Root system data for type F¶
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sage.combinat.root_system.type_F.
AmbientSpace
¶ The lattice behind \(F_4\). The computations are based on Bourbaki, Groupes et Algebres de Lie, Ch. 4,5,6 (planche VIII).
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class
sage.combinat.root_system.type_F.
CartanType
¶ 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_crystallographic
EXAMPLES:
sage: ct = CartanType(['F',4]) sage: ct ['F', 4] sage: ct._repr_(compact = True) 'F4' sage: ct.is_irreducible() True sage: ct.is_finite() True sage: ct.is_crystallographic() True sage: ct.is_simply_laced() False sage: ct.dual() ['F', 4] relabelled by {1: 4, 2: 3, 3: 2, 4: 1} sage: ct.affine() ['F', 4, 1]
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AmbientSpace
¶ The lattice behind \(F_4\). The computations are based on Bourbaki, Groupes et Algebres de Lie, Ch. 4,5,6 (planche VIII).
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ascii_art
(label=<function CartanType.<lambda>>, node=None)¶ Return an ascii art representation of the extended Dynkin diagram.
EXAMPLES:
sage: print(CartanType(['F',4]).ascii_art(label = lambda x: x+2)) O---O=>=O---O 3 4 5 6 sage: print(CartanType(['F',4]).ascii_art(label = lambda x: x-2)) O---O=>=O---O -1 0 1 2
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coxeter_number
()¶ Return the Coxeter number associated with
self
.EXAMPLES:
sage: CartanType(['F',4]).coxeter_number() 12
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dual
()¶ Return the dual Cartan type.
This uses that \(F_4\) is self-dual up to relabelling.
EXAMPLES:
sage: F4 = CartanType(['F',4]) sage: F4.dual() ['F', 4] relabelled by {1: 4, 2: 3, 3: 2, 4: 1} sage: F4.dynkin_diagram() O---O=>=O---O 1 2 3 4 F4 sage: F4.dual().dynkin_diagram() O---O=>=O---O 4 3 2 1 F4 relabelled by {1: 4, 2: 3, 3: 2, 4: 1}
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dual_coxeter_number
()¶ Return the dual Coxeter number associated with
self
.EXAMPLES:
sage: CartanType(['F',4]).dual_coxeter_number() 9
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dynkin_diagram
()¶ Returns a Dynkin diagram for type F.
EXAMPLES:
sage: f = CartanType(['F',4]).dynkin_diagram() sage: f O---O=>=O---O 1 2 3 4 F4 sage: sorted(f.edges()) [(1, 2, 1), (2, 1, 1), (2, 3, 2), (3, 2, 1), (3, 4, 1), (4, 3, 1)]
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