Root system data for type E¶
- class sage.combinat.root_system.type_E.AmbientSpace(root_system, baseRing)[source]¶
- Bases: - AmbientSpace- The lattice behind E6, E7, or E8. The computations are based on Bourbaki, Groupes et Algèbres de Lie, Ch. 4,5,6 (planche V-VII). - dimension()[source]¶
- EXAMPLES: - sage: e = RootSystem(['E',6]).ambient_space() sage: e.dimension() 8 - >>> from sage.all import * >>> e = RootSystem(['E',Integer(6)]).ambient_space() >>> e.dimension() 8 
 - fundamental_weights()[source]¶
- EXAMPLES: - sage: e = RootSystem(['E',6]).ambient_space() sage: e.fundamental_weights() Finite family {1: (0, 0, 0, 0, 0, -2/3, -2/3, 2/3), 2: (1/2, 1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2), 3: (-1/2, 1/2, 1/2, 1/2, 1/2, -5/6, -5/6, 5/6), 4: (0, 0, 1, 1, 1, -1, -1, 1), 5: (0, 0, 0, 1, 1, -2/3, -2/3, 2/3), 6: (0, 0, 0, 0, 1, -1/3, -1/3, 1/3)} - >>> from sage.all import * >>> e = RootSystem(['E',Integer(6)]).ambient_space() >>> e.fundamental_weights() Finite family {1: (0, 0, 0, 0, 0, -2/3, -2/3, 2/3), 2: (1/2, 1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2), 3: (-1/2, 1/2, 1/2, 1/2, 1/2, -5/6, -5/6, 5/6), 4: (0, 0, 1, 1, 1, -1, -1, 1), 5: (0, 0, 0, 1, 1, -2/3, -2/3, 2/3), 6: (0, 0, 0, 0, 1, -1/3, -1/3, 1/3)} 
 - negative_roots()[source]¶
- The negative roots. - EXAMPLES: - sage: e = RootSystem(['E',6]).ambient_space() sage: e.negative_roots() [(-1, -1, 0, 0, 0, 0, 0, 0), (-1, 0, -1, 0, 0, 0, 0, 0), (-1, 0, 0, -1, 0, 0, 0, 0), (-1, 0, 0, 0, -1, 0, 0, 0), (0, -1, -1, 0, 0, 0, 0, 0), (0, -1, 0, -1, 0, 0, 0, 0), (0, -1, 0, 0, -1, 0, 0, 0), (0, 0, -1, -1, 0, 0, 0, 0), (0, 0, -1, 0, -1, 0, 0, 0), (0, 0, 0, -1, -1, 0, 0, 0), (1, -1, 0, 0, 0, 0, 0, 0), (1, 0, -1, 0, 0, 0, 0, 0), (1, 0, 0, -1, 0, 0, 0, 0), (1, 0, 0, 0, -1, 0, 0, 0), (0, 1, -1, 0, 0, 0, 0, 0), (0, 1, 0, -1, 0, 0, 0, 0), (0, 1, 0, 0, -1, 0, 0, 0), (0, 0, 1, -1, 0, 0, 0, 0), (0, 0, 1, 0, -1, 0, 0, 0), (0, 0, 0, 1, -1, 0, 0, 0), (-1/2, -1/2, -1/2, -1/2, -1/2, 1/2, 1/2, -1/2), (-1/2, -1/2, -1/2, 1/2, 1/2, 1/2, 1/2, -1/2), (-1/2, -1/2, 1/2, -1/2, 1/2, 1/2, 1/2, -1/2), (-1/2, -1/2, 1/2, 1/2, -1/2, 1/2, 1/2, -1/2), (-1/2, 1/2, -1/2, -1/2, 1/2, 1/2, 1/2, -1/2), (-1/2, 1/2, -1/2, 1/2, -1/2, 1/2, 1/2, -1/2), (-1/2, 1/2, 1/2, -1/2, -1/2, 1/2, 1/2, -1/2), (-1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, -1/2), (1/2, -1/2, -1/2, -1/2, 1/2, 1/2, 1/2, -1/2), (1/2, -1/2, -1/2, 1/2, -1/2, 1/2, 1/2, -1/2), (1/2, -1/2, 1/2, -1/2, -1/2, 1/2, 1/2, -1/2), (1/2, -1/2, 1/2, 1/2, 1/2, 1/2, 1/2, -1/2), (1/2, 1/2, -1/2, -1/2, -1/2, 1/2, 1/2, -1/2), (1/2, 1/2, -1/2, 1/2, 1/2, 1/2, 1/2, -1/2), (1/2, 1/2, 1/2, -1/2, 1/2, 1/2, 1/2, -1/2), (1/2, 1/2, 1/2, 1/2, -1/2, 1/2, 1/2, -1/2)] - >>> from sage.all import * >>> e = RootSystem(['E',Integer(6)]).ambient_space() >>> e.negative_roots() [(-1, -1, 0, 0, 0, 0, 0, 0), (-1, 0, -1, 0, 0, 0, 0, 0), (-1, 0, 0, -1, 0, 0, 0, 0), (-1, 0, 0, 0, -1, 0, 0, 0), (0, -1, -1, 0, 0, 0, 0, 0), (0, -1, 0, -1, 0, 0, 0, 0), (0, -1, 0, 0, -1, 0, 0, 0), (0, 0, -1, -1, 0, 0, 0, 0), (0, 0, -1, 0, -1, 