Root system data for type E¶
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sage.combinat.root_system.type_E.
AmbientSpace
¶ The lattice behind E6, E7, or E8. The computations are based on Bourbaki, Groupes et Algebres de Lie, Ch. 4,5,6 (planche V-VII).
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class
sage.combinat.root_system.type_E.
CartanType
(n)¶ Bases:
sage.combinat.root_system.cartan_type.CartanType_standard_finite
,sage.combinat.root_system.cartan_type.CartanType_simple
,sage.combinat.root_system.cartan_type.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]
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AmbientSpace
¶ The lattice behind E6, E7, or E8. The computations are based on Bourbaki, Groupes et Algebres de Lie, Ch. 4,5,6 (planche V-VII).
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ascii_art
(label=<function CartanType.<lambda>>, node=None)¶ 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
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coxeter_number
()¶ 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
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dual_coxeter_number
()¶ 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
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dynkin_diagram
()¶ Returns a Dynkin diagram for type E.
EXAMPLES:
sage: e = CartanType(['E',6]).dynkin_diagram() sage: e O 2 | | O---O---O---O---O 1 3 4 5 6 E6 sage: sorted(e.edges()) [(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() sage: e O 2 | | O---O---O---O---O---O 1 3 4 5 6 7 E7 sage: sorted(e.edges()) [(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() sage: e O 2 | | O---O---O---O---O---O---O 1 3 4 5 6 7 8 E8 sage: sorted(e.edges()) [(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)]
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