Bases: sage.combinat.root_system.ambient_space.AmbientSpace
EXAMPLES:
sage: e = RootSystem(['A',3]).ambient_lattice()
sage: s = e.simple_reflections()
sage: L = RootSystem(['A',3]).coroot_lattice()
sage: e.has_coerce_map_from(L)
True
sage: e(L.simple_root(1))
(1, -1, 0, 0)
EXAMPLES:
sage: e = RootSystem(['D',3]).ambient_space()
sage: e.dimension()
3
EXAMPLES:
sage: RootSystem(['D',4]).ambient_space().fundamental_weights()
Finite family {1: (1, 0, 0, 0), 2: (1, 1, 0, 0), 3: (1/2, 1/2, 1/2, -1/2), 4: (1/2, 1/2, 1/2, 1/2)}
EXAMPLES:
sage: RootSystem(['D',4]).ambient_space().negative_roots()
[(-1, 1, 0, 0),
(-1, 0, 1, 0),
(0, -1, 1, 0),
(-1, 0, 0, 1),
(0, -1, 0, 1),
(0, 0, -1, 1),
(-1, -1, 0, 0),
(-1, 0, -1, 0),
(0, -1, -1, 0),
(-1, 0, 0, -1),
(0, -1, 0, -1),
(0, 0, -1, -1)]
EXAMPLES:
sage: RootSystem(['D',4]).ambient_space().positive_roots()
[(1, 1, 0, 0),
(1, 0, 1, 0),
(0, 1, 1, 0),
(1, 0, 0, 1),
(0, 1, 0, 1),
(0, 0, 1, 1),
(1, -1, 0, 0),
(1, 0, -1, 0),
(0, 1, -1, 0),
(1, 0, 0, -1),
(0, 1, 0, -1),
(0, 0, 1, -1)]
Note that indexing starts at 0.
EXAMPLES:
sage: e = RootSystem(['D',3]).ambient_space()
sage: e.root(0, 1, 1, 1)
(-1, -1, 0)
sage: e.root(0, 0, 1, 1)
(-1, 0, 0)
EXAMPLES:
sage: RootSystem(['D',4]).ambient_space().simple_roots()
Finite family {1: (1, -1, 0, 0), 2: (0, 1, -1, 0), 3: (0, 0, 1, -1), 4: (0, 0, 1, 1)}
Bases: sage.combinat.root_system.cartan_type.CartanType_standard_finite, sage.combinat.root_system.cartan_type.CartanType_simply_laced
EXAMPLES:
sage: ct = CartanType(['D',4])
sage: ct
['D', 4]
sage: ct._repr_(compact = True)
'D4'
sage: ct.is_irreducible()
True
sage: ct.is_finite()
True
sage: ct.is_crystallographic()
True
sage: ct.is_simply_laced()
True
sage: ct.dual()
['D', 4]
sage: ct.affine()
['D', 4, 1]
sage: ct = CartanType(['D',2])
sage: ct.is_irreducible()
False
sage: ct.dual()
['D', 2]
sage: ct.affine()
Traceback (most recent call last):
...
ValueError: ['D', 2, 1] is not a valid Cartan type
TESTS:
sage: TestSuite(ct).run()
alias of AmbientSpace
Returns a ascii art representation of the extended Dynkin diagram
EXAMPLES:
sage: print CartanType(['D',3]).ascii_art()
O 3
|
|
O---O
1 2
sage: print CartanType(['D',4]).ascii_art()
O 4
|
|
O---O---O
1 2 3
sage: print CartanType(['D',4]).ascii_art(label = lambda x: x+2)
O 6
|
|
O---O---O
3 4 5
sage: print CartanType(['D',6]).ascii_art(label = lambda x: x+2)
O 8
|
|
O---O---O---O---O
3 4 5 6 7
Return the Coxeter number associated with self.
EXAMPLES:
sage: CartanType(['D',4]).coxeter_number()
6
Return the dual Coxeter number associated with self.
EXAMPLES:
sage: CartanType(['D',4]).dual_coxeter_number()
6
Returns a Dynkin diagram for type D.
EXAMPLES:
sage: d = CartanType(['D',5]).dynkin_diagram(); d
O 5
|
|
O---O---O---O
1 2 3 4
D5
sage: sorted(d.edges())
[(1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 1), (3, 4, 1), (3, 5, 1), (4, 3, 1), (5, 3, 1)]
sage: d = CartanType(['D',4]).dynkin_diagram(); d
O 4
|
|
O---O---O
1 2 3
D4
sage: sorted(d.edges())
[(1, 2, 1), (2, 1, 1), (2, 3, 1), (2, 4, 1), (3, 2, 1), (4, 2, 1)]
sage: d = CartanType(['D',3]).dynkin_diagram(); d
O 3
|
|
O---O
1 2
D3
sage: sorted(d.edges())
[(1, 2, 1), (1, 3, 1), (2, 1, 1), (3, 1, 1)]
sage: d = CartanType(['D',2]).dynkin_diagram(); d
O O
1 2
D2
sage: sorted(d.edges())
[]
Implements CartanType_abstract.is_atomic()
is atomic, like all
, despite being non irreducible.
EXAMPLES:
sage: CartanType(["D",2]).is_atomic()
True
sage: CartanType(["D",2]).is_irreducible()
False