0, 0, 0), (0, 0, 0, -1, -1, 0, 0, 0), (1, -1, 0, 0, 0, 0, 0, 0), (1, 0, -1, 0, 0, 0, 0, 0), (1, 0, 0, -1, 0, 0, 0, 0), (1, 0, 0, 0, -1, 0, 0, 0), (0, 1, -1, 0, 0, 0, 0, 0), (0, 1, 0, -1, 0, 0, 0, 0), (0, 1, 0, 0, -1, 0, 0, 0), (0, 0, 1, -1, 0, 0, 0, 0), (0, 0, 1, 0, -1, 0, 0, 0), (0, 0, 0, 1, -1, 0, 0, 0), (-1/2, -1/2, -1/2, -1/2, -1/2, 1/2, 1/2, -1/2), (-1/2, -1/2, -1/2, 1/2, 1/2, 1/2, 1/2, -1/2), (-1/2, -1/2, 1/2, -1/2, 1/2, 1/2, 1/2, -1/2), (-1/2, -1/2, 1/2, 1/2, -1/2, 1/2, 1/2, -1/2), (-1/2, 1/2, -1/2, -1/2, 1/2, 1/2, 1/2, -1/2), (-1/2, 1/2, -1/2, 1/2, -1/2, 1/2, 1/2, -1/2), (-1/2, 1/2, 1/2, -1/2, -1/2, 1/2, 1/2, -1/2), (-1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, -1/2), (1/2, -1/2, -1/2, -1/2, 1/2, 1/2, 1/2, -1/2), (1/2, -1/2, -1/2, 1/2, -1/2, 1/2, 1/2, -1/2), (1/2, -1/2, 1/2, -1/2, -1/2, 1/2, 1/2, -1/2), (1/2, -1/2, 1/2, 1/2, 1/2, 1/2, 1/2, -1/2), (1/2, 1/2, -1/2, -1/2, -1/2, 1/2, 1/2, -1/2), (1/2, 1/2, -1/2, 1/2, 1/2, 1/2, 1/2, -1/2), (1/2, 1/2, 1/2, -1/2, 1/2, 1/2, 1/2, -1/2), (1/2, 1/2, 1/2, 1/2, -1/2, 1/2, 1/2, -1/2)] 
 - positive_roots()[source]¶
- These are the roots positive w.r. to lexicographic ordering of the basis elements (e1<…<e4). - EXAMPLES: - sage: e = RootSystem(['E',6]).ambient_space() sage: e.positive_roots() [(1, 1, 0, 0, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0, 0, 0), (1, 0, 0, 1, 0, 0, 0, 0), (1, 0, 0, 0, 1, 0, 0, 0), (0, 1, 1, 0, 0, 0, 0, 0), (0, 1, 0, 1, 0, 0, 0, 0), (0, 1, 0, 0, 1, 0, 0, 0), (0, 0, 1, 1, 0, 0, 0, 0), (0, 0, 1, 0, 1, 0, 0, 0), (0, 0, 0, 1, 1, 0, 0, 0), (-1, 1, 0, 0, 0, 0, 0, 0), (-1, 0, 1, 0, 0, 0, 0, 0), (-1, 0, 0, 1, 0, 0, 0, 0), (-1, 0, 0, 0, 1, 0, 0, 0), (0, -1, 1, 0, 0, 0, 0, 0), (0, -1, 0, 1, 0, 0, 0, 0), (0, -1, 0, 0, 1, 0, 0, 0), (0, 0, -1, 1, 0, 0, 0, 0), (0, 0, -1, 0, 1, 0, 0, 0), (0, 0, 0, -1, 1, 0, 0, 0), (1/2, 1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (1/2, 1/2, 1/2, -1/2, -1/2, -1/2, -1/2, 1/2), (1/2, 1/2, -1/2, 1/2, -1/2, -1/2, -1/2, 1/2), (1/2, 1/2, -1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (1/2, -1/2, 1/2, 1/2, -1/2, -1/2, -1/2, 1/2), (1/2, -1/2, 1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (1/2, -1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (1/2, -1/2, -1/2, -1/2, -1/2, -1/2, -1/2, 1/2), (-1/2, 1/2, 1/2, 1/2, -1/2, -1/2, -1/2, 1/2), (-1/2, 1/2, 1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, 1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, 1/2, -1/2, -1/2, -1/2, -1/2, -1/2, 1/2), (-1/2, -1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, -1/2, 1/2, -1/2, -1/2, -1/2, -1/2, 1/2), (-1/2, -1/2, -1/2, 1/2, -1/2, -1/2, -1/2, 1/2), (-1/2, -1/2, -1/2, -1/2, 1/2, -1/2, -1/2, 1/2)] sage: e.rho() (0, 1, 2, 3, 4, -4, -4, 4) sage: E8 = RootSystem(['E',8]) sage: e = E8.ambient_space() sage: e.negative_roots() [(-1, -1, 0, 0, 0, 0, 0, 0), (-1, 0, -1, 0, 0, 0, 0, 0), (-1, 0, 0, -1, 0, 0, 0, 0), (-1, 0, 0, 0, -1, 0, 0, 0), (-1, 0, 0, 0, 0, -1, 0, 0), (-1, 0, 0, 0, 0, 0, -1, 0), (-1, 0, 0, 0, 0, 0, 0, -1), (0, -1, -1, 0, 0, 0, 0, 0), (0, -1, 0, -1, 