Bases: sage.combinat.root_system.ambient_space.AmbientSpace
EXAMPLES:
sage: e = RootSystem(['G',2]).ambient_space(); e
Ambient space of the Root system of type ['G', 2]
One can not construct the ambient lattice because the simple coroots have rational coefficients:
sage: e.simple_coroots()
Finite family {1: (0, 1, -1), 2: (1/3, -2/3, 1/3)}
sage: e.smallest_base_ring()
Rational Field
TESTS:
sage: TestSuite(e).run()
sage: [WeylDim(['G',2],[a,b]) for a,b in [[0,0], [1,0], [0,1], [1,1]]] # indirect doctest
[1, 7, 14, 64]
EXAMPLES:
sage: e = RootSystem(['G',2]).ambient_space()
sage: e.dimension()
3
EXAMPLES:
sage: CartanType(['G',2]).root_system().ambient_space().fundamental_weights()
Finite family {1: (1, 0, -1), 2: (2, -1, -1)}
EXAMPLES:
sage: CartanType(['G',2]).root_system().ambient_space().negative_roots()
[(0, -1, 1), (-1, 2, -1), (-1, 1, 0), (-1, 0, 1), (-1, -1, 2), (-2, 1, 1)]
EXAMPLES:
sage: CartanType(['G',2]).root_system().ambient_space().positive_roots()
[(0, 1, -1), (1, -2, 1), (1, -1, 0), (1, 0, -1), (1, 1, -2), (2, -1, -1)]
EXAMPLES:
sage: CartanType(['G',2]).root_system().ambient_space().simple_roots()
Finite family {1: (0, 1, -1), 2: (1, -2, 1)}
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(['G',2])
sage: ct
['G', 2]
sage: ct._repr_(compact = True)
'G2'
sage: ct.is_irreducible()
True
sage: ct.is_finite()
True
sage: ct.is_crystalographic()
True
sage: ct.is_simply_laced()
False
sage: ct.dual()
['G', 2]^*
sage: ct.affine()
['G', 2, 1]
alias of AmbientSpace
Returns an ascii art representation of the Dynkin diagram
EXAMPLES:
sage: print CartanType(['G',2]).ascii_art(label = lambda x: x+2)
3
O=<=O
3 4
Returns a Dynkin diagram for type G.
EXAMPLES:
sage: g = CartanType(['G',2]).dynkin_diagram()
sage: g
3
O=<=O
1 2
G2
sage: sorted(g.edges())
[(1, 2, 1), (2, 1, 3)]