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)}
TESTS:
sage: TestSuite(e).run()
EXAMPLES:
sage: e = RootSystem(['C',3]).ambient_space()
sage: e.dimension()
3
EXAMPLES:
sage: RootSystem(['C',3]).ambient_space().fundamental_weights()
Finite family {1: (1, 0, 0), 2: (1, 1, 0), 3: (1, 1, 1)}
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)]
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)]
Note that indexing starts at 0.
EXAMPLES:
sage: e = RootSystem(['C',3]).ambient_space()
sage: e.root(0, 1, 1, 1)
(-1, -1, 0)
EXAMPLES:
sage: RootSystem(['C',3]).ambient_space().simple_roots()
Finite family {1: (1, -1, 0), 2: (0, 1, -1), 3: (0, 0, 2)}
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_crystalographic
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_crystalographic()
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]
alias of AmbientSpace
Returns 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
Types B and C are in duality:
EXAMPLES:
sage: CartanType(["C", 3]).dual()
['B', 3]
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: e = c.edges(); e.sort(); e
[(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())
[]