0, 0, 0, 0), (0, -1, 0, 0, -1, 0, 0, 0), (0, -1, 0, 0, 0, -1, 0, 0), (0, -1, 0, 0, 0, 0, -1, 0), (0, -1, 0, 0, 0, 0, 0, -1), (0, 0, -1, -1, 0, 0, 0, 0), (0, 0, -1, 0, -1, 0, 0, 0), (0, 0, -1, 0, 0, -1, 0, 0), (0, 0, -1, 0, 0, 0, -1, 0), (0, 0, -1, 0, 0, 0, 0, -1), (0, 0, 0, -1, -1, 0, 0, 0), (0, 0, 0, -1, 0, -1, 0, 0), (0, 0, 0, -1, 0, 0, -1, 0), (0, 0, 0, -1, 0, 0, 0, -1), (0, 0, 0, 0, -1, -1, 0, 0), (0, 0, 0, 0, -1, 0, -1, 0), (0, 0, 0, 0, -1, 0, 0, -1), (0, 0, 0, 0, 0, -1, -1, 0), (0, 0, 0, 0, 0, -1, 0, -1), (0, 0, 0, 0, 0, 0, -1, -1), (1, -1, 0, 0, 0, 0, 0, 0), (1, 0, -1, 0, 0, 0, 0, 0), (1, 0, 0, -1, 0, 0, 0, 0), (1, 0, 0, 0, -1, 0, 0, 0), (1, 0, 0, 0, 0, -1, 0, 0), (1, 0, 0, 0, 0, 0, -1, 0), (1, 0, 0, 0, 0, 0, 0, -1), (0, 1, -1, 0, 0, 0, 0, 0), (0, 1, 0, -1, 0, 0, 0, 0), (0, 1, 0, 0, -1, 0, 0, 0), (0, 1, 0, 0, 0, -1, 0, 0), (0, 1, 0, 0, 0, 0, -1, 0), (0, 1, 0, 0, 0, 0, 0, -1), (0, 0, 1, -1, 0, 0, 0, 0), (0, 0, 1, 0, -1, 0, 0, 0), (0, 0, 1, 0, 0, -1, 0, 0), (0, 0, 1, 0, 0, 0, -1, 0), (0, 0, 1, 0, 0, 0, 0, -1), (0, 0, 0, 1, -1, 0, 0, 0), (0, 0, 0, 1, 0, -1, 0, 0), (0, 0, 0, 1, 0, 0, -1, 0), (0, 0, 0, 1, 0, 0, 0, -1), (0, 0, 0, 0, 1, -1, 0, 0), (0, 0, 0, 0, 1, 0, -1, 0), (0, 0, 0, 0, 1, 0, 0, -1), (0, 0, 0, 0, 0, 1, -1, 0), (0, 0, 0, 0, 0, 1, 0, -1), (0, 0, 0, 0, 0, 0, 1, -1), (-1/2, -1/2, -1/2, -1/2, -1/2, -1/2, -1/2, -1/2), (-1/2, -1/2, -1/2, -1/2, -1/2, 1/2, 1/2, -1/2), (-1/2, -1/2, -1/2, -1/2, 1/2, -1/2, 1/2, -1/2), (-1/2, -1/2, -1/2, -1/2, 1/2, 1/2, -1/2, -1/2), (-1/2, -1/2, -1/2, 1/2, -1/2, -1/2, 1/2, -1/2), (-1/2, -1/2, -1/2, 1/2, -1/2, 1/2, -1/2, -1/2), (-1/2, -1/2, -1/2, 1/2, 1/2, -1/2, -1/2, -1/2), (-1/2, -1/2, -1/2, 1/2, 1/2, 1/2, 1/2, -1/2), (-1/2, -1/2, 1/2, -1/2, -1/2, -1/2, 1/2, -1/2), (-1/2, -1/2, 1/2, -1/2, -1/2, 1/2, -1/2, -1/2), (-1/2, -1/2, 1/2, -1/2, 1/2, -1/2, -1/2, -1/2), (-1/2, -1/2, 1/2, -1/2, 1/2, 1/2, 1/2, -1/2), (-1/2, -1/2, 1/2, 1/2, -1/2, -1/2, -1/2, -1/2), (-1/2, -1/2, 1/2, 1/2, -1/2, 1/2, 1/2, -1/2), (-1/2, -1/2, 1/2, 1/2, 1/2, -1/2, 1/2, -1/2), (-1/2, -1/2, 1/2, 1/2, 1/2, 1/2, -1/2, -1/2), (-1/2, 1/2, -1/2, -1/2, -1/2, -1/2, 1/2, -1/2), (-1/2, 1/2, -1/2, -1/2, -1/2, 1/2, -1/2, -1/2), (-1/2, 1/2, -1/2, -1/2, 1/2, -1/2, -1/2, -1/2), (-1/2, 1/2, -1/2, -1/2, 1/2, 1/2, 1/2, -1/2), (-1/2, 1/2, -1/2, 1/2, -1/2, -1/2, -1/2, -1/2), (-1/2, 1/2, -1/2, 1/2, -1/2, 1/2, 1/2, -1/2), (-1/2, 1/2, -1/2, 1/2, 1/2, -1/2, 1/2, -1/2), (-1/2, 1/2, -1/2, 1/2, 1/2, 1/2, -1/2, -1/2), (-1/2, 1/2, 1/2, -1/2, -1/2, -1/2, -1/2, -1/2), (-1/2, 1/2, 1/2, -1/2, -1/2, 1/2, 1/2, -1/2), (-1/2, 1/2, 1/2, -1/2, 1/2, -1/2, 1/2, -1/2), (-1/2, 1/2, 1/2, -1/2, 1/2, 1/2, -1/2, -1/2), (-1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2, -1/2), (-1/2, 1/2, 1/2, 1/2, -1/2, 1/2, -1/2, -1/2), (-1/2, 1/2, 1/2, 1/2, 1/2, -1/2, -1/2, -1/2), (-1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, -1/2), (1/2, -1/2, -1/2, -1/2, -1/2, -1/2, 1/2, -1/2), (1/2, -1/2, -1/2, -1/2, -1/2, 1/2, -1/2, -1/2), (1/2, -1/2, -1/2, -1/2, 1/2, -1/2, -1/2, -1/2), (1/2, -1/2, -1/2, -1/2, 1/2, 1/2, 1/2, -1/2), (1/2, -1/2, -1/2, 1/2, -1/2, -1/2, -1/2, -1/2), (1/2, -1/2, -1/2, 1/2, -1/2, 1/2, 1/2, -1/2), (1/2, -1/2, -1/2, 1/2, 1/2, -1/2, 1/2, -1/2), (1/2, -1/2, -1/2, 1/2, 1/2, 1/2, -1/2, -1/2), (1/2, -1/2, 1/2, -1/2, -1/2, -1/2, -1/2, -1/2), (1/2, -1/2, 1/2, -1/2, -1/2, 1/2, 1/2, -1/2), (1/2, -1/2, 1/2, -1/2, 1/2, -1/2, 1/2, -1/2), (1/2, -1/2, 1/2, -1/2, 1/2, 1/2, -1/2, -1/2), (1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 1/2, -1/2), (1/2, -1/2, 1/2, 1/2, -1/2, 1/2, -1/2, -1/2), (1/2, -1/2, 1/2, 1/2, 1/2, -1/2, -1/2, -1/2), (1/2, -1/2, 1/2, 1/2, 1/2, 1/2, 1/2, -1/2), (1/2, 1/2, -1/2, -1/2, -1/2, -1/2, -1/2, -1/2), (1/2, 1/2, -1/2, -1/2, -1/2, 1/2, 1/2, -1/2), (1/2, 1/2, -1/2, -1/2, 1/2, -1/2, 1/2, -1/2), (1/2, 1/2, -1/2, -1/2, 1/2, 1/2, -1/2, -1/2), (1/2, 1/2, -1/2, 1/2, -1/2, -1/2, 1/2, -1/2), (1/2, 1/2, -1/2, 1/2, -1/2, 1/2, -1/2, -1/2), (1/2, 1/2, -1/2, 1/2, 1/2, -1/2, -1/2, -1/2), (1/2, 1/2, -1/2, 1/2, 1/2, 1/2, 1/2, -1/2), (1/2, 1/2, 1/2, -1/2, -1/2, -1/2, 1/2, -1/2), (1/2, 1/2, 1/2, -1/2, -1/2, 1/2, -1/2, -1/2), (1/2, 1/2, 1/2, -1/2, 1/2, -1/2, -1/2, -1/2), (1/2, 1/2, 1/2, -1/2, 1/2, 1/2, 1/2, -1/2), (1/2, 1/2, 1/2, 1/2, -1/2, -1/2, -1/2, -1/2), (1/2, 1/2, 1/2, 1/2, -1/2, 1/2, 1/2, -1/2), (1/2, 1/2, 1/2, 1/2, 1/2, -1/2, 1/2, -1/2), (1/2, 1/2, 1/2, 1/2, 1/2, 1/2, -1/2, -1/2)] sage: e.rho() (0, 1, 2, 3, 4, 5, 6, 23) - >>> from sage.all import * >>> e = RootSystem(['E',Integer(6)]).ambient_space() >>> e.positive_roots() [(1, 1, 0, 0, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0, 0, 0), (1, 0, 0, 1, 0, 0, 0, 0), (1, 0, 0, 0, 1, 0, 0, 0), (0, 1, 1, 0, 0, 0, 0, 0), (0, 1, 0, 1, 0, 0, 0, 0), (0, 1, 0, 0, 1, 0, 0, 0), (0, 0, 1, 1, 0, 0, 0, 0), (0, 0, 1, 0, 1, 0, 0, 0), (0, 0, 0, 1, 1, 0, 0, 0), (-1, 1, 0, 0, 0, 0, 0, 0), (-1, 0, 1, 0, 0, 0, 0, 0), (-1, 0, 0, 1, 0, 0, 0, 0), (-1, 0, 0, 0, 1, 0, 0, 0), (0, -1, 1, 0, 0, 0, 0, 0), (0, -1, 0, 1, 0, 0, 0, 0), (0, -1, 0, 0, 1, 0, 0, 0), (0, 0, -1, 1, 0, 0, 0, 0), (0, 0, -1, 0, 1, 0, 0, 0), (0, 0, 0, -1, 1, 0, 0, 0), (1/2, 1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (1/2, 1/2, 1/2, -1/2, -1/2, -1/2, -1/2, 1/2), (1/2, 1/2, -1/2, 1/2, -1/2, -1/2, -1/2, 1/2), (1/2, 1/2, -1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (1/2, -1/2, 1/2, 1/2, -1/2, -1/2, -1/2, 1/2), (1/2, -1/2, 1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (1/2, -1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (1/2, -1/2, -1/2, -1/2, -1/2, -1/2, -1/2, 1/2), (-1/2, 1/2, 1/2, 1/2, -1/2, -1/2, -1/2, 1/2), (-1/2, 1/2, 1/2, -1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, 1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, 1/2, -1/2, -1/2, -1/2, -1/2, -1/2, 1/2), (-1/2, -1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2), (-1/2, -1/2, 1/2, -1/2, -1/2, -1/2, -1/2, 1/2), (-1/2, -1/2, -1/2, 1/2, -1/2, -1/2, -1/2, 1/2), (-1/2, -1/2, -1/2, -1/2, 1/2, -1/2, -1/2, 1/2)] >>> e.rho() (0, 1, 2, 3, 4, -4, -4, 4) >>> E8 = RootSystem(['E',Integer(8)]) >>> e = E8.ambient_space() >>> e.negative_roots() [(-1, -1, 0, 0, 0, 0, 0, 0), (-1, 0, -1, 0, 0, 0, 0, 0), (-1, 0, 0, -1, 0, 0, 0, 0), (-1, 0, 0, 0, -1, 0, 0, 0), (-1, 0, 0, 0, 0, -1, 0, 0), (-1, 0, 0, 0, 0, 0, -1, 0), (-1, 0, 0, 0, 0, 0, 0, -1), (0, -1, -1, 0, 0, 0, 0, 0), (0, -1, 0, -1, 0, 0, 0, 0), (0, -1, 0, 0, -1, 0, 0, 0), (0, -1, 0, 0, 0, -1, 0, 0), (0, -1, 0, 0, 0, 0, -1, 0), (0, -1, 0, 0, 0, 0, 0, -1), (0, 0, -1, -1, 0, 0, 0, 0), (0, 0, -1, 0, -1, 0, 0, 0), (0, 0, -1, 0, 0, -1, 0, 0), (0, 0, -1, 0, 0, 0, -1, 0), (0, 0, -1, 0, 0, 0, 0, -1), (0, 0, 0, -1, -1, 0, 0, 0), (0, 0, 0, -1, 0, -1, 0, 0), (0, 0, 0, -1, 0, 0, -1, 0), (0, 0, 0, -1, 0, 0, 0, -1), (0, 0, 0, 0, -1, -1, 0, 0), (0, 0, 0, 0, -1, 0, -1, 0), (0, 0, 0, 0, -1, 0, 0, -1), (0, 0, 0, 0, 0, -1, -1, 0), (0, 0, 0, 0, 0, -1, 0, -1), (0, 0, 0, 0, 0, 0, -1, -1), (1, -1, 0, 0, 0, 0, 0, 0), (1, 0, -1, 0, 0, 0, 0, 0), (1, 0, 0, -1, 0, 0, 0, 0), (1, 0, 0, 0, -1, 0, 0, 0), (1, 0, 0, 0, 0, -1, 0, 0), (1, 0, 0, 0, 0, 0, -1, 0), (1, 0, 0, 0, 0, 0, 0, -1), (0, 1, -1, 0, 0, 0, 0, 0), (0, 1, 0, -1, 0, 0, 0, 0), (0, 1, 0, 0, -1, 0, 0, 0), (0, 1, 0, 0, 0, -1, 0, 0), (0, 1, 0, 0, 0, 0, -1, 0), (0, 1, 0, 0, 0, 0, 0, -1), (0, 0, 1, -1, 0, 0, 0, 0), (0, 0, 1, 0, -1, 0, 0, 0), (0, 0, 1, 0, 0, -1, 0, 0), (0, 0, 1, 0, 0, 0, -1, 0), (0, 0, 1, 0, 0, 0, 0, -1), (0, 0, 0, 1, -1, 0, 0, 0), (0, 0, 0, 1, 0, -1, 0, 0), (0, 0, 0, 1, 0, 0, -1, 0), (0, 0, 0, 1, 0, 0, 0, -1), (0, 0, 0, 0, 1, -1, 0, 0), (0, 0, 0, 0, 1, 0, -1, 0), (0, 0, 0, 0, 1, 0, 0, -1), (0, 0, 0, 0, 0, 1, -1, 0), (0, 0, 0, 0, 0, 1, 0, -1), (0, 0, 0, 0, 0, 0, 1, -1), (-1/2, -1/2, -1/2, -1/2, -1/2, -1/2, -1/2, -1/2), (-1/2, -1/2, -1/2, -1/2, -1/2, 1/2, 1/2, -1/2), (-1/2, -1/2, -1/2, -1/2, 1/2, -1/2, 1/2, -1/2), (-1/2, -1/2, -1/2, -1/2, 1/2, 1/2, -1/2, -1/2), (-1/2, -1/2, -1/2, 1/2, -1/2, -1/2, 1/2, -1/2), (-1/2, -1/2, -1/2, 1/2, -1/2, 1/2, -1/2, -1/2), (-1/2, -1/2, -1/2, 1/2, 1/2, -1/2, -1/2, -1/2), (-1/2, -1/2, -1/2, 1/2, 1/2, 1/2, 1/2, -1/2), (-1/2, -1/2, 1/2, -1/2, -1/2, -1/2, 1/2, -1/2), (-1/2, -1/2, 1/2, -1/2, -1/2, 1/2, -1/2, -1/2), (-1/2, -1/2, 1/2, -1/2, 1/2, -1/2, -1/2, -1/2), (-1/2, -1/2, 1/2, -1/2, 1/2, 1/2, 1/2, -1/2), (-1/2, -1/2, 1/2, 1/2, -1/2, -1/2, -1/2, -1/2), (-1/2, -1/2, 1/2, 1/2, -1/2, 1/2, 1/2, -1/2), (-1/2, -1/2, 1/2, 1/2, 1/2, -1/2, 1/2, -1/2), (-1/2, -1/2, 1/2, 1/2, 1/2, 1/2, -1/2, -1/2), (-1/2, 1/2, -1/2, -1/2, -1/2, -1/2, 1/2, -1/2), (-1/2, 1/2, -1/2, -1/2, -1/2, 1/2, -1/2, -1/2), (-1/2, 1/2, -1/2, -1/2, 1/2, -1/2, -1/2, -1/2), (-1/2, 1/2, -1/2, -1/2, 1/2, 1/2, 1/2, -1/2), (-1/2, 1/2, -1/2, 1/2, -1/2, -1/2, -1/2, -1/2), (-1/2, 1/2, -1/2, 1/2, -1/2, 1/2, 1/2, -1/2), (-1/2, 1/2, -1/2, 1/2, 1/2, -1/2, 1/2, -1/2), (-1/2, 1/2, -1/2, 1/2, 1/2, 1/2, -1/2, -1/2), (-1/2, 1/2, 1/2, -1/2, -1/2, -1/2, -1/2, -1/2), (-1/2, 1/2, 1/2, -1/2, -1/2, 1/2, 1/2, -1/2), (-1/2, 1/2, 1/2, -1/2, 1/2, -1/2, 1/2, -1/2), (-1/2, 1/2, 1/2, -1/2, 1/2, 1/2, -1/2, -1/2), (-1/2, 1/2, 1/2, 1/2, -1/2, -1/2, 1/2, -1/2), (-1/2, 1/2, 1/2, 1/2, -1/2, 1/2, -1/2, -1/2), (-1/2, 1/2, 1/2, 1/2, 1/2, -1/2, -1/2, -1/2), (-1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, -1/2), (1/2, -1/2, -1/2, -1/2, -1/2, -1/2, 1/2, -1/2), (1/2, -1/2, -1/2, -1/2, -1/2, 1/2, -1/2, -1/2), (1/2, -1/2, -1/2, -1/2, 1/2, -1/2, -1/2, -1/2), (1/2, -1/2, -1/2, -1/2, 1/2, 1/2, 1/2, -1/2), (1/2, -1/2, -1/2, 1/2, -1/2, -1/2, -1/2, -1/2), (1/2, -1/2, -1/2, 1/2, -1/2, 1/2, 1/2, -1/2), (1/2, -1/2, -1/2, 1/2, 1/2, -1/2, 1/2, -1/2), (1/2, -1/2, -1/2, 1/2, 1/2, 1/2, -1/2, -1/2), (1/2, -1/2, 1/2, -1/2, -1/2, -1/2, -1/2, -1/2), (1/2, -1/2, 1/2, -1/2, -1/2, 1/2, 1/2, -1/2), (1/2, -1/2, 1/2, -1/2, 1/2, -1/2, 1/2, -1/2), (1/2, -1/2, 1/2, -1/2, 1/2, 1/2, -1/2, -1/2), (1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 1/2, -1/2), (1/2, -1/2, 1/2, 1/2, -1/2, 1/2, -1/2, -1/2), (1/2, -1/2, 1/2, 1/2, 1/2, -1/2, -1/2, -1/2), (1/2, -1/2, 1/2, 1/2, 1/2, 1/2, 1/2, -1/2), (1/2, 1/2, -1/2, -1/2, -1/2, -1/2, -1/2, -1/2), (1/2, 1/2, -1/2, -1/2, -1/2, 1/2, 1/2, -1/2), (1/2, 1/2, -1/2, -1/2, 1/2, -1/2, 1/2, -1/2), (1/2, 1/2, -1/2, -1/2, 1/2, 1/2, -1/2, -1/2), (1/2, 1/2, -1/2, 1/2, -1/2, -1/2, 1/2, -1/2), (1/2, 1/2, -1/2, 1/2, -1/2, 1/2, -1/2, -1/2), (1/2, 1/2, -1/2, 1/2, 1/2, -1/2, -1/2, -1/2), (1/2, 1/2, -1/2, 1/2, 1/2, 1/2, 1/2, -1/2), (1/2, 1/2, 1/2, -1/2, -1/2, -1/2, 1/2, -1/2), (1/2, 1/2, 1/2, -1/2, -1/2, 1/2, -1/2, -1/2), (1/2, 1/2, 1/2, -1/2, 1/2, -1/2, -1/2, -1/2), (1/2, 1/2, 1/2, -1/2, 1/2, 1/2, 1/2, -1/2), (1/2, 1/2, 1/2, 1/2, -1/2, -1/2, -1/2, -1/2), (1/2, 1/2, 1/2, 1/2, -1/2, 1/2, 1/2, -1/2), (1/2, 1/2, 1/2, 1/2, 1/2, -1/2, 1/2, -1/2), (1/2, 1/2, 1/2, 1/2, 1/2, 1/2, -1/2, -1/2)] >>> e.rho() (0, 1, 2, 3, 4, 5, 6, 23) 
 - root(i1, i2=None, i3=None, i4=None, i5=None, i6=None, i7=None, i8=None, p1=0, p2=0, p3=0, p4=0, p5=0, p6=0, p7=0, p8=0)[source]¶
- Compute an element of the underlying lattice, using the specified elements of the standard basis, with signs dictated by the corresponding ‘pi’ arguments. We rely on the caller to provide the correct arguments. This is typically used to generate roots, although the generated elements need not be roots themselves. We assume that if one of the indices is not given, the rest are not as well. This should work for E6, E7, E8. - EXAMPLES: - sage: e = RootSystem(['E',6]).ambient_space() sage: [ e.root(i, j, p3=1) for i in range(e.n) for j in range(i+1, e.n) ] [(1, 1, 0, 0, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0, 0, 0), (1, 0, 0, 1, 0, 0, 0, 0), (1, 0, 0, 0, 1, 0, 0, 0), (1, 0, 0, 0, 0, 1, 0, 0), (1, 0, 0, 0, 0, 0, 1, 0), (1, 0, 0, 0, 0, 0, 0, 1), (0, 1, 1, 0, 0, 0, 0, 0), (0, 1, 0, 1, 0, 0, 0, 0), (0, 1, 0, 0, 1, 0, 0, 0), (0, 1, 0, 0, 0, 1, 0, 0), (0, 1, 0, 0, 0, 0, 1, 0), (0, 1, 0, 0, 0, 0, 0, 1), (0, 0, 1, 1, 0, 0, 0, 0), (0, 0, 1, 0, 1, 0, 0, 0), (0, 0, 1, 0, 0, 1, 0, 0), (0, 0, 1, 0, 0, 0, 1, 0), (0, 0, 1, 0, 0, 0, 0, 1), (0, 0, 0, 1, 1, 0, 0, 0), (0, 0, 0, 1, 0, 1, 0, 0), (0, 0, 0, 1, 0, 0, 1, 0), (0, 0, 0, 1, 0, 0, 0, 1), (0, 0, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 1, 0, 1, 0), (0, 0, 0, 0, 1, 0, 0, 1), (0, 0, 0, 0, 0, 1, 1, 0), (0, 0, 0, 0, 0, 1, 0, 1), (0, 0, 0, 0, 0, 0, 1, 1)] - >>> from sage.all import * >>> e = RootSystem(['E',Integer(6)]).ambient_space() >>> [ e.root(i, j, p3=Integer(1)) for i in range(e.n) for j in range(i+Integer(1), e.n) ] [(1, 1, 0, 0, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0, 0, 0), (1, 0, 0, 1, 0, 0, 0, 0), (1, 0, 0, 0, 1, 0, 0, 0), (1, 0, 0, 0, 0, 1, 0, 0), (1, 0, 0, 0, 0, 0, 1, 0), (1, 0, 0, 0, 0, 0, 0, 1), (0, 1, 1, 0, 0, 0, 0, 0), (0, 1, 0, 1, 0, 0, 0, 0), (0, 1, 0, 0, 1, 0, 0, 0), (0, 1, 0, 0, 0, 1, 0, 0), (0, 1, 0, 0, 0, 0, 1, 0), (0, 1, 0, 0, 0, 0, 0, 1), (0, 0, 1, 1, 0, 0, 0, 0), (0, 0, 1, 0, 1, 0, 0, 0), (0, 0, 1, 0, 0, 1, 0, 0), (0, 0, 1, 0, 0, 0, 1, 0), (0, 0, 1, 0, 0, 0, 0, 1), (0, 0, 0, 1, 1, 0, 0, 0), (0, 0, 0, 1, 0, 1, 0, 0), (0, 0, 0, 1, 0, 0, 1, 0), (0, 0, 0, 1, 0, 0, 0, 1), (0, 0, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 1, 0, 1, 0), (0, 0, 0, 0, 1, 0, 0, 1), (0, 0, 0, 0, 0, 1, 1, 0), (0, 0, 0, 0, 0, 1, 0, 1), (0, 0, 0, 0, 0, 0, 1, 1)] 
 - simple_root(i)[source]¶
- There are computed as what Bourbaki calls the Base:
- a1 = e2-e3, a2 = e3-e4, a3 = e4, a4 = 1/2*(e1-e2-e3-e4) 
 - EXAMPLES: - sage: LE6 = RootSystem(['E',6]).ambient_space() sage: LE6.simple_roots() Finite family {1: (1/2, -1/2, -1/2, -1/2, -1/2, -1/2, -1/2, 1/2), 2: (1, 1, 0, 0, 0, 0, 0, 0), 3: (-1, 1, 0, 0, 0, 0, 0, 0), 4: (0, -1, 1, 0, 0, 0, 0, 0), 5: (0, 0, -1, 1, 0, 0, 0, 0), 6: (0, 0, 0, -1, 1, 0, 0, 0)} - >>> from sage.all import * >>> LE6 = RootSystem(['E',Integer(6)]).ambient_space() >>> LE6.simple_roots() Finite family {1: (1/2, -1/2, -1/2, -1/2, -1/2, -1/2, -1/2, 1/2), 2: (1, 1, 0, 0, 0, 0, 0, 0), 3: (-1, 1, 0, 0, 0, 0, 0, 0), 4: (0, -1, 1, 0, 0, 0, 0, 0), 5: (0, 0, -1, 1, 0, 0, 0, 0), 6: (0, 0, 0, -1, 1, 0, 0, 0)} 
 
- class sage.combinat.root_system.type_E.CartanType(n)[source]¶
- Bases: - CartanType_standard_finite,- CartanType_simple,- CartanType_simply_laced- EXAMPLES: - sage: ct = CartanType(['E',6]) sage: ct ['E', 6] sage: ct._repr_(compact = True) 'E6' sage: ct.is_irreducible() True sage: ct.is_finite() True sage: ct.is_affine() False sage: ct.is_crystallographic() True sage: ct.is_simply_laced() True sage: ct.affine() ['E', 6, 1] sage: ct.dual() ['E', 6] - >>> from sage.all import * >>> ct = CartanType(['E',Integer(6)]) >>> ct ['E', 6] >>> ct._repr_(compact = True) 'E6' >>> ct.is_irreducible() True >>> ct.is_finite() True >>> ct.is_affine() False >>> ct.is_crystallographic() True >>> ct.is_simply_laced() True >>> ct.affine() ['E', 6, 1] >>> ct.dual() ['E', 6] - AmbientSpace[source]¶
- alias of - AmbientSpace
 - ascii_art(label=None, node=None)[source]¶
- Return a ascii art representation of the extended Dynkin diagram. - EXAMPLES: - sage: print(CartanType(['E',6]).ascii_art(label = lambda x: x+2)) O 4 | | O---O---O---O---O 3 5 6 7 8 sage: print(CartanType(['E',7]).ascii_art(label = lambda x: x+2)) O 4 | | O---O---O---O---O---O 3 5 6 7 8 9 sage: print(CartanType(['E',8]).ascii_art(label = lambda x: x+1)) O 3 | | O---O---O---O---O---O---O 2 4 5 6 7 8 9 - >>> from sage.all import * >>> print(CartanType(['E',Integer(6)]).ascii_art(label = lambda x: x+Integer(2))) O 4 | | O---O---O---O---O 3 5 6 7 8 >>> print(CartanType(['E',Integer(7)]).ascii_art(label = lambda x: x+Integer(2))) O 4 | | O---O---O---O---O---O 3 5 6 7 8 9 >>> print(CartanType(['E',Integer(8)]).ascii_art(label = lambda x: x+Integer(1))) O 3 | | O---O---O---O---O---O---O 2 4 5 6 7 8 9 
 - coxeter_number()[source]¶
- Return the Coxeter number associated with - self.- EXAMPLES: - sage: CartanType(['E',6]).coxeter_number() 12 sage: CartanType(['E',7]).coxeter_number() 18 sage: CartanType(['E',8]).coxeter_number() 30 - >>> from sage.all import * >>> CartanType(['E',Integer(6)]).coxeter_number() 12 >>> CartanType(['E',Integer(7)]).coxeter_number() 18 >>> CartanType(['E',Integer(8)]).coxeter_number() 30 
 - dual_coxeter_number()[source]¶
- Return the dual Coxeter number associated with - self.- EXAMPLES: - sage: CartanType(['E',6]).dual_coxeter_number() 12 sage: CartanType(['E',7]).dual_coxeter_number() 18 sage: CartanType(['E',8]).dual_coxeter_number() 30 - >>> from sage.all import * >>> CartanType(['E',Integer(6)]).dual_coxeter_number() 12 >>> CartanType(['E',Integer(7)]).dual_coxeter_number() 18 >>> CartanType(['E',Integer(8)]).dual_coxeter_number() 30 
 - dynkin_diagram()[source]¶
- Return a Dynkin diagram for type E. - EXAMPLES: - sage: # needs sage.graphs sage: e = CartanType(['E',6]).dynkin_diagram(); e O 2 | | O---O---O---O---O 1 3 4 5 6 E6 sage: e.edges(sort=True) [(1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1), (4, 2, 1), (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1), (6, 5, 1)] sage: e = CartanType(['E',7]).dynkin_diagram(); e O 2 | | O---O---O---O---O---O 1 3 4 5 6 7 E7 sage: e.edges(sort=True) [(1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1), (4, 2, 1), (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1), (6, 5, 1), (6, 7, 1), (7, 6, 1)] sage: e = CartanType(['E',8]).dynkin_diagram(); e O 2 | | O---O---O---O---O---O---O 1 3 4 5 6 7 8 E8 sage: e.edges(sort=True) [(1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1), (4, 2, 1), (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1), (6, 5, 1), (6, 7, 1), (7, 6, 1), (7, 8, 1), (8, 7, 1)] - >>> from sage.all import * >>> # needs sage.graphs >>> e = CartanType(['E',Integer(6)]).dynkin_diagram(); e O 2 | | O---O---O---O---O 1 3 4 5 6 E6 >>> e.edges(sort=True) [(1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1), (4, 2, 1), (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1), (6, 5, 1)] >>> e = CartanType(['E',Integer(7)]).dynkin_diagram(); e O 2 | | O---O---O---O---O---O 1 3 4 5 6 7 E7 >>> e.edges(sort=True) [(1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1), (4, 2, 1), (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1), (6, 5, 1), (6, 7, 1), (7, 6, 1)] >>> e = CartanType(['E',Integer(8)]).dynkin_diagram(); e O 2 | | O---O---O---O---O---O---O 1 3 4 5 6 7 8 E8 >>> e.edges(sort=True) [(1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1), (4, 2, 1), (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1), (6, 5, 1), (6, 7, 1), (7, 6, 1), (7, 8, 1), (8, 7, 1)]