2,C) highest-weight modules |
| SO(3) | In physics | Section 32.5, Example 32.13: sl(2,C) highest-weight modules |
| Aᵣ | Complex algebra dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Aᵣ | Coxeter number h | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Aᵣ | Dual Coxeter number h∨ | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Aᵣ | Root-length convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Aᵣ | Representation convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Bᵣ | Complex algebra dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Bᵣ | Coxeter number h | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Bᵣ | Dual Coxeter number h∨ | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Bᵣ | Root-length convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Bᵣ | Representation convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Cᵣ | Complex algebra dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Cᵣ | Coxeter number h | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Cᵣ | Dual Coxeter number h∨ | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Cᵣ | Root-length convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Cᵣ | Representation convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Dᵣ | Complex algebra dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Dᵣ | Coxeter number h | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Dᵣ | Dual Coxeter number h∨ | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Dᵣ | Root-length convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Dᵣ | Representation convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₆ | Complex algebra dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₆ | Coxeter number h | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₆ | Dual Coxeter number h∨ | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₆ | Root-length convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₆ | Representation convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₇ | Complex algebra dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₇ | Coxeter number h | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₇ | Dual Coxeter number h∨ | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₇ | Root-length convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₇ | Representation convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₈ | Complex algebra dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₈ | Coxeter number h | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₈ | Dual Coxeter number h∨ | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₈ | Root-length convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₈ | Representation convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| F₄ | Complex algebra dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| F₄ | Coxeter number h | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| F₄ | Dual Coxeter number h∨ | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| F₄ | Root-length convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| F₄ | Representation convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| G₂ | Complex algebra dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| G₂ | Coxeter number h | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| G₂ | Dual Coxeter number h∨ | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| G₂ | Root-length convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| G₂ | Representation convention | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| sl(2, C) | Complex dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| sl(3, C) | Complex dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| sl(4, C) | Complex dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| sl(5, C) | Complex dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| so(5, C) ≅ sp(4, C) | Complex dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| sp(6, C) | Complex dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| so(8, C) | Complex dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| so(10, C) | Complex dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| e₆(C) | Complex dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| e₇(C) | Complex dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| e₈(C) | Complex dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| f₄(C) | Complex dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| g₂(C) | Complex dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| sl(2, C) ⊕ sl(2, C) | Complex dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| C (zero bracket) | Complex dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| sl(2, C) ⊕ C | Complex dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| SU(4) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| PSU(4) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| SU(5) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| U(2) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| SO(4) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| SO(5) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| SO(6) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Sp(2) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Sp(3) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Spin(8) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| SO(8) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| PSO(8) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Spin(10) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| SO(10) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| G₂ compact | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| F₄ compact | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₆ compact, simply connected | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₇ compact, simply connected | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₈ compact | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₆ compact, adjoint | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| E₇ compact, adjoint | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| SL(2, R) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| PSL(2, R) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Universal cover of SL(2, R) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| SL(3, R) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| SU(2,2) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| Sp(4, R) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| SO₀(1,3) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| SO₀(2,4) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |
| SO₀(2,3) | Real dimension | Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers |