Root system data for type C¶
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class
sage.combinat.root_system.type_C.
AmbientSpace
(root_system, base_ring, index_set=None)¶ Bases:
sage.combinat.root_system.ambient_space.AmbientSpace
EXAMPLES:
sage: e = RootSystem(['C',2]).ambient_space(); e Ambient space of the Root system of type ['C', 2]
One cannot construct the ambient lattice because the fundamental coweights have rational coefficients:
sage: e.smallest_base_ring() Rational Field sage: RootSystem(['B',2]).ambient_space().fundamental_weights() Finite family {1: (1, 0), 2: (1/2, 1/2)}
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dimension
()¶ EXAMPLES:
sage: e = RootSystem(['C',3]).ambient_space() sage: e.dimension() 3
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fundamental_weight
(i)¶ EXAMPLES:
sage: RootSystem(['C',3]).ambient_space().fundamental_weights() Finite family {1: (1, 0, 0), 2: (1, 1, 0), 3: (1, 1, 1)}
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negative_roots
()¶ EXAMPLES:
sage: RootSystem(['C',3]).ambient_space().negative_roots() [(-1, 1, 0), (-1, 0, 1), (0, -1, 1), (-1, -1, 0), (-1, 0, -1), (0, -1, -1), (-2, 0, 0), (0, -2, 0), (0, 0, -2)]
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positive_roots
()¶ EXAMPLES:
sage: RootSystem(['C',3]).ambient_space().positive_roots() [(1, 1, 0), (1, 0, 1), (0, 1, 1), (1, -1, 0), (1, 0, -1), (0, 1, -1), (2, 0, 0), (0, 2, 0), (0, 0, 2)]
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root
(i, j, p1, p2)¶ Note that indexing starts at 0.
EXAMPLES:
sage: e = RootSystem(['C',3]).ambient_space() sage: e.root(0, 1, 1, 1) (-1, -1, 0)
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simple_root
(i)¶ EXAMPLES:
sage: RootSystem(['C',3]).ambient_space().simple_roots() Finite family {1: (1, -1, 0), 2: (0, 1, -1), 3: (0, 0, 2)}
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class
sage.combinat.root_system.type_C.
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_crystallographic
EXAMPLES:
sage: ct = CartanType(['C',4]) sage: ct ['C', 4] sage: ct._repr_(compact = True) 'C4' sage: ct.is_irreducible() True sage: ct.is_finite() True sage: ct.is_crystallographic() True sage: ct.is_simply_laced() False sage: ct.affine() ['C', 4, 1] sage: ct.dual() ['B', 4] sage: ct = CartanType(['C',1]) sage: ct.is_simply_laced() True sage: ct.affine() ['C', 1, 1]
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AmbientSpace
¶ alias of
AmbientSpace
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ascii_art
(label=<function CartanType.<lambda> at 0x7f96d34e4af0>, node=None)¶ Return a ascii art representation of the extended Dynkin diagram.
EXAMPLES:
sage: print(CartanType(['C',1]).ascii_art()) O 1 sage: print(CartanType(['C',2]).ascii_art()) O=<=O 1 2 sage: print(CartanType(['C',3]).ascii_art()) O---O=<=O 1 2 3 sage: print(CartanType(['C',5]).ascii_art(label = lambda x: x+2)) O---O---O---O=<=O 3 4 5 6 7
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coxeter_number
()¶ Return the Coxeter number associated with
self
.EXAMPLES:
sage: CartanType(['C',4]).coxeter_number() 8
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dual
()¶ Types B and C are in duality:
EXAMPLES:
sage: CartanType(["C", 3]).dual() ['B', 3]
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dual_coxeter_number
()¶ Return the dual Coxeter number associated with
self
.EXAMPLES:
sage: CartanType(['C',4]).dual_coxeter_number() 5
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dynkin_diagram
()¶ Returns a Dynkin diagram for type C.
EXAMPLES:
sage: c = CartanType(['C',3]).dynkin_diagram() sage: c O---O=<=O 1 2 3 C3 sage: c.edges(sort=True) [(1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 2)] sage: b = CartanType(['C',1]).dynkin_diagram() sage: b O 1 C1 sage: sorted(b.edges()) []